*hypf*

Contents

1. Natural Numbers
2. Ring of Integers
3. Rational Numbers
4. Real numbers
5. Complex numbers

Natural Numbers

Natural Numbers are pretty known. Every person who has at least a few fingers left is able to count up, mostly up to ten. This gives us a rough idea what natural numbers are like. Apparently the number one belongs to them, and we can now define that each natural number is a successor of another natural number. This gives us the sequence 1, 2, 3, 4, 5, 6, ... up to infinity (which is no number, though). We can now do pretty much already. We can define addition and multiplication on this set of numbers. They both work fine and without restrictions. We can add 5 to 876 or we could multiply 2345 by 123. The result is always another natural number. We can even define a limited variant of subtraction. Feel free to subtract 5 from 12. Or 234 from 876. There is just one problem here: What if you try subtracting 4 from 1?

Ring of Integers

To overcome one of the limitations of the natural numbers we need subtraction without restrictions. You could represent them as cut-off fingers or similar if you like. We now have negative numbers that are denoted by a minus sign in front of them: —5. All of a sudden calculations like 4 subtracted from 1 have results. Valid results inside the ring of integers (the name implies an algebraic structure named ring that has certain properties which are relevant here but a bit complicated to explain). It may be surprising to notice that the number of elements in the field of integers is exactly the same as the number of natural numbers. After some fun subtraction tasks we may notice that there is another operation still missing. An operation that even has valid results within the natural numbers: Division. You can divide the 15 by 3 and get 5. Or you could divide 225 by 15 and get 15. The reason why I didn't say anything about them in the first place was that natural numbers and the ring of integers are quite similar to each other so mostly they are just integers. This implies negative and positive numbers and zero. Division poses yet another problem, though: What if we want to divide 22 by 7?

Rational Numbers

Rational numbers derive their name from ratio. And that's all what they are about. They simply represent ratios between numbers. So every rational number consists of two integers (positive or negative): a numerator p and a denumerator q which must not be zero. They are commonly written as a fraction in the form p/q which simply means p divided by q. Rational numbers, however, also correspond to either integers or decimal fractions. So the fraction 3/5 can also be written as 0.2 as well as the fraction 0.1 corresponds to 1/10. Problems arise when trying to write fractions like 1/3 as decimal fraction: 0.3333333333333333333333333333333... ad infinitum. There is a shorthand for such thing where you can simply draw a bar over the sequence of numbers that repeats. Now we have some really cool means of expressing all kinds of numbers. One might think there are already enough of those. With the rational numbers we can find another rational in between two others, no matter how close we choose them. So even in the interval between one and two there is an infinite number of rationals (the same infinity again like with integers; and surprisingly as well, it's also exactly the same as the number of all rationals; there are ways to show that, I won't here). There is still one downside: We can't divide by zero. This is an issue that isn't resolvable since any number divided by zero could equal any other number. This can easily be seen since every number multiplied by zero equals zero itself. So the opposite direction isn't really defined. Yet, there are still things not present in the rational numbers. To create such a number you could draw a line that contains your rational numbers. Then you mark the one and zero on it and construct a square above that line. Draw the diagonal line between the lower left and the upper right point of that square and use a compass to transfer that length onto the line. Voilà, you just constructed a completely non-rational, thus irrational, number.

Real numbers

The most famous irrational number is the square root of two, sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of it while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans.

We have shown that there are numbers that can't be expressed with rational numbers, notably the square root of two in the chapter before. There is no easy way of defining which numbers are irrational. So far irrationality has not even been proven for numbers like π raised to the power of the square root of two, although they are most likely irrational. All rational and irrational numbers together are called Real numbers. There is still an infinite number of them although there are more real numbers than rationals. Real numbers allow almost all mathematical operations on them, only partially restricted. This includes powers, logarithms, roots and more. The square root, however, does not allow for negative arguments. So what is the root of minus one?

Complex numbers

The answer is pretty simple: No one really cares about what the square root of minus one might actually might look like. It simply was defined as i. Or more precise to avoid confusion: i² was defined as —1. This avoids trying to evaluate the square root of —1 within the real numbers where this isn't defined. Complex numbers consist of two parts: A real part and an imaginary part which both are real numbers. A complex number z may be written as z = a + b · i. Where a is the real part and b the imaginary part of the number. Complex numbers may be represented by an angle and a length instead of two lengths. The following sketch may illustrate this: There also are so-called complex conjugates that are simply formed by negating the imaginary part thus mirroring the point in the complex plane at the real axis. Arithmetic is a bit complex, hence the name of the set, maybe. There are a few nice things, like the n-th root of a complex number always yields n results within the complex plane that all lie on a circle. Addition of two complex numbers is simply the concatenation of the arrows that may be drawn between the origin and the number in the complex plane. Multiplication of two complex numbers is actually a combination of rotation and scaling, where multiplication of a complex number and a real number only consists of scaling. And so forth ...