Numbers are simple.
I don’t understand anything.
I AM ALL UP IN THIS PICTURE. I WANT THIS AS A POSTER!!!!!!
Okay but let’s talk number progression.
The number zero didn’t arise naturally for centuries. No one ever needed to count how many somethings they had, when they had none of those somethings. But counting was the first use for numbers. 1, 2, 3.. the list goes on.
This was pretty nice, but one big use for numbers was measuring things. If you’ve ever tried to find the area of a triangle, you might recall the formal is A = bh/2, or “half the base times the height”.
This introduces something quite perplexing though, the concept of a half. Further, a whole number system that could be represented as ratios of two whole numbers, 1/2, 1/3, 597/426, etc. These are the “Rationals” above, once you include negatives.
I can’t remember the full details, as numeracy was invented by Indians, Sumerians, and other societies, all independently, but the progress was surprisingly similar.
For, once they had comprehended the ideas of measuring, Geometry was very soon to follow. Relationships between the lengths of shapes, and their internal area, was very important.
A lot of canals were trapezium shaped, so in order to estimate the cost of digging canals, employers would have to be able to calculate the volume of dirt to be removed.
With Geometry, came equations. Expressions that displayed the relationships between the measurements of shapes. It wasn’t long before equations were invented, that couldn’t be solved with the current number system.
For example, x + 5 = 2. Nowadays, we can solve it and see x = -3, but in days of old, there was no concept of negative numbers. Although it didn’t arise naturally, such numbers were indeed useful to add our number system, to represent debt, or decay.
Further still, there were equations we could not solve. These were the polynomials.
The problem first arose when followers of Pythagorus (who didn’t actually prove the Theorem of Pythagorus first, that was the Chinese. He just made it popular), who knew that given a right-angled triangle with both sides adjacent to the right angle being of length 1, would give a hypotenuse of length square root of 2.
But you see, the square root of 2 can NOT be written as a fraction, as people commonly believed. This fact was proved, was very contraversial, and I’m pretty sure someone got poisoned then drowned at sea for trying to publish the proof.
In fact there are infinitely many unique numbers that can not be written as fractions. The polynomial x^2 - p = 0, where p is a prime number, will always have what is called “irrational roots”, solutions to x that can not be written as fractions. These irrationals are strictly Algebraic though; they are solutions to algebraic equations. Thus your Real Algebraic numbers.
Given an Algebraic Irrational number, if you were to try to write it out as a decimal, it would be an infinite string of seemingly random digits.
"Like Pi!", you might say.
Well, Pi is different still. There is no polynomial equation (with integer coefficients to your x values) that has Pi as a solution. Same with the famous constant e. These are so irrationally irrational, that you can’t write out an equation to which they are a solution. Yet they are still useful numbers, that crop up occasionally, and by adding them to our number system, it is finally called Complete. “Complete” has a very rigorous mathematical definition, but the main idea is that there are no holes. This gives us the Real number line, which is indeed a continuous line with no holes.
But that’s not all! x^2 + 1 = 0 has no solution in the Real number line. Which is where the Complex plane comes into play. We invent a solution, some number i, such that i^2 = -1. When we add this to our number system, and close it under addition and multiplication, we finally have what is called an Algebraically closed field. Every single equation, finite or even an infinite equation, will have solutions in the complex plane.
Also, it was a huge step forward at the time to even represent an unknown quantity in an equation, as we now do all the time with the familiar x. It’s become so routine that it’s easy to forget how big an achievement it was to come up with the idea the very first time. And along with this advance came the realization that x can be manipulated like any other normal, “actual” number and this helped make algebra a real area of study.
I’m pretty sure these two authors are both mathematicians, so it’s all accurate - the first book even has really understandable primers on stuff from a college level course on algebra - so you get a really good idea of what’s being talked about.