Types of numbers

There are an infinite number of numbers, both simple like 0 and complicated like e*i+pi. Between two numbers, in fact, there are an infinite number more numbers. People classify numbers based on their properties. Below is a tree diagram that shows the types of numbers. This picture is not very good because it was made with zero budget using Scratch Editor. It is recommended you first learn about the four basic operations and order of operations before progressing to the section below.

Let's go over each category, starting from the most inner category.

Natural numbers

Natural numbers, or counting numbers, are number 1, 2, 3, 4... that extends to infinity. As the tree shows, all natural numbers are whole numbers, integers, rational numbers, real numbers, and numbers.

Examples of natural numbers: 5, 87, 254, 999999999999999999999999999

0

0 is a one-number category, and 0 simply means that there is literally nothing. It is a whole number but not a natural number. 0 multiplied by any finite real number is 0, if put to any real number power is 0. Any real number divided by 0 is infinity, except 0 itself, where 0/0 is undefined.

Examples of 0: 0 (Wow so cool right?)

Whole numbers

Whole numbers are all numbers 0, 1, 2, 3... which also infinitely extends. As you can see with the chart, whole numbers are just every natural number with 0 as an addition.

Examples of whole numbers: 0, 5, 87, 482

Negative integers

Negative numbers are all numbers that are below zero, or nothing. Negative integers follow the domain -1, -2, -3, -4... In real life, if you have negative something, that means you are in debt. A negative number would be solutions to questions like "What is 5-9," where the answer would be -4.  Negative numbers also have a few properties, which will be shown below. 

Examples of negative integers: -3, -5, -135, -45782734982759

Negative arithmetic

Can you add negative numbers? If you are wonder about this, then you have come to the right section. Here is a list showing addition        equations. Can you guess what will some next?

1+5=6

1+4=5

1+3=4

1+2=3

1+1=2

1+0=1

1+(-1)=?

If you observed that these values were constantly decreasing by one in each progressing row, and you guessed 0 as your answer, then you are correct. You may have also observed that 1-1 is zero. Yes, there is a rule that y+(-x)=y-x. This is because -x simply means 0-x. Similarly, the rule also applies for subtraction, being y-(-x)=y+x. This is where the phrase "two negatives make a positive" originates. 

Continuing with the operations, multiplying and dividing with negatives also is not the most intuitive thing ever. Let's start with multiplication, the easier of the two. Below is a table that will help you find answers.

3*3=9

3*2=6

3*1=3

3*0=0

3*(-1)=?

If you guessed -3, then you are correct. A positive number multiplied by a negative number is a negative number, but what about a negative number multiplied by another negative multiplier? Below is another table showing products.

(-3)*3=-9

(-3) *2=-6

(-3)*1=-3

(-3)*0=0

(-3)*(-1)=?

The correct answer to that is 3 again, two negatives make a positive. You can think of this like a light switch that is default set to "off", or positive. Multiplying by a positive number does not flip the switch, but a multiplying by a negative number flips the switch once. Because of this, the switch is set to "on", or negative in this analogy. However, multiply it by another negative number, it turns back to "off". This forms a pattern, that if you multiply an odd number of negative numbers together, you get a negative product, but if you multiply an even number of negative numbers together, then you get a positive product. Using this, multiplying by negative numbers is made possible, and all of this previous stuff also applies to division, because the two operations are similar. This concludes the short section on arithmetic with negative numbers.

Integers

Integers are all numbers that can be written without a fraction, decimal, or percent. They contain all whole numbers as well as all the negative integers the previous section talked about. Apart from that, integers are not all that interesting.

Examples of integers: 499, 1304, 3409, 98907

Fractions/Decimals

Fractions are another way to express division, and in this case, non-integer fractions are discussed, fractions where the numerator cannot be cleanly divided by the denominator. They can also be represented as decimals.

Examples of fractions: 1/2, 539/3, 40005/2, 35789274489527/438

Rational numbers

Rational numbers are numbers that can be, lets say, counted to, or read from a piece of paper. For example, you can't read a number that goes on forever, like pi, or a never-ending decimal. Rational numbers can ALWAYS be written as fractions, where the numerator and denominator are integers, it is in simplest form, and the denominator not zero. Rational numbers include all integers and all fractions.

Examples of rational numbers: 1/7, 1/6, 342, 3999349888888888889320000008493.49384938495014

Irrational numbers

As the root of the name suggests, irrational numbers are not rational numbers. This means that they cannot be expressed as a normal fraction, meaning that they are never-ending, without a pattern. For example, the square root of two is irrational, since it cannot be put as a fraction. (Pi is also an irrational number)

Examples of irrational numbers: Square root of 2, pi, 8 times the square root of 5, 8pi, e, golden ratio number

Real numbers

Real numbers are well... real. The best way to think about real numbers is that is you can plot it on a one-dimensional number line, then it is a real number. If you can have x liters of water, weigh x kilograms or drive x kilometers, x is a real number (also metric is better fools). A number is a real number if and only if it is a rational or irrational number.

Examples of real numbers: 1, pi, 38pi, 2891/341

Imaginary numbers

Imaginary numbers are numbers that aren't real. For example, you can't get the square root of a negative number, since squared numbers are always positive. The letter i can be used to express an imaginary number. Dividing something by zero gives you an imaginary number, since you can't see how many 0's are in a number.

Examples of imaginary numbers: i, 3i, 3938019i/293012