Studying Mathematics Thread

[Whateverist]

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

But much less elegant.  Still it is far more intriguing.

[Kernel Sohcahtoa]

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

I appreciate your comment, Hammy; however, I'm no where near skilled enough at math to be considered a nerd.  At best, I'm just some guy who finds math interesting, and I am very appreciative of those people who are talented at it.

[GrandizerII]

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

I appreciate your comment, Hammy; however, I'm no where near skilled enough at math to be considered a nerd.  At best, I'm just some guy who finds math interesting, and I am very appreciative of those people who are talented at it.

You are just as good as I am at maths, if not better. Actually, from the stuff you've posted, you definitely better.

[Edwardo Piet]

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

I appreciate your comment, Hammy; however, I'm no where near skilled enough at math to be considered a nerd.  At best, I'm just some guy who finds math interesting, and I am very appreciative of those people who are talented at it.

Kernel if you're not good at maths then I'm extremely stupid at maths.

[LadyForCamus]

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.

What’s the point, Grand?  Apparently math is useless now.  😉

[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.

What’s the point, Grand?  Apparently math is useless now.  😉

Yeah, the theists sure got us there. I guess I just like to play around with useless concepts.

[polymath257]

OK, complex numbers. These are numbers of the form x+i*y where x and y are real numbers and i*i=-1.

ALL the usual rules of algebra still apply, so

(2+3*i)(4-5*i)=2*4 -2*5*i +3*i*4 -3*5*i*i =8 -10*i +12*i -15*(-1) =23+2*i.

Now, if we want (and we should), we can identify the complex number x+i*y with the point of the plane (x,y). In this scheme, addition becomes ordinary vector addition, but multiplication turns out to *add* angles and multiply lengths of the vectors.

So, for example, multiplication by i corresponds to a 90 degree rotation. Since two such multiplications flip the direction of a vector, we get i*i=-1.

The complex numbers have a variety of uses. They simplify notation for many phenomena involving wave motion and are used in electronics extensively because of this. Quantum mechanics has wave functions that are complex valued and this is essential to explain some of the strange behavior of the quantum world.

We can consider the Gaussian integers, which are numbers of the form m+n*i where both m and n are integers. It turns out that many of the classical properties of the integers still hold for Gaussian integers, including things like unique factorization into primes. These turn out to be useful even for questions about ordinary integers and many of their early uses were for that purpose.

[GrandizerII]

We can consider the Gaussian integers, which are numbers of the form m+n*i where both m and n are integers. It turns out that many of the classical properties of the integers still hold for Gaussian integers, including things like unique factorization into primes. These turn out to be useful even for questions about ordinary integers and many of their early uses were for that purpose.

So Gaussian integers are complex numbers, but with integers only involved instead of just any real numbers.

[polymath257]

We can consider the Gaussian integers, which are numbers of the form m+n*i where both m and n are integers. It turns out that many of the classical properties of the integers still hold for Gaussian integers, including things like unique factorization into primes. These turn out to be useful even for questions about ordinary integers and many of their early uses were for that purpose.

So Gaussian integers are complex numbers, but with integers only involved instead of just any real numbers.

Exactly.

[robvalue]

I find the application of complex numbers to be fascinating.

Once you extend the real numbers to the complex number, any polynomial* with rational coefficients is guaranteed to have at least one root. (Polynomials are the sum of natural powers of a variable, with multipliers called coefficients, such as ax^3 + bx^2 + cx + d. A root is a value of x which makes the whole thing equal zero.)

*excluding those which are just a constant, i.e. no nonzero powers of x appear