# The Theory of Numbers

Subject:

### MATHEMATICS

Term:

First Term

Week:

Week 2

Class:

JSS 3 / BASIC 9

Previous lesson: Pupils have previous knowledge of

### CONVERTING FROM BASE TWO TO ANOTHER BASE (BINARY NUMBERS)

that was taught in their previous lesson

Topic:

The Theory of Numbers

1. Types of numbers
2. Rational and non-rational numbers; the concept of Pi; Recurring and terminating decimals.
3. Proportion (direct and inverse proportion, reciprocals). (d) Word problems.

Behavioural Objectives:

At the end of the lesson, learners will be able to

• explain types of numbers
• define rational and irrational number
• explain direct and indirect proportion
• solve simple sums on the topics stated above

Instructional Materials:

• Wall charts
• Pictures
• Related Online Video
• Flash Cards

Methods of Teaching:

• Class Discussion
• Group Discussion
• Asking Questions
• Explanation
• Role Modelling
• Role Delegation

Reference Materials:

• Scheme of Work
• Online Information
• Textbooks
• Workbooks
• 9 Year Basic Education Curriculum
• Workbooks

CONTENT:

WEEK 2

CONTENT: Number System:

• Types of numbers
• Rational and non-rational numbers; the concept of Pi; Recurring and terminating decimals.
• Proportion (direct and inverse proportion, reciprocals). (d) Word problems.

Numbers: Definitions

• 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.
• Complex numbers: These are the numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is an imaginary number. E.g. 3 + 4i, -5 – 2i, 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….
2. 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

x =  , which is the rational number.

1. b) Let x = 4.142 142 142…. (1)

Multiply both sides of the equation by 1000

1000x = 4142.142 142 142               (2)

999x = 4138

Divide both sides by 999

x =

So, 4.142 142 142…. = , which is a rational number.

When a decimal number is repeated, or has a repeating pattern, it is called a recurring decimal. In order to express a recurring decimal as a rational number, we can use the following steps:

1) Find the repeating pattern in the decimal number.

2) Express the repeating pattern as a fraction. The denominator of the fraction will be a number with as many zeros as there are digits in the repeating pattern.

3) Add the fraction to the non-repeating part of the decimal number.

For example, let’s express the recurring decimal 0.3333… as a rational number.

The first step is to find the repeating pattern, which in this case is 3.

The second step is to express the repeating pattern as a fraction. Since there are an infinite number of 3s in the decimal, we can use the fraction 3/9.

The third and final step is to add the fraction to the non-repeating part of the decimal number, which in this case is 0.

Therefore, the rational number equivalent of 0.3333… is 3/9.

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.

Non-rational numbers are those numbers that cannot be expressed as a rational number. They include all of the real numbers that are not rational, such as √2 (square root of 2) and π (pi). Non-rational numbers cannot be written as a ratio of two integers, and they often have an infinite or repeating decimal expansion. Some examples of non-rational numbers are shown below.

Non-rational numbers are often used in mathematical and scientific applications, where they can help to describe situations that cannot be easily expressed as a rational number. In many cases, non-rational numbers can provide a more accurate description of a real-world situation than a rational number could. For example, the decimal expansion of π (pi) is often used to calculate the circumference of a circle, as it is more accurate than using the rational number 3.14.

Non-rational numbers can also be used in art and music to create interesting and complex patterns that would be difficult to achieve with rational numbers alone. In

### THE CONCEPT OF Pi (π)

Pi is the ratio of a circle’s circumference to its diameter. It is also the ratio of a sphere’s surface area to its volume. Pi is an irrational number, which means that it cannot be expressed as a rational number (a number that can be expressed as a fraction). Pi is also a transcendental number, which means that it is not the root of any polynomial equation with integer coefficients.

Pi is believed to have first been calculated by the Greek mathematician Archimedes, who approximated it to be 3.14. The first known exact calculation of pi was done by the Chinese mathematician Zu Chongzhi, who approximated it to be 3.141592653589793238462643383279502884197169399375105820974944592307816406286 208998628034825342117067982148086513282306647093844609550582231725359408128481 117450284102701938521105559644622948954930381964428810975665933446128475648233 77836172069466019305, in the 5th century.

Pi is an important number in many fields, including mathematics, physics, and engineering. It is also a popular topic of discussion among mathematicians and physicists.

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

π =  =  =  , where r is the radius of the circle.

Early mathematicians made effort to obtain the exact value of π. Some of these attempts are indicated in the table below:

 Inventor Year Value of π Egyptians About 1600 BC = 3.1604938… Jews About 1000 BC 3 Archimedes (a Greek) 287 – 212 BC = 3.14128571… Chinese 500 AD = 3.1415929

Archimedes attempt was the most famous. He obtained his value of pi by drawing squares inside and outside a circle of radius r. He then used this along with the fact that area of a circle is πr2.

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.

In mathematics, two variables are in direct proportion if a change in one is always accompanied by a change in the other, and both changes are in the same direction. In other words, the variables increase or decrease together.

The formula for direct proportion is:

y = kx

where:

y is the dependent variable (the one that changes in response to a change in the other variable)

x is the independent variable (the one that changes first)

k is the proportionality constant (the number that relates the two variables).

For example, if we know that y = 2x then we can say that y is directly proportional to x. This means that if x increases by 1 unit, then y will increase by 2 units. Similarly, if x decreases by 1 unit, then y will decrease by 2 units.

We can also use the direct proportionality formula to calculate one variable if we know the other variable and the proportionality constant. For example, if we know that y = 2x and we want to find out what y is when x = 6, we can simply substitute the values into the formula:

y = 2x

y = 2(6)

y = 12

This means that when x = 6, y = 12.

Direct proportionality is a special case of proportionality where the proportionality constant is always 1. In other words, if two variables are in direct proportion then they vary in the same way and at the same rate.

Evaluation

1. What is the mathematical definition of direct proportion?

2. Give examples of situations in which direct proportion applies.

3. How do you calculate direct proportions?

4. What is the inverse of direct proportion?

5. What are the real-world applications of direct proportion?

INVERSE PROPORTION

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

In mathematics, inverse proportion is a relationship between two variables in which one variable increases as the other decreases. In other words, if two variables are inversely proportional, then as one variable increases, the other decreases.

There are many real-world examples of inverse proportion. For example, the amount of time it takes to complete a task decreases as the number of people working on the task increases. Another example is the relationship between the length of a rope and the amount of weight it can support. The longer the rope, the more weight it can support.

Inverse proportion can be represented by an equation in which one variable is inversely proportional to another variable. For example, the equation y = k/x represents an inverse proportion relationship between the variables y and x. In this equation, k is a constant.

Inverse proportion can also be represented by a graph. For example, the graph of the equation y = k/x would look like a line that decreases as it goes from left to right.

PRACTICAL APPLICATION OF PROPORTION

Example:

1. If 20 laborers can clear a school field for 28 days. How many days would it take 10 laborers to clear the field?

Solution:

20 labourers used 28days

1 labourer will use 20 days

10 labours can clear same field in days.

1. 10 men can build a wall in 12 days. How long will it take 4 men?

Solution:

10 men built the wall in 12 days.

1 man will build it in 120 days

4 men will build it in days.

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.

Examples:

Translate the following word problems into numerical expressions.

1. There are 26 students in Ms. Nguyen’s class. Thirteen of them are girls. What fraction of the class is made up of girls?

There are _____13_____ students in Ms. Nguyen’s class. _____26_____ of them are girls. What fraction of the class is made up of girls?

2. The sharks in the aquarium are fed 2 times a day. If there are 12 sharks in the aquarium, how many times will they be fed in a week?

The sharks in the aquarium are fed _____2_____ times a day. If there are _____12_____ sharks in the aquarium, how many times will they be fed in a week?

3. In a race, Paul ran 3/5 of a mile. How many miles did he run?

In a race, Paul ran _____3/5_____ of a mile. How many miles did he run?

4. A recipe for punch calls for 1/2 gallon of juice. How many quarts of juice does this recipe require?

A recipe for punch calls for _____1/2_____ gallon of juice. How many quarts of juice does this recipe require?

5. There are 4 feet in a yard. How many yards are in 16 feet?

There are _____4_____ feet in a yard. How many yards are in _____16_____ feet?

1. From the sum of 78 and 129 subtract 264

Solution:

(78

1. What must be multiplied by 0.75 to obtain 6?

Solution:

Let the number be

1. Add 18 to the negative difference between 56 and 45.

Solution:

(

SOLVING WORD PROBLEMS.

Example:

1. Find one quuarter of the positive difference between 15 and 55.

Solution:

1. Subtract 15 from the product of 5 and 15, then divide the result by 3.

Solution:

PRACTICE EXERCISE

1. Divide 42 by the sum of 2 and 4.
2. Find one –eighth of the sum 18 and the product of 6 and 12.
3. A bottle of water can fill five cups of capacity 200ml, or four cups of capacity 250ml.
4. a) Does the number of cups vary directly or inversely with their capacity?
5. b) How many cups of capacity 100ml could the bottle fill?
6. A bus travelling at steady speed, takes 2 ½ hours for a certain journey. How long will a car take if it travels at three times the speed of the bus?
7. Identify the rational and non-rational numbers among the following
8. a) b) (√17)2 c)

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

PRESENTATION:

Step 1:

The subject teacher revises the previous topic

Step 2:

He or she introduces the new topic

Step 3:

The class teacher allows the pupils to give their own examples and he corrects them when the needs arise

CONCLUSION:

The subject goes round to mark the pupil’s notes. He does the necessary corrections

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