There are infinitely many numbers, and infinitely many ways to combine and manipulate those numbers.
Mathematicians often represent numbers in a line. Pick a point on the line, and this represents a number.
At the end of the day, though, almost all of the numbers that we use are based on a handful of extremely important numbers that sit at the foundation of all of math.
What follows are the eight numbers you actually need to build the number line, and to do just about anything quantitative.
Zero
In The Beginning, There Was Zero.
Zero represents the absence of things. Zero is also an essential element of our number system. We use zero as a placeholder when writing numbers with more than one digit — zero lets me know the difference between having 2 dollars and 20 dollars.
Zero as a number on its own is also extremely important in math. Zero is the "additive identity"— any time I add a number to zero, I get that number back: 3 + 0 = 3. This property of zero is a central aspect of arithmetic and algebra. Zero sits in the middle of the number line, separating the positive numbers from the negative numbers, and is thus the starting point for building our number system.
One
We Can't Get Very Far Just Having Zero, So We Turn To One.
As zero was the additive identity, one is the multiplicative identity — take any number and multiply it by one, and you get that number back. 5 x 1 is just 5.
Just using one, we can start to build up the number line. In particular, we can use one to get the natural numbers: 0, 1, 2, 3, 4, 5, and so on. We keep adding one to itself to get these other numbers: 2 is 1 + 1, 3 is 1 + 2, 4 is 1 + 3, and we keep going, right on out to infinity.
The natural numbers are our most basic numbers. We use them to count things. We can also do arithmetic with the natural numbers: if I add or multiply together any two natural numbers, I get another natural number. I can also sometimes, but not always, subtract two natural numbers, or divide one natural number by another: 10 - 6 = 4, and 12 ÷ 4 = 3. Just using zero and one, and our basic arithmetic operations, we can already do a good amount of math just using the natural numbers.
Negative One
Natural Numbers Are Pretty Great, But They Are Also Quite Limited.
To start with, it is not always possible to subtract two naturals and get another natural. If all I have to work with are these counting numbers, I have no idea how to parse a statement like 3 - 8.
One of the wonderful things about math is that, when we are confronted with a limitation like this, we can just expand the system we are working with to remove the limitation. To allow for subtraction, we add -1 to our growing number line. -1 brings with it all the other negative whole numbers, since multiplying a positive number by -1 gives the negative version of that number: -3 is just -1 x 3. By bringing in negative numbers, we have solved our subtraction problem. 3 - 8 is just -5. Putting together the positive numbers, zero, and our new negative numbers, we get the integers, and we can always subtract two integers from each other and get an integer as the result. The integers provide the anchor points for the number line.
The negative numbers are useful in representing deficits — if I owe the bank $500, I can think of my bank balance as being -500. We also use negative numbers when we have some scaled quantity where values below zero are possible, such as temperature. In the frozen wasteland of my hometown of Buffalo, we would get a few winter days each year down in the -20° range.
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