Number system

• An integer is a whole number which could be positive (1, 2, 3…), negative (-1, -2, -3 …) or zero (0).
• An even number is an integer which is divisible by 2, e.g. -4, -2, 6, 12, 46, etc.
• An odd number is an integer which is not divisible by 2, e.g. -5, 3, 13, 15, etc.
• A prime number has only two factors, 1 and itself. It is only divisible by 1 and itself. E.g. 2, 3, 5, 7, 11, etc.
• A rational number is one which is expressed as a quotient of two integers. It can be expressed as exact fractions or ratios. E.g.  , , etc.
• Non-rational: Numbers which cannot be written as exact fractions are called irrational or non-rational numbers. E.g. √2, √7, π, etc.

RATIONAL NUMBERS

A rational number is one which is expressed as a quotient of two integers. It can be expressed as exact fractions or ratios in the form , where x and y are integers and y ≠ 0.

Examples of rational numbers

1. All integers (positive and negative), E.g. 5, 34, -28, -145, etc.
2. All recurring decimals, E.g. 0.165165165…, 1.3333, 2.666666, etc.
3. All terminating decimals, E.g. 0.625, 23.5, 3.474 925, etc.

Note that a recurring decimal can be expressed with a dot on the recurring digits, e.g. 0.666666 = 0.6^.

EXPRESSING RECURRING DECIMALS AS RATIONAL NUMBERS

EXAMPLE: Write the following as rational numbers

1. a) 0.373 737 37…. b) 4.142 142 142….

Solution

1. a) Let x = 0.373 737 37…. (1)

Multiply both sides of the equation (1) by 100:

100x = 37.373 737…….             (2)

Subtract equation (1) from (2):

99x = 37

Divide both sides by 99

Multiply both sides of the equation by 1000

1000x = 4142.142 142 142               (2)

999x = 4138

Divide both sides by 999

NON-RATIONAL NUMBERS

Non-rational: Numbers which cannot be written as exact fractions are called irrational or non-rational numbers. Non-rational numbers, when expressed as decimals are non-terminating or recurring. Examples are √2, π, √5, etc.

THE CONCEPT OF Pi (π)

Pi or π is the ratio of the circumference of a circle to its diameter.

CLASS ACTIVITY

1. Which of the following are rational and which are non-rational numbers?
2. a) 3.142 b) √17 c)√8 d) π e) 0.2222222
3. Write the following recurring decimals as rational numbers.
4. a) 3.191 919 191…… b) 3.283 283 283…..

PROPORTION

Proportion describes the relationship between two quantities such that change (increase or decrease) will lead to corresponding change (increase or decrease) in the other. There are two types of proportion, direct and indirect proportion.

DIRECT PROPORTION

When increase in one quantity leads to corresponding increase in the other or when decrease in one quantity leads to corresponding increase in the other, then the quantities are said to be in direct proportion. For example, the more fuel added to a burning fire, the more it burns.

INVERSE PROPORTION

If the proportion is such that when one quantity increases, the other decreases proportionally, then the proportion is said to be inverse.

PRACTICAL APPLICATION OF PROPORTION

EVALUATION:

1. A nurse counted 30 heart beats in 25 seconds. How many times will the heart beat in 1 minutes.
2. 8 men plough a farmland in 15 days. How many days will it take 5 men working at the same rate?

TRANSLATION OF WORD PROBLEMS INTO NUMERICAL EXPRESSIONS.

The following terms are commonly used in words expressions.

• SUM: The sum of two or more numbers is the result obtained when they are added together.

• DIFFERENCE: The difference between two numbers is the result obtained when one of the numbers is subtracted from the other.

• POSITIVE DIFFERENCE: This implies larger number smaller number.

• NEGATIVE DIFFERENCE: This implies  larger number

• NOTE: When the nature of the difference required is not stated, we consider the positive difference.

• PRODUCT: When two or more numbers are multiplied together, the result obtained is known as the product of the numbers.

• QUOTIENT: The quotient of two numbers is the result obtained by dividing one number by another.

ASSIGNMENT

Translate the following word problems into numerical expressions.

1. When a number is trebled and five times the number is subtracted, the result is 3.7
2. The sum of three consecutive numbers is 108.
3. Divide the sum of 12 and by the sum of  and 4.
4. If is doubled and 7 subtracted, the result is 23.
5. The difference between certain number and is 15.

KEYWORDS

• Consecutive
• Rational
• Non-rational
• Proportion