Number systems (natural numbers, integers, rational and irrational numbers, real numbers) — KCSE Mathematics

A number line is a visual representation of numbers in a straight line. It helps in comparing and ordering different types of numbers such as integers and rational numbers. Key features of a number line:

• It extends infinitely in both directions.
• Each point on the line corresponds to a unique number.
• Positive numbers are to the right of zero, while negative numbers are to the left.

To represent integers:

• Plot whole numbers like -3, 0, and 5 on the number line.

To represent rational numbers:

• Include fractions like 1/2 or -3/4. For example, 1/2 lies between 0 and 1.

Comparing and ordering:

• To compare numbers, identify their positions on the number line. The number further to the right is greater.
• For instance, to compare -2 and 3, -2 is to the left of 3, hence -2 < 3.

Ordering numbers:

• Arrange numbers in ascending order by plotting them on the number line and reading from left to right.

In this lesson, we will apply the four operations (addition, subtraction, multiplication, and division) on integers and state the properties of real numbers.

1. Addition and Subtraction of Integers:

   • Example: (-3 + 5 = 2) and (-7 - 2 = -9)
   • The sum of two integers can be positive, negative, or zero.

2. Multiplication of Integers:

   • Example: (-4 \times 3 = -12) and (2 \times -5 = -10)
   • The product of two integers is positive if both are positive or negative, and negative if one is positive and the other is negative.

3. Division of Integers:

   • Example: (-10 \div 2 = -5) and (8 \div -4 = -2)
   • Division by zero is undefined.

Properties of Real Numbers:

• Commutative Property: (a + b = b + a) and (ab = ba)
• Associative Property: ((a + b) + c = a + (b + c)) and ((ab)c = a(bc))
• Distributive Property: (a(b + c) = ab + ac)

Number systems are essential in mathematics, and they categorize numbers based on their properties. Here are the main types:

• Natural Numbers: These are positive integers starting from 1 (1, 2, 3, ...). They do not include zero or negative numbers.
• Integers: This set includes all whole numbers, both positive and negative, as well as zero (..., -3, -2, -1, 0, 1, 2, 3, ...).
• Rational Numbers: These can be expressed as a fraction where both the numerator and denominator are integers, and the denominator is not zero (e.g., 1/2, -3, 0.75).
• Irrational Numbers: These cannot be expressed as simple fractions. Their decimal expansions are non-terminating and non-repeating (e.g., √2, π).
• Real Numbers: This set includes all rational and irrational numbers, representing all possible values along the number line.

Understanding these categories helps in performing various mathematical operations and solving problems effectively.