Types of numbers

Digits - the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, used to create numbers in the base 10 decimal number system.

Numerals - the symbols used to denote the natural numbers. The Arabic numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are those used in the Hindu-Arabic number system to define numbers.

As mathematics teachers, we need to know about the different types of numbers that we are dealing with.

There are numbers like 1, 2, 3, ... etc.,  ones like 0.33333... , or ones like 5/7.

We introduce students to these gradually, and each new type comes with its own uses, and its own challenges.

The main types of numbers used in school mathematics are listed below:

Natural Numbers (N),The counting numbers {1, 2, 3, 4, 5, …} are commonly called natural numbers. (These numbers also called positive numbers, counting numbers, or natural numbers).

Whole Numbers (W). Natural numbers with zero (0) are also called whole numbers This is the set of  natural numbers, with zero, i.e., {0, 1, 2, 3, 4, 5, …}.

Integers (Z). Positive and negative counting numbers, as well as zero: {..., −3, −2, −1, 0, 1, 2, 3, ...}. 

This is the set of all whole numbers (Non-Negative Integers) plus all the negatives (or opposites aur additive inverse) of the natural numbers, i.e., {..., −3, −2, −1, 0, 1, 2, 3, ...}

Look the set {..., -3, -2, -1, 0, 1, 2, 3, ...}

Here are the whole numbers {1,2,3,...}, negative whole numbers {..., -3,-2,-1} and zero {0}. 

These are further divided into the even numbers, odd numbers, prime numbers and composite numbers.

Rational numbers (Q). Numbers that can be expressed as a ratio (p/q) of an integer to a non-zero integer. All integers are rational, but there are rational numbers that are not integers, such as −1/9.

These all are the fractions where the top (numerator) and bottom numbers denominator) are integers; e.g., 1/2, 3/4, 7/2, –4/3, 4/1 [Note: The denominator cannot be 0, but the numerator can be].

Real numbers (R), Numbers that can represent a distance along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true.

(These numbers are also called measuring numbers or measurement numbers or Scalars). This includes all numbers that can be written as a decimal (or decimal fraction). 

These includes fractions written in decimal form e.g., 0.5, 0.75 2.35, ⁻0.073, 0.3333, or 2.142857. 

It also includes all the irrational numbers such as π, √2 etc. Every real number corresponds to a point on the number line.

Irrational numbers: Real numbers that are not rational.

Purely imaginary numbers: Numbers that equal the product of a real number and the square root of −1. The number 0 is both real and purely imaginary.

Complex numbers: Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers.

1. Students generally start with the counting numbers (N).
2. They are then introduced to 0, and this gives them the whole numbers (W).
3. The integers are avoided initially, even though simple subtraction could lead to negative numbers ( e.g., 3 – 4 = ⁻1).
4. Simple unit fractions are the next group of numbers that are met i.e., {1/2, 1/3, 1/4, 1/5 ... }, then other fractions (e.g., 3/4, 4/9, 7/2, 3/100, ­­⁻1/2 etc.) which are known as the rational numbers (Q).
5. We next move onto decimal numbers (such as 0.3,  0.32, ⁻2.7). These can be called decimal fractions, because they can be written in a fractional form (e.g., 3/10, 32/100, ⁻27/10).
6. These expand to the real numbers (R), which include irrational numbers such as π, √2. An irrational number cannot be represented as a fraction (i.e., a rational number). π can be represented with numerals, i.e., 3.14159265 ... ; however the digits go on infinitely but there is no pattern to them