In mathematics, we can classify all numbers into groups based on their shared characteristics and type. Many of the classifications are subcategories of each other, so often a single number belongs to more than one group. These classifications include natural numbers, whole numbers, integers, rational and irrational numbers, and real and complex numbers.
Now, we will look at each group of numbers individually.
Natural Numbers
Natural numbers are defined as non-negative counting numbers. The set of natural numbers is denoted by the symbol, N.
The natural numbers include 1, 2, 3, 4, 5, etc. Typically the number zero is not included in the set of natural numbers, however many mathematicians debate this fact.
Whole Numbers
The set of whole numbers includes all the natural numbers together with zero. Whole numbers do not include any negative values.
Whole numbers are easily identified because they are not decimals and they are not fractions, they are simply whole values.
Integers
Integers extend N, the set of all natural numbers, by including the negative counting numbers and zero. The set of all integers is denoted by the symbol, Z. All integers are whole numbers, or they are the negative equivalent of a whole number.
The set of integers would include: Z = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4 …}.
Rational Numbers
A rational number is the ratio or quotient of an integer and another non-zero integer. The set of all rational numbers is denoted by the symbol, Q.
Rational numbers can be integers, fractions or decimals which terminate or repeat a pattern.
All rational numbers are of the form, a/b such that b is not equal to b. Examples of rational numbers are: -100, -20 ¼, -1.5, 0, 1, 1.5, 1 ½, 1.75, 1/3.
Irrational Numbers
Irrational numbers are numbers which cannot be represented as a fraction. All irrational numbers are decimal values which never terminate or repeat.
Some examples of irrational numbers are √2, √3, and e.
Real Numbers
Real numbers are all the numbers on a number line. The set of all real numbers denoted by the symbol, R, and consists of all the rational numbers (which includes all natural numbers, whole numbers and integers) and all of the irrational numbers.
Imaginary Numbers
An imaginary number is a number whose square is a negative real number, and is denoted by the symbol i, such that i² = -1. Some examples of imaginary numbers are: -5i, 3i and 7.5i.
Complex Numbers
A complex number consists of two parts; a real number and an imaginary number, and is expressed in the form a + bi. Remember that “i” is the notation for the imaginary part of a number.
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