Studying Mathematics Thread

[GrandizerII]

Hey, so this is a general mathematics thread where you can post anything mathematical you feel like posting that may be educational to at least some of us. You can post something basic or advanced, up to you. Choose your hypothetical target audience and share your knowledge.

I'm going to go with something very, very basic in this post, just to kick things off. Note I have no degree of any sort in mathematics, and especially not in anything to do with the pedagogical aspect of it. So it's possible I may use the wrong terms for this and that, or fail to describe things very accurately and satisfactorily, but I am bored, so hence this thread.

Numbers are ... numbers ... like 0 ... -5 ... 2.56 ... "pi" ... and so on.

You have natural numbers, like 1, 2, 3, 4, 5, 6, 7, and so on ... natural because they look "clean", perhaps. In other words, no "-" and no "decimal points" required. So 3 and 67894834865305 are natural numbers, but -5 and 6.123 are not.

Note: 0 may or may not be considered a natural number (there is a bit of debate about this), but for all practical purposes, it doesn't seem to matter much.

But when it comes to whole numbers, 0 is definitely an example. Whole numbers are pretty much equivalent to natural numbers, except they definitely include 0 as well.

Then we have negative numbers like -9 and -45678454545.

Integers are all the numbers that are either 0, negative or positive, but without decimal points required to represent them.

So -5 is an integer, 6 is an integer, 0 is an integer, but 3.15 is not an integer because there is a decimal point required to represent it literally in writing.

Note that 1.0 is still an integer even though there is a decimal point in there. This is because 1.0 is nevertheless the same as 1, and so doesn't really require the decimal point to represent it in writing. Same with 6.00 and -8.00000 and such (all integers).

What this means is all natural numbers and all whole numbers are also integers.

Whole numbers:
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Integers:
{..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

Now for rational numbers, what are they?

They are all the numbers that can be represented as a fraction that has an integer for its numerator and an integer for its denominator.

Rational numbers include whole numbers such as 0 (which can be represented as 0/1 or 0/2 or 0/345676688), 1 (which can be represented as 2/2, 1/1, 3/3), 2.5 (which can be represented as 5/2).

Basically, rational numbers include all the numbers that are integers and also all the numbers with decimal points required that happen to have a finite number of digits after the decimal point or an infinite but repeating successive sequence of digits after the decimal point.

For example:
-101 is an integer, therefore it is a rational number.

5.567 has a finite number of digits after the decimal point, therefore it is a rational number.

788545.567678567678567678... has an infinite number of digits after the decimal point, but there is nevertheless a repeating sequence of digits occurring successively (the sequence being '567678' which repeats over and over). Therefore, it is a rational number.

Remember that all rational numbers can be represented as fractions. 1/3 = 0.3333333... is a rational number (note the infinite but repeating successive sequence of '3' after the decimal point).

Note: 0.1989898... is also a rational number because even though there is a 1 that is not part of the repeated sequence of digits after the decimal point, the number itself still nevertheless satisfies one of the criteria for being a rational number.

Note also: all natural numbers are rational, all whole numbers are rational, and all integers are rational numbers.

"pi" is a number that is not rational. It cannot be represented as a fraction that has integers only. And its literal representation as 3.14159... has an infinite number of digits after the decimal point but no repeating sequence occurring infinitely successively. "pi" is irrational, as opposed to rational.

And finally, the real numbers are all the numbers that include all the examples above, including irrational numbers such as "pi".

So all natural numbers, all whole numbers, all integers, and all rational and irrational numbers are real numbers.

Then there are the imaginary and complex numbers, but let's leave that for another post.

Your turn.

[Silver]

https://goo.gl/images/BQrLkN

https://goo.gl/images/Y95WFV

[Tiberius]

0.999... has the same numeric value as 1.

[polymath257]

Not bad. One point: in the definition of rational numbers, we do not allow 0 in the denominator (no division by 0).

One issue that comes up in this is the meaning of infinite decimal expansions. So, for example, when we write
pi=3.141592653589793....
what, precisely, is being said?

One way to answer this is through the concept of a limit. In essence, a limit describes what a process is getting closer and closer to. In this case, what we have is a sequence of *rational approximations*
3
3.1
3.14
3.141
3.1415
3.14159
...

And the idea is that these rational approximations are getting closer and closer to some fixed irrational number, which happens to be pi. There are many *other* sequences of rational approximations, by the way. And to tell when two different rational approximations represent the same real number, we just ask if the approximations themselves are getting closer and closer to the other.

So, for example, and a source of many internet discussions, consider the representation
1.99999999.....
where the sequence of 9's is infinite. As was noted above, since the sequence repeats in a cycle, this number is rational. But we can do better. Consider the successive approximations
1
1.9
1.99
1.999
1.9999
1.99999
...

and ask "Is there is some *fixed* number that these are better and better approximations to?" A bit of thought will readily give
the answer as 2. And, in fact,
2=1.99999.....

This confuses many people because they say the right hand side 'never gets there'. And that is even correct, in a sense: the sequence of approximations never has a term equal to 2. But that wasn't the question. The question is what fixed number these are approximating better and better. And there is precisely one such number and that number is 2.

Now, we can do algebraic manipulations to reach the same answer:
If x=1.9999...., then
10x=19.999999....
We can subtract and get
9x=10x-x=19.9999... -1.9999..... =18,
so x=2.

The main reason I don't like this is that it ignores the underlying meaning of the notation. The notation *means* the value of the limit. That is one, fixed number. It just has two different *representations* in terms of decimals. But that is fine, it also has more than one representation in terms of fractions: 2=2/1=4/2=6/3... As long as we recognize this ambiguity in our notation, there is no problem with the numbers themselves.

[robvalue]

Prime numbers are natural numbers p so that:

p>1; If a natural number n>0 divides exactly into p, then n=1 or n=p

Another way of saying this is that each p has exactly two factors among the natural numbers.

So why isn't 1 allowed to be a prime number? It's because we then wouldn't have unique factorization into primes for natural numbers. For example:

12 = 2 * 2 * 3 = 2^2 * 3

is a unique factorization. But if we allow 1 to be prime, we have

12 = 1 * 2^2 * 3 = 1^2 * 2^2 * 3 = 1^3 * 2^2 * 3 =...

[polymath257]

Prime numbers are natural numbers p so that:

p>1; If a natural number n>0 divides exactly into p, then n=1 or n=p

Another way of saying this is that each p has exactly two factors among the natural numbers.

So why isn't 1 allowed to be a prime number? It's because we then wouldn't have unique factorization into primes for natural numbers. For example:

12 = 2 * 2 * 3 = 2^2 * 3

is a unique factorization. But if we allow 1 to be prime, we have

12 = 1 * 2^2 * 3 = 1^2 * 2^2 * 3 = 1^3 * 2^2 * 3 =...

And if we go to integers, we have to deal with things like

12=(-2)*2*(-3)

That makes life a bit more interesting!

Past this, there is the notion of a Gaussian integer: a complex number of the form m+n*i where i*i=-1.
Many of the results that are true for integers are also true for Gaussian integers, but which numbers are prime can be different.

For example, 5=(2+i)*(2-i) is no longer prime! But, it turns out, 7 is still prime!

[Edwardo Piet]

Prime numbers are natural numbers p so that:

p>1; If a natural number n>0 divides exactly into p, then n=1 or n=p

Another way of saying this is that each p has exactly two factors among the natural numbers.

So why isn't 1 allowed to be a prime number? It's because we then wouldn't have unique factorization into primes for natural numbers. For example:

12 = 2 * 2 * 3 = 2^2 * 3

is a unique factorization. But if we allow 1 to be prime, we have

12 = 1 * 2^2 * 3 = 1^2 * 2^2 * 3 = 1^3 * 2^2 * 3 =...

I think you may like Kernel Sohcahtoa's posts around here Rob He's the local math nerd. In a good way.

[polymath257]

I think you my like Kernel Sohcahtoa's posts around here Rob He's the local math nerd. In a good way.

*The* local math nerd? I'll see his nerdom and raise him a Banach space.

[Fireball]

Someone I know got on their knees to Lebesgue for a solution on a test in his calculus class.

[Alexmahone]

Q: What was Benoit B. Mandelbrot's middle name?

A: Benoit B. Mandelbrot