Number Systems

What are Number Systems

Number systems are a way to represent quantities or values.

---

Sets of Numbers

We can view number sets through the lens of human evolution. natural numbers were used when human beings were still hunter gatherers, and thought of gathering and grouping physical things, such as apples. Then humans evolved to manage forms of surplus and scarcity, resulting in Integers, where some may have an excess of apples, and some may have a scarcity of apples (leading to zeros and negatives). Then human beings developed apple pies, but apple pies couldn’t all be eaten by one person at once! So they invented Rational Numbers to divide the apple pie into even pieces for the family. Then, human beings wanted to understand how big apple pies could get, and whether it was worth making many small apple pies or one big apple pie, hence they had to understand ratios and circles, leading to the discovery of $\pi$, hence the discovery of Real Numbers. But then Rafael Bombelli in the 1500’s asked the question “what happens when we $\sqrt{-1}$?” and came up with conventions to handle these numbers, leading to Complex Numbers.

Natural Numbers

The natural numbers are represented by the sign $\mathbb{N}$. The intuition behind natural numbers is to represent quantities of physical things in reality. These are the first types of numbers that human beings considered. The natural numbers do not contain zero!

$$\mathbb{N}=\{1,2,3,\dots ,\infty \}$$

For example, when counting the number of apples in a basket, you start from one and count to the number of apples that there are.

Note

Natural numbers have no end. If you reach a maximum, then you can always ‘add another’. For example, if you have 1000 apples in your basket, you can always add one more. Although this may not be physically possible, it is theoretically possible.

Subsets of Natural Numbers

1. Prime Numbers

Integers

Integers are represented by the sign $\mathbb{Z}$. They are similar to natural numbers, but include negatives and zero.

$$\mathbb{Z}=\{-\infty ,\dots ,-3,-2,-1,0,1,2,3,\dots ,\infty \}$$

Integers are useful for describing things that whole. One person may have an excess of three apples $(3)$, and another may have a need for two apples, but doesn’t have them yet $(-2)$. Integers can also be useful for describing things that ‘step’, such as digital signals.

Rational Numbers

Rational numbers are represented by the sign $\mathbb{Q}$. Quotients represent fractions, being parts of whole numbers.

$$\mathbb{Q}=\{\frac{m}{n}:m,n\in \mathbb{Z},n\ne 0\}$$

The above means that rational numbers are a fraction, with $m$ and $n$ being integers, and $n$ cannot be zero. Such numbers include the following:

$$\mathbb{Q}\in \{1.0001,0.5,2.5,8.49,10.9999\}$$

Real Numbers

Real numbers are represented by the sign $\mathbb{R}$. Real numbers contain all of the above + irrational numbers. That means numbers such as $\pi$ and $e$ and $\sqrt{2}$ . Real numbers can be thought of in different ways, but commonly are represented by a number line.

Complex Numbers

Complex numbers are represented by the sign $\mathbb{C}$. Complex numbers contain an imaginary component, which is a multiple of $\sqrt{-1}$, often combined with a real component. An imaginary number $z$ is made up of the following:

$$z=x+y\cdot i$$

where

$$x\in \mathbb{R}\text{and}y\cdot i\in \mathbb{C}$$

In other words, a complex number $z$ has a real component $x$ and a complex component $i\cdot y$. In the image below, you can see the horizontal plane representing real numbers, and the vertical plane representing complex numbers.

---

Base-N Systems

Base-2 (Binary)

Base-8 (Octary)

Base-10 (Decimal)

Base-16 (Hexadecimal)

Base-60 (Sexagesimal)