Types of Numbers

Published: Jul 20th, 2020

While this is not formally introduced into your curriculum, knowing the different types of numbers and what they denote is an important part of mathematical literacy. This section will introduce you to the numbers you will encounter regularly in grade 11, 12, and early university mathematics.

Types of Numbers

  1. Natural Numbers (\(\mathbb{N}*\)): These are the counting numbers. They are the numbers we use to count objects. In set notation, the natural numbers are: \(\mathbb{N}* = \left\{1, 2, 3, 4, 5, ... ,\infty\ \right\}\). Note that these numbers exclude zero!
  2. Whole Numbers (\(\mathbb{N}\)): These are the same as the natural numbers, except that they also include zero. In set notation, the whole numbers are: \(\mathbb{N}* = \left\{0, 1, 2, 3, 4, 5, ... ,\infty\ \right\}\).
  3. Integer Numbers (\(\mathbb{Z}\)): Hey! Stop being so negative. Leave that to the integer numbers. The integer numbers contain both the positive and negative of whole numbers. In set notation, these are \(\left\{ ..., -1, -2, -3, 0, 1, 2, 3, ... \right\}\);
  4. Rational Numbers (\(\mathbb{Q}\)): These are the numbers that can be represented as a fraction of two integers (All integers are allowed, except that the denominator cannot be zero). These can be represented as decimals or fractions. Examples: \(\frac{1}{3} = 0.\overline{3333}\)\(\frac{0}{1} = 0\)\(\frac{-1}{5}= -0.2\);
  5. Irrational Numbers (\(\mathbb{I}\)): These are numbers that cannot be expressed as a ratio of two integers. For example, \(\pi, e, \sqrt{2}\) are all rational numbers as they cannot be represented as a fraction of two integers. 
  6. Real Numbers (\(\mathbb{R}\)): Real numbers combine rational and irrational numbers. For example, between 0 and 1, there are infinite numbers, including \(0.1, 0.0001, 0.000001, \frac{1}{4}, \frac{1}{\sqrt{2}}\). When we use set notation to write the domain of a function, we typically write \(\left\{x\epsilon \mathbb{R} | restrictions\right\}\), which indicates that "x is a member of a real number family, as long as we are within the defined restrictions"
  7. Complex Numbers (\(\mathbb{C}\)): Sometimes, the solution to an equation is not a real number. For example, the square root of a negative number is a complex number denoted by the symbol \(i \), where \(i^2 = -1\). A complex number is an expression of the form \(a+bi\), where \(a\) and \(b\) are both real numbers.

We can represent the relation between these numbers diagrammatically as follows:

 

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