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When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required, you can neglect them as being zero for all practical purposes.

This is fine for those that are applied mathematicians, physicists, actuaries etc, that just want it as a tool to use in their work. But mathematicians, while conceding it is OK to start that way, eventually will need to rectify being handwavey and be logically sound. In calculus, that is sometimes called doing your ‘epsilonics’. This is code for studying what is called real analysis:

http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

I posted the above link in case the reader does not know real analysis, and since this is a beginner-level thread I don't expect them to, wanted to see what it is about. Just peek at it - I will not be using it. Any analysis ideas I will explicitly state. Instead, I will be making the idea of infinitesimal logically sound. About 1960, mathematicians (notably Abraham Robinson) did something nifty. They created hyperreal numbers, which have real numbers plus actual infinitesimals. These are numbers y with a very strange property. If x is any positive real number -x<y<x. Normally 0 is the only number with that property - but in the hyperreals, there are actual positive numbers not equal to zero that is less than any positive real number. That way, the infinitesimal approach can be justified without logical issues. It's also more in line with how many are likely to do calculus in practice. Even though I know real analysis, I hardly ever use it - instead use infinitesimals. After reading this, you can continue doing it knowing it is logically sound. I could give some links, and it would be advisable to read some texts or lecture notes later, but I will instead explain hyperreals in this post. Also, many books introduce, IMHO, unnecessary ideas, such as ultrafilters, making understanding them more complex than necessary.

Since the reader is not expected to have done real analysis, nobody probably defined precisely real numbers. For mathematicians, that is a no-no; you encounter the problem of talking about something that may not exist. There are several ways of doing it. I will use sequences because that is what is used for hyperreals. However, we need a special kind of sequence called a Cauchy sequence.

A Cauchy sequence is simply a sequence of numbers Xn such that for any e > 0, an N can be found if m and n are greater than N, then |Xm-Xn| < e. It is an idea from real analysis, but it is all we need. Intuitively, the terms of the sequence eventually get closer and closer together with no constraint on how close they get. You would likely say for n very large, the terms are so close to each other that they are, for all practical purposes, equal, i.e. the sequence converges to a number. And that is the general idea - except we are careful in the definition and don't have to specify that number because we are using it to define the real number it converges to.

The reals are all the Cauchy sequences of rational numbers.

Two reals, A and B, are equal if An - Bn converges to zero. This means in analysis lingo for any e > 0 an N can be found such that for any n > N |An - Bn| < e.

We define A < B as an N can be found such that for any m > N, Am < Bm. Similarly, for A > B.

We can easily define the usual operations of addition, multiplication, etc. A + B = An + Bn. A*B = An*Bn. Similarly, for subtraction. But not division. 1/Xn may not be Cauchy. We need to ensure 1/Xn is Cauchy first, which I show below for those interested provided the sequence does not converge to 0. Hence if Bn does not converge to zero division is defined by An/Bn.

The hyperreals are constructed similarly to the reals but use any sequence of reals, with a different definition of equality, > and <.

The hyperreals are all the sequences of real numbers.

If F(X) is a function defined on the reals, then that can easily be extended to the hyperreals by F(X) = F(Xn).

A + B = An + Bn. A*B = An*Bn. Similar definitions for subtraction and division without worrying about the division issue in constructing reals.

Here is the other difference from the construction of the reals. Two hyperreals, A and B, are equal if An = Bn except for a finite number of terms.

We define A < B as Am < Bm except for a finite number of terms. Similarly, for A > B.

If X is a real number, then the sequence Xn = X is the hyperreal of the real number X. Any sequence equal to X is obviously also the real number X.

Now we can show that the hyperreals contain infinitesimals. Let X be any positive real number. Let b be the hyperreal bn = 1/n. Then regardless of what value X is, we can find an N such that 1/n < X for any n > N. Hence, by the definition of < in the hyperreals, b<X for any positive real number, hence b is infinitesimal.

This implies some rather strange things. We have for the reals .9999999999999…… = 1. But what about the hyperreals? .9999999….. is the sequence .9 .99 .999 ………. But every term is less than 1. Thus .99999999….. < 1. However, 1 - .99999999999...... is the sequence = .1 .01 .001 ……. = a1 a2 … an …. Hence for any positive real number X, we can find N such that for n > N then an < X. Hence .9999999…. differs infinitesimally from 1.

Also, we have infinitesimals smaller than other infinitesimals, eg 1/n^2 < 1/n, except when n = 1.

Hyperreals also contain infinite numbers larger than any real number. Let A be the sequence n. If X is any real number there is an N such for all n > N n > X. Again we have infinitely large numbers greater than other infinitely large numbers because except for n = 1 n^2 > n.

If a is an infinitesimal 1/a is infinitely large. This follows from a/a = (1/a)*a = 1. If 1/a is infinitesimal then multiplying by an infinitesimal will give another infinitesimal. If 1/a is a real number, multiplying it by an infinitesimal still gives another infinitesimal. This means 1/a must be infinite. And conversely, if a is infinite 1/a is infinitesimal.

Let's see how it is used in calculus. The derivative is simply (f(x + dx) - f(x))/dx - dx infinitesimal, nothing more is needed except to say we can, of course, neglect the infinitesimal part of the answer when working in the reals which would normally be done. Why? If a is infinitesimal and x is any positive real number then -x<a<x. But in the reals, only one number has that property - 0.

For the integral, let A be the area under f(x) from a to b. A = sigma (from a to b) f(x)*delta(x) + e(delta(x)) with e an error term that becomes 0 when delta(x) is zero. Of course, delta(x) can't be zero, however, it can be infinitesimally close to zero in which case the error should be infinitesimally close to zero. A = sigma (from a to b) f(x)*dx + e(dx) where dx is the infinitesimal (delta(x), delta(x)/2, delta(x)/3 .........). Neglecting infinitesimals, we have A = integral (a to b) f(x)dx.

The reader may find it instructive and fun to go through a usual handwavy infinitesimal calculus treatment and apply the hyperreals to it.

This is fine for those that are applied mathematicians, physicists, actuaries etc, that just want it as a tool to use in their work. But mathematicians, while conceding it is OK to start that way, eventually will need to rectify being handwavey and be logically sound. In calculus, that is sometimes called doing your ‘epsilonics’. This is code for studying what is called real analysis:

http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

I posted the above link in case the reader does not know real analysis, and since this is a beginner-level thread I don't expect them to, wanted to see what it is about. Just peek at it - I will not be using it. Any analysis ideas I will explicitly state. Instead, I will be making the idea of infinitesimal logically sound. About 1960, mathematicians (notably Abraham Robinson) did something nifty. They created hyperreal numbers, which have real numbers plus actual infinitesimals. These are numbers y with a very strange property. If x is any positive real number -x<y<x. Normally 0 is the only number with that property - but in the hyperreals, there are actual positive numbers not equal to zero that is less than any positive real number. That way, the infinitesimal approach can be justified without logical issues. It's also more in line with how many are likely to do calculus in practice. Even though I know real analysis, I hardly ever use it - instead use infinitesimals. After reading this, you can continue doing it knowing it is logically sound. I could give some links, and it would be advisable to read some texts or lecture notes later, but I will instead explain hyperreals in this post. Also, many books introduce, IMHO, unnecessary ideas, such as ultrafilters, making understanding them more complex than necessary.

Since the reader is not expected to have done real analysis, nobody probably defined precisely real numbers. For mathematicians, that is a no-no; you encounter the problem of talking about something that may not exist. There are several ways of doing it. I will use sequences because that is what is used for hyperreals. However, we need a special kind of sequence called a Cauchy sequence.

A Cauchy sequence is simply a sequence of numbers Xn such that for any e > 0, an N can be found if m and n are greater than N, then |Xm-Xn| < e. It is an idea from real analysis, but it is all we need. Intuitively, the terms of the sequence eventually get closer and closer together with no constraint on how close they get. You would likely say for n very large, the terms are so close to each other that they are, for all practical purposes, equal, i.e. the sequence converges to a number. And that is the general idea - except we are careful in the definition and don't have to specify that number because we are using it to define the real number it converges to.

The reals are all the Cauchy sequences of rational numbers.

Two reals, A and B, are equal if An - Bn converges to zero. This means in analysis lingo for any e > 0 an N can be found such that for any n > N |An - Bn| < e.

We define A < B as an N can be found such that for any m > N, Am < Bm. Similarly, for A > B.

We can easily define the usual operations of addition, multiplication, etc. A + B = An + Bn. A*B = An*Bn. Similarly, for subtraction. But not division. 1/Xn may not be Cauchy. We need to ensure 1/Xn is Cauchy first, which I show below for those interested provided the sequence does not converge to 0. Hence if Bn does not converge to zero division is defined by An/Bn.

The hyperreals are constructed similarly to the reals but use any sequence of reals, with a different definition of equality, > and <.

The hyperreals are all the sequences of real numbers.

If F(X) is a function defined on the reals, then that can easily be extended to the hyperreals by F(X) = F(Xn).

A + B = An + Bn. A*B = An*Bn. Similar definitions for subtraction and division without worrying about the division issue in constructing reals.

Here is the other difference from the construction of the reals. Two hyperreals, A and B, are equal if An = Bn except for a finite number of terms.

We define A < B as Am < Bm except for a finite number of terms. Similarly, for A > B.

If X is a real number, then the sequence Xn = X is the hyperreal of the real number X. Any sequence equal to X is obviously also the real number X.

Now we can show that the hyperreals contain infinitesimals. Let X be any positive real number. Let b be the hyperreal bn = 1/n. Then regardless of what value X is, we can find an N such that 1/n < X for any n > N. Hence, by the definition of < in the hyperreals, b<X for any positive real number, hence b is infinitesimal.

This implies some rather strange things. We have for the reals .9999999999999…… = 1. But what about the hyperreals? .9999999….. is the sequence .9 .99 .999 ………. But every term is less than 1. Thus .99999999….. < 1. However, 1 - .99999999999...... is the sequence = .1 .01 .001 ……. = a1 a2 … an …. Hence for any positive real number X, we can find N such that for n > N then an < X. Hence .9999999…. differs infinitesimally from 1.

Also, we have infinitesimals smaller than other infinitesimals, eg 1/n^2 < 1/n, except when n = 1.

Hyperreals also contain infinite numbers larger than any real number. Let A be the sequence n. If X is any real number there is an N such for all n > N n > X. Again we have infinitely large numbers greater than other infinitely large numbers because except for n = 1 n^2 > n.

If a is an infinitesimal 1/a is infinitely large. This follows from a/a = (1/a)*a = 1. If 1/a is infinitesimal then multiplying by an infinitesimal will give another infinitesimal. If 1/a is a real number, multiplying it by an infinitesimal still gives another infinitesimal. This means 1/a must be infinite. And conversely, if a is infinite 1/a is infinitesimal.

Let's see how it is used in calculus. The derivative is simply (f(x + dx) - f(x))/dx - dx infinitesimal, nothing more is needed except to say we can, of course, neglect the infinitesimal part of the answer when working in the reals which would normally be done. Why? If a is infinitesimal and x is any positive real number then -x<a<x. But in the reals, only one number has that property - 0.

For the integral, let A be the area under f(x) from a to b. A = sigma (from a to b) f(x)*delta(x) + e(delta(x)) with e an error term that becomes 0 when delta(x) is zero. Of course, delta(x) can't be zero, however, it can be infinitesimally close to zero in which case the error should be infinitesimally close to zero. A = sigma (from a to b) f(x)*dx + e(dx) where dx is the infinitesimal (delta(x), delta(x)/2, delta(x)/3 .........). Neglecting infinitesimals, we have A = integral (a to b) f(x)dx.

The reader may find it instructive and fun to go through a usual handwavy infinitesimal calculus treatment and apply the hyperreals to it.

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