#mathsmonday — Public Fediverse posts

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

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1/6
#MathsMonday
This week #scaffolding, as used in #Mathematics (and other) #textbooks, as I'll be referring to this in two upcoming #Maths threads. The name is taken from the scaffolding used when buildings are being erected, and removed when construction is complete. In #Math textbooks (and worksheets) this takes the form of hints/tips at the start of a chapter/exercise, to help the student understand the process, and no longer appear once the student has thus learnt how to do it un-aided...

1/6
#MathsMonday
This week #scaffolding, as used in #Mathematics (and other) #textbooks, as I'll be referring to this in two upcoming #Maths threads. The name is taken from the scaffolding used when buildings are being erected, and removed when construction is complete. In #Math textbooks (and worksheets) this takes the form of hints/tips at the start of a chapter/exercise, to help the student understand the process, and no longer appear once the student has thus learnt how to do it un-aided...

1/6
#MathsMonday
This week #scaffolding, as used in #Mathematics (and other) #textbooks, as I'll be referring to this in two upcoming #Maths threads. The name is taken from the scaffolding used when buildings are being erected, and removed when construction is complete. In #Math textbooks (and worksheets) this takes the form of hints/tips at the start of a chapter/exercise, to help the student understand the process, and no longer appear once the student has thus learnt how to do it un-aided...

1/6
#MathsMonday
This week #scaffolding, as used in #Mathematics (and other) #textbooks, as I'll be referring to this in two upcoming #Maths threads. The name is taken from the scaffolding used when buildings are being erected, and removed when construction is complete. In #Math textbooks (and worksheets) this takes the form of hints/tips at the start of a chapter/exercise, to help the student understand the process, and no longer appear once the student has thus learnt how to do it un-aided...

1/6
#MathsMonday
This week #scaffolding, as used in #Mathematics (and other) #textbooks, as I'll be referring to this in two upcoming #Maths threads. The name is taken from the scaffolding used when buildings are being erected, and removed when construction is complete. In #Math textbooks (and worksheets) this takes the form of hints/tips at the start of a chapter/exercise, to help the student understand the process, and no longer appear once the student has thus learnt how to do it un-aided...

Time for another round of examining the interesting beliefs of SmartmanApps and #debunking the #disinformation this #MathsMonday.

We saw last time that his view of mathematics is at odds with that of the mainstream: I enumerated the standard axioms of the real numbers and proved that there can be no number 0.999… that is simultaneously less than 1 and greater than 0.9, 0.99, and any such finite decimal truncation of 0.999….

His idea is that 1 is “the limit of” 0.999…, but not the exact value of it. But what exactly is a limit? Let’s have a look at what our Smart Friend calls a limit:

> The limit is the number which [the sequence] never reaches

(see https://dotnet.social/@SmartmanApps/116303201093245275 I am not quite certain he intends this language to apply to all sequences, but I have not seen other descriptions from him)

This is simply inadequate, and it’s worth seeing why such poor explanations are inadequate with some examples.

* The sequence (1, 1, 1, …) *never reaches* the number 2, so is 2 the limit? The same applies for every number greater than 1, so are all of them the limit? The wording “the number” implies that the limit should be unique.
* On the other hand, it *does* reach 1, which intuition says ought to be the limit of this sequence.
* The sequence (1, 2, 3, 4, …) will exceed every number eventually and so, I guess, “reaches” every number. Again the wording “the number” suggests that such a number always exists, but apparently does not.

It may be that the Genius has a more precise idea of limit lurking in his mind, but to tease it out we’d have to interrogate him about these (and probably other) examples, and most likely anyone who tried would get blocked before they could complete their investigation.

The usual definition can be seen clearly from the early 19th century, due to Bolzano, though its roots go back further. That definition is:

> For a sequence (a_0, a_1, a_2, …), and a real number A, if for every real number ε > 0, there is some natural number N such that for every n > N, |a_n - A| < ε, then we say that A is the limit of the sequence (a_0, a_1, a_2, …).

This is quite a mouthful, and first year mathematics undergraduates spend quite some time getting the hang of it. A characteristic of the definition is the alternating *quantifiers*, which are written out “for every” and “there is” here (but would normally be written with symbols). It took mathematicians some time to come up with this modern version of quantified mathematical language.

Nevertheless we can put it into simpler language, at the cost of a little precision: **the limit of a sequence is A if the sequence gets as close to A as we like and remains that close forever**. It’s important that we keep that “remains that close” in. It’s important that neither of these ways of describing the concept assume a limit exists, because not all sequences have a limit. It is quite easy to prove directly from the definition and the properties of the real numbers that:

* A constant sequence (a, a, a, …) has a as its limit
* If a sequence has a limit, the limit is unique
* The sequence (1, 2, 3, …) does not have a limit

The ordinary way of proving such basic facts is via our friends Completeness and the Archimedean property. Our pal has explicitly rejected these (by affirming the existence of infinitesimals) and so does not have them available for this purpose.

To see this practically, how should we prove that the limit of the sequence (0.9, 0.99, 0.999, …) is 1? The ordinary way would be to appeal to the definition:

1. Pick any positive distance ε. By the Archimedean property, ε > 1/N > 1/10^N for some N
2. If n > N then 1-0.99…99 (with n nines) is less than 1/10^N < ε, so 1 is the limit.

The astute reader will notice this argument is very similar to the one from last week. But if infinitesimals exist, we *cannot do this*: the first step is, in fact, false. If ε is infinitesimal, then there will not be any N such that ε > 1/N! That is in fact what it means to be infinitesimal!

Specifically, if ε = 1 - 0.999…, which the Smart Man says is greater than zero, this argument falls down; we cannot get the sequence (0.9, 0.99, 0.999, …) to be ε-close to 1 if ε is infinitesimal by looking to some point far enough into the sequence: for any n, the nth term 1/10^n away from 1, and 1/10^n is larger than ε.

From further reading of SmartmanApps’ posts, I suspect he might want to object that if we continue the sequence “to infinity” the difference becomes infinitesimal. I should be very clear here: sequences as here defined and as used by him cannot be continued “to infinity”. The defining rule for this sequence is that the nth term is 1 - 1/10^n, something which makes sense and is defined for *natural numbers* n, and because infinity is not a natural number and 10^∞ is not defined, we can’t just continue like that. The only way would be to make a *definition* of what 1 - 1/10^∞ means, i.e. to *choose* what happens at this continuation; there are no axioms governing rational numbers that force us to give a certain value to this expression.

Another potential objection is that I have used the “wrong” definition of a limit, but:

1. You can find this definition in every single textbook and set of lecture notes on real analysis
2. You can find this definition (written in an old fashioned way of "variables that take on successive values" rather than sequences) in the 120-year old algebra textbooks he loves to cite
3. We saw multiple problems with his broken pseudo-definition that make it useless

so it’s up to him to provide a correct one. One could try as a first attempt to replace “for every real number greater than zero” with “for every non-infinitesimal real number greater than zero”. But without Completeness, basic facts like the uniqueness of limits, on which many more important theorems rest, would still be false.

It’s fun to explore what happens to mathematics if throw out some of its founding principles, though it does make doing anything useful with it hard. Forget working out anything truly useful like calculus without Completeness, or something precise to replace it!

Next time I plan to look a bit more at infinitesimals and how you can treat them rigorously.

#maths #math #mathematics

# Introduction

Right, it's time to start the big #debunk! This is a story of #mathematics, #education and one Smart Man. It is a debunking of the arrogant proclamations of SmartmanApps (a maths teacher who lacks both knowledge and pedagogical skill) as well as a lesson in the limitations of education, and hopefully along the way I’ll highlight some interesting areas of mathematics that are not illuminated by a high school #maths education, but which nevertheless are quite accessible. The intended level is that of a curious high-school student. I will attempt only to give the necessary details, so if you are left feeling like you need a reference, or more information, let me know and I’ll gladly point you in the right direction.

I acknowledge the audience for this will be small, but it nevertheless needs to be done. (It's certainly unlikely that the user in question will read it, as despite at first having an apparently limitless capacity to reply with new claims, and raining scorn on those who gave up replying or blocked him for their own sanity, he nowadays seems to block anyone who engages with him for more than a handful of posts - including me).

The first topic (in the thread below to avoid clogging your feed) will simply be the real numbers, and how the number 0.999… illuminates the mathematics of them.

Oh, and since it's fitting... #MathsMonday :)

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#MathsMonday #Mathematics
This rubbish article https://www.scientificamerican.com/article/mathematicians-cant-agree-on-whether-0-999-equals-1/ popped up in my feed a few times, and I've already debunked the various points, but will cover it with specific links for each (non-)point.

"Mathematicians can’t agree on whether 0.999... equals 1" - yes they can, it's not, as per division, limits, infinite decimals, and other #Maths topics, all found in #Math textbooks

"by Manon Bischoff" - "is a theoretical Physicist". Maybe just stay in your lane dude... 🙄

@binford2k @TechDesk @404mediaco @emanuelmaiberg
"nobody asked you whether one out of 65,380,334 pages was correct or not" - and I never said anything about one page, but MANY pages 🙄

"That’s 0.000001529511917% of Wikipedia" - that's a strawman

"You have a hell of a long way to go before you’ve supported your “full of misinformation” claim" - go ahead and search for #MathsMonday to find a whole bunch more (and that's only for Maths)!

1/5
#MathsMonday #Mathematics #Math
This week I'm coming back to the topic of 0.(3) only being approximately equal to 1/3, which I discussed previously at https://dotnet.social/@SmartmanApps/115207065189190900

At the time, I knew this was true simply from knowing doing the division always leaves a remainder of 1. Since then I've now seen 2 #Maths textbooks which explicitly spell this out, that all non-terminating decimals are only approximations, and that only terminating decimals are exactly equal to fractions...

1/6
Further extending this #MathsMonday on the laughable Cantor #Mathematics claim that the "infinite sets" of the Naturals and the Evens are "the same size", let's accept that #Math argument for a moment, and prove by contradiction that it can't be true...

If sets can be infinite, that means not only do they come with the #Maths bracket form of set notation, and cardinality, but also set operations, like Union and Intersection, and ways to depict those sets/operations... with Venn diagrams...

1/9
Coming back this #MathsMonday to Cantor (who has regrettably been filling my #Mathematics timeline again), we have https://www.quantamagazine.org/how-can-infinity-come-in-many-sizes-20260223/ which lays out his (non-)proof in layman's terms, so let's look at the specifics of the #Maths...

"Aristotle rejected the existence of the infinite entirely; to him, infinity was simply a limit that could never be reached, not a true mathematical entity" - yep, and this is what is still taught about #Math limits and infinity today...

@everton137
Yes, me. Not every week, but most weeks, I write a mini-thread about Maths - #MathsMonday - because I've seen enough #disinformation #misinformation on #Wikipedia to know the only true way to spread the word is to have a place where no-one else can back out your literal textbook quotes (notably the Wiki Maths pages in question never cite any actual textbooks 🙄 ). Here's an index thread of everything I have posted so far... https://dotnet.social/@SmartmanApps/110968910722113903

1/x
#MathsMonday #Maths #Math
Over time I've saved many screenshots of #AI #slop #aiSlop stuffing up #Mathematics big time, and on occasion I've had cause to reshare them, and at times I have cursed that I can only attach 4 pics per post. Then I realised, what am I worried about - just post them all in a thread and then I can just link to the thread (or individual screenshots), and can add to it as more come up 🙂 P.S. feel free to reply with more

I hereby present to you, AI's greatest 5hits...

@RoRo
I'm a high school Maths teacher/tutor, if that's of any help (I've found trying to change things in a school to be extremely difficult - the "we've always done it this way" mentality - so not sure I can help you much in that regard). All my Maths posts are made using the #MathsMonday hashtag (and various other Maths hashtags, but you can find all of my specific ones under MathsMonday, which only a couple of other teachers have ever used here)

1/5
This #MathsMonday I'm expanding on #scalars in #Mathematics. Previously we discussed that integers in #Maths are discrete, and used for counting things. In #Math we also have the Real Numbers, which is a bit of a misnomer since we don't usually use them for counting! It's more a case of people conflating them with numbers, because both use numerals. Rather, the Reals are used as multipliers, to provide a scale (though sometimes used as a fraction's decimal equivalent)…

1/7
#Maths #Math
This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS!

Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing!

In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

1/7
#Maths #Math
This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS!

Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing!

In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

1/7
#Maths #Math
This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS!

Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing!

In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

1/7
#Maths #Math
This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS!

Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing!

In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

1/7
#Maths #Math
This #MathsMonday Why there isn't an #infinite amount of #numbers between 0 and 1 on the #numberline AKA The number-line isn't a TARDIS!

Firstly, one thing to point out is that people often conflate numbers with #numerals. This is important because #scalars also use numerals. Even though both numbers and scalars use numerals, they aren't the same thing!

In #Mathematics, especially #arithmetic, we use numbers to count things. They enumerate how many things there are...

1/7
This air-quote #Mathematics unquote article https://www.quantamagazine.org/a-new-bridge-links-the-strange-math-of-infinity-to-computer-science-20251121/ keeps appearing in my feed, and I initially made some comments the first time, debunking it from a #Maths point-of-view, but given how it keeps popping up I think I need to do a more thorough #MathsMonday thread about it

Firstly, the author is a Physics journo, so you can take what he says about #math with a grain of salt (for some reason I see a lot of them doing this overreach, instead of checking with a Mathematician)…

1/6
#MathsMonday
#Mathematics #Maths #Math
This week I want to come back to the "proof" that 0.(9)=1, which I debunked at https://dotnet.social/@SmartmanApps/112711027003047449. At the time, I said there was insufficient information, because it was unclear whether 1 was meant to be interpreted as the integer 1, or as a limits of accuracy 1. i.e. 1±½. People are taking it as the integer 1, in which case I can now thoroughly debunk that argument with that interpretation...

1/10
#MathsMonday
This #Mathematics article came across my feed, about how #Maths #teachers teach #Math - https://teamone.msuurbanstem.org/wp-content/uploads/2014/07/Skemp-Relational-Instrumental-clean-copy-AT-1978.pdf - where it is quite clear that the author didn't bother asking any actual teachers about it(!), and the assumption made is completely wrong (could've saved himself all that effort if he'd just checked his #facts first #misinformation). First I'm going to address some of the wrong claims, then I will go through the actual reasons, which we DO actually teach...

#MathsMonday #Mathematics
This has been in my feed a couple of times, but with only 1 hashtag, total(!), so I'm mostly just boosting visibility (by adding more hashtags), but I will take the chance to say, yet again, no, #AI does NOT know how to do #Maths ! It doesn't know how to count (how many R's in Strawberry), it doesn't know how to reason (1=0), and this article shows, yet again, it doesn't know how to DO #Math, it only knows how to find references (in this case)

https://the-decoder.com/leading-openai-researcher-announced-a-gpt-5-math-breakthrough-that-never-happened/

1/5
#MathsMonday #Mathematics #Maths #Math
So we've seen that 0.(3) is merely a decimal APPROXIMATION of 1/3, due to the limitations of using Base 10. It's also true, for the same reason, that 1 is merely an integer APPROXIMATION of 0.(9), since 9 is infinitely recurring. If we think of 0.(9) as being 0.9+0.09+0.009+... this leads some to make a false equivalence argument, which is wrong in 2 ways...

@ricmac
I'm pro-facts, and anti-AI because I'm also a #Mathematics #teacher and #AI is horribly bad at doing Maths (can't reason). Students (or anyone at all) should NOT be using AI to learn Maths (or do their homework)! I just posted today about how AI is bad at tutoring. I run 2 (intermittent) series here, #MathsMonday and #FactFriday

Related: Programmers are also horribly bad at Maths, as evident by almost every e #calculator in existence gives wrong answers. See https://dotnet.social/@SmartmanApps/111164851485070719

@ricmac
I'm pro-facts, and anti-AI because I'm also a #Mathematics #teacher and #AI is horribly bad at doing Maths (can't reason). Students (or anyone at all) should NOT be using AI to learn Maths (or do their homework)! I just posted today about how AI is bad at tutoring. I run 2 (intermittent) series here, #MathsMonday and #FactFriday

Related: Programmers are also horribly bad at Maths, as evident by almost every e #calculator in existence gives wrong answers. See https://dotnet.social/@SmartmanApps/111164851485070719

@ricmac
I'm pro-facts, and anti-AI because I'm also a #Mathematics #teacher and #AI is horribly bad at doing Maths (can't reason). Students (or anyone at all) should NOT be using AI to learn Maths (or do their homework)! I just posted today about how AI is bad at tutoring. I run 2 (intermittent) series here, #MathsMonday and #FactFriday

Related: Programmers are also horribly bad at Maths, as evident by almost every e #calculator in existence gives wrong answers. See https://dotnet.social/@SmartmanApps/111164851485070719

@ricmac
I'm pro-facts, and anti-AI because I'm also a #Mathematics #teacher and #AI is horribly bad at doing Maths (can't reason). Students (or anyone at all) should NOT be using AI to learn Maths (or do their homework)! I just posted today about how AI is bad at tutoring. I run 2 (intermittent) series here, #MathsMonday and #FactFriday

Related: Programmers are also horribly bad at Maths, as evident by almost every e #calculator in existence gives wrong answers. See https://dotnet.social/@SmartmanApps/111164851485070719

@ricmac
I'm pro-facts, and anti-AI because I'm also a #Mathematics #teacher and #AI is horribly bad at doing Maths (can't reason). Students (or anyone at all) should NOT be using AI to learn Maths (or do their homework)! I just posted today about how AI is bad at tutoring. I run 2 (intermittent) series here, #MathsMonday and #FactFriday

Related: Programmers are also horribly bad at Maths, as evident by almost every e #calculator in existence gives wrong answers. See https://dotnet.social/@SmartmanApps/111164851485070719

1/6 #Mathematics #Math
I'm coming back to the topic of 0.(9) (i.e. 9 infinitely recurring) not being equal to 1, even though I already discussed the "proof" (which isn't a proof) at https://dotnet.social/@SmartmanApps/112711027003047449, as I've found there are many gaslighters insisting they are equal (including this wrong meme), so I will do a thorough debunking over several #MathsMonday weeks, and I've made a thread index at https://dotnet.social/@SmartmanApps/115207044364101854, starting this week with some revision of #Maths Base systems...

#MathsMonday 0.(9) isn't 1 thread index
#Mathematics #Maths #Math
1 0.999... =1 "proof" not a proof https://dotnet.social/@SmartmanApps/112711027003047449
2 Base systems https://dotnet.social/@SmartmanApps/115207057163375953
3 Limit of infinite sum false equivalence argument https://dotnet.social/@SmartmanApps/115365301522477363
4 Thorough debunking of the alleged "proof" https://dotnet.social/@SmartmanApps/115603719725224574
5 Further proof that 0.(3) is only approximately equal to 1/3 https://dotnet.social/@SmartmanApps/116316765092652161

1/6
#MathsMonday #Math
There's been a buzz - mostly from Tech Bros - about AI getting gold medals in the International #Mathematics Olympiad, including a remark "Now this *is* impressive". No it isn't, here's why...

TL;DR computers aren't human. Shocking, I know. 😂 The standards are established based on what's reasonable for humans to achieve in #Maths. Humans have limited ability to think, and fallible memory. In contrast computers have vast amounts of processing power and infallible memory...

@aolawani
Absolutely. Need to counter all the #Disinformati on Youtube and elsewhere. I've not been able to keep #MathsMonday going for a while, but plan to get back to it once life lets up for a minute...

#MathsMonday
#Chinese #Maths #Math #Mathematics
I've seen this video a couple of times, but I never saw any explanation.... so I decided to work it out for myself how they did that! 🙂 Feel free to watch it and see if you can work it out yourself, but if you can't (or don't want to), then here's how this works (scroll down for reveal)...

First some pronumerals, then the steps - a=97, b=94, c=3, d=6
- c=100-a
- d=100-b
- the first 2 digits is a-d
- the last 2 digits is cxd

1/7
#MathsMonday #Maths #Math
I want to discuss some #Mathematics terminology, as I've seen some people tripping up on it, particularly around what an "Expression" is, which I'll get to shortly...

First up, Pronumeral, but also commonly called "Variable", though strictly speaking that's not true, since sometimes Pronumerals are constants, like in a linear equation, y=mx+b, where "m" (gradient) and "b" (y-intercept) are both constants. In this case "pro" means "substituted for"...

4/4 (only 1 post today, but replying to my previous memes thread)
#MathsMonday
I made a new order of operations meme this week. Enjoy! 🙂
#Mathematics #Maths #Math

1/2
#MathsMonday
#Mathematics #Math
This week mainly just a new resource to share with you all that I came across this week, as a result of another of those ridiculous claims made in arguments about #Maths. In this case there was a familiar "division is just the inverse of multiplication" claim, which I knew wasn't right. I mean we know that multiplication is repeated addition - 2x3=2+2+2 - so does that mean division is repeated subtraction? 😂 Or...

1/3
#MathsMonday #Maths #Math
I was reminded this week about the #algebraic #Mathematics #proof of #Pythagoras theorem, which I always thought was pretty cool, so I thought I'd share it this week for those who haven't seen it. I'd seen the one where you cut out shapes, but I always found that one a little boring, but being a numbers person I really liked this proof, when I finally saw it!

You'll see in the image we have a big square which is made up of 4 triangles and a smaller square...

1/3
#MathsMonday
Previously I have written about the problem with e-calcs at https://dotnet.social/@SmartmanApps/111164851485070719 in which I pointed out that the #Wolfram Alpha calculator is one of those which gives the wrong answer for #Mathematics order of operations questions. This week someone pointed out that you CAN get a right answer, if you use #Math Input mode, so I tried it out, henceforth are my findings on this little #Maths experiment, for future reference, should anyone need it... 🙂

1/7
#MathsMonday
Should've known better than to think my last instalment would end the #Mathematics order of operations arguments. The #Math deniers always making up new excuses to avoid being wrong (sigh). So, this week I'm debunking the claim that the #Maths order of operations rules are dependent on notation (despite having already covered that the rules are universal - https://dotnet.social/@SmartmanApps/112019943359120289). Spoiler alert: they're not...

1/8
I thought I had run out of false #Mathematics claims to talk about for #MathsMonday, but good ol' human nature 😂 I've seen yet more made-up #Maths rules by people not willing to admit they were wrong (which I just happened to write about for #FactFriday last week https://dotnet.social/@SmartmanApps/112654473881719394) - in this case "the #Math order of operations (and other) rules only exist to save us writing brackets". Yeah, nah...

1/8
I thought I had run out of false #Mathematics claims to talk about for #MathsMonday, but good ol' human nature 😂 I've seen yet more made-up #Maths rules by people not willing to admit they were wrong (which I just happened to write about for #FactFriday last week https://dotnet.social/@SmartmanApps/112654473881719394) - in this case "the #Math order of operations (and other) rules only exist to save us writing brackets". Yeah, nah...

1/8
I thought I had run out of false #Mathematics claims to talk about for #MathsMonday, but good ol' human nature 😂 I've seen yet more made-up #Maths rules by people not willing to admit they were wrong (which I just happened to write about for #FactFriday last week https://dotnet.social/@SmartmanApps/112654473881719394) - in this case "the #Math order of operations (and other) rules only exist to save us writing brackets". Yeah, nah...

1/8
I thought I had run out of false #Mathematics claims to talk about for #MathsMonday, but good ol' human nature 😂 I've seen yet more made-up #Maths rules by people not willing to admit they were wrong (which I just happened to write about for #FactFriday last week https://dotnet.social/@SmartmanApps/112654473881719394) - in this case "the #Math order of operations (and other) rules only exist to save us writing brackets". Yeah, nah...