#PreAlgebra - What are the different kinds of numbers (Natural, Whole, Integer,...)

Question

What are the different kinds of numbers (Natural, Whole, Integer,...) and where do they come from?

Answer

See below for a discussion:

Analysis

The groupings of numbers and the development of them is a fascinating walk through history. Let's take a brief walk down that path:

• We can start with the number 1

• One shows existence, it shows something I can point to - someone can hand something to someone else and say "here's this thing". You have to think really really basic representation stuff here (which operates below where essentially all of us do), but even if it never really happened, one serves as that bedrock number from which everything else is going to arise. As a side note, the development of the number 0 took much much longer to form (like thousands of years after the development of the number 1).

• Counting/Natural/Whole numbers

• Once we had the number 1, people would be able to count things with tick marks (essentially little number 1s) and add them up. If there's a tick mark and a tick mark... well that's tick tick, right? At some point, it'd be simpler to count, not by saying tick tick tick tick tick.... but by having a word that associated with the number of ticks. Thus came 2, 3, 4, etc... These are therefore called the Counting Numbers. And I believe they get the name Natural Numbers because they naturally arise from the number 1. Whole Numbers are the counting numbers with the number 0 included. The symbol for these kinds of number is .

•  Integers 

• Integers are a result of subtraction. The basic thought goes like this: I have 2 things. What happens if I subtract 3 things? In the world of "I have 2 chickens. And you want to take away 3?", 2 - 3 makes no sense. However, in many other applications, the use of negative numbers is very useful. And that is what the integers are - the counting numbers that are both above zero (positive) and below zero (negative). The symbol for these kinds of numbers is .

•  Rational numbers 

• Rational numbers are a result of division. We start with an integer (say 4) and divide it into a number of pieces (say 7). And in fact this is the definition of a rational number - any number that can be expressed as a fraction of integers. The symbol for these kinds of numbers is .

•  Real numbers 

• Real numbers are all the numbers that sit on a number line and are symbolized with . There are two types of numbers on the number line - Rational (discussed above) and Irrational numbers. 
• Irrational numbers are those numbers that can't be expressed in terms of a fraction using integers. These kinds of numbers have endless strings of digits, such as more famous ones like  and others like . Irrational numbers are symbolized by .

•  Complex numbers 

• Complex numbers consist of two parts - a real part and an imaginary number part that is a real number multiplied by the square root of -1. An example of a complex number is.

• Imaginary numbers came about from needing to take the square roots of negative numbers. Consider - the square root(s) of a number is/are the numbers we can multiply together to find the number under the square root. For instance, take . We can multiply 2 by itself, so 2 X 2 = 4, or we can multiply -2 by itself, so -2 X -2 = 4. We're good so far. So what happens if we have . There are no two numbers that we can multiply together that will get us to -4. So what to do? Well, if we were to rewrite it this way, , we know what the square root of 4 is - it's 2. So what to do with the square root of -1? Assign it a symbol and move on, and the symbol is the small letter i. 

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Questions and comments always welcome!