MATHEMATICS O LEVEL(FORM ONE) NOTES – NUMBERS

NUMBERS

What are numbers?

Numbers are symbols or words that represent the quantity of something. For example, in form 1B there are forty-four students.

i.e. 44 students

Numbers are represented by symbols called numerals.

Each symbol in a numeral is called a digit.

For example, in 256 there are three digits: 2, 5, and 6.

There are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Each digit in a numeral has a value called place value.

Example: in 316, 6 is called ones and can be written as 6¹, 1 is called tens and is written as 1 × 10, and 3 is called hundreds written as 3 × 100.

Therefore, 316 can be written as 3 × 100 + 1 × 10 + 6 × 1. This form is called expanded form.

The system where numbers are in groups of ten is called base ten numeration or decimal system numeration. The place value of every digit in decimal system numeration is ten times the place value of the next digit to the right.

Table of place values:

Exercise 1.1

1. Write each of the following numbers in expanded form:
   • (a) 94 =
   • (b) 7019 =
   • (c) 50 =
   • (d) 303 =
   • (e) 5003 =
   • (f) 500 =
   • (g) 999 =
   • (h) 5 =
   • (i) 5000 =
2. Find the place value of the following:
   • (a) 513 (1) – The place value of 1 is
   • (b) 357999 (5) – The place value of 5 is
   • (c) 50149 (5) – The place value of 5 is
   • (d) 8665 (8) – The place value of 8 is
   • (e) 227 (7) – The place value of 7 is
   • (f) 900412 (4) – The place value of 4 is
3. Write the numerals for each of the following:
   • (a) 9 × 100 + 0 × 10 + 1 =
   • (b) 5 × 10000 + 5 × 1000 + 5 × 100 + 5 × 10 + 5 =
   • (c) 6 × 100000 + 8 × 10000 + 0 × 1000 + 1 × 100 + 7 × 10 + 0 =
   • (d) 5 × 100 + 0 × 10 + 1 =

1:2 NATURAL AND WHOLE NUMBERS

Natural numbers

Natural numbers are the ones that begin with 1, 2, 3 to infinity.

They are counting numbers and are denoted by N.

Number line of natural numbers:

Whole numbers

Whole numbers are the ones that begin with 0, 1, 2, 3 to infinity. They are denoted by W on a number line.

Even, odd and prime numbers

Even: Natural numbers which are divisible by 2 without remainder.

Example: 2, 4, 6, 8, 10, 12.

Odd: Natural numbers which are not divisible by 2. When divided by 2, they give a remainder.

Example: 1, 3, 5, 7, 9, 11, …

Prime numbers: Natural numbers which are divisible only by one and themselves. They have exactly two factors.

Example: 2, 3, 5, 7, 11, 13, …

Numbers up to one billion:

EXERCISE 1:2

1. For the given numbers below, write down which are:
   • (a) Even
   • (b) Prime
   • (c) Odd

   9, 12, 15, 17, 25, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71.

2. Write down the prime numbers between 70 and 90.
3. Write down a number which is even and prime.
4. Among the numbers 3, 5, 7, 9, 11, 13, what is not a prime number?
5. Show even, odd, and prime numbers less than 10 on separate number lines.
6. (a) Given any point as a number on a number line for N or W, can you always name another one to the right of it?
7. Given any two points as numbers one after another on a number line for N or W, can you find the whole numbers between them?
8. Do the points representing N and W on a number line completely fill the number line?
9. How does N differ from W?

More examples

1. Write the following numerals in words:
   • (a) 72 – Seventy two
   • (b) 10368 – Ten thousand three hundred and sixty-eight
   • (c) 1152 – One thousand one hundred and fifty-two
   • (d) 144 – One hundred and forty-four
   • (e) 573921 – Five hundred seventy-three thousand nine hundred twenty-one
   • (f) 952675 – Nine hundred fifty-two thousand six hundred seventy-five
   • (g) 105,451,225 – One hundred five million four hundred fifty-one thousand two hundred twenty-five
2. Express the statement in numerals:

   Nine billion eight hundred million and two hundred = 9,800,000,200.

3. Write down the largest four-digit number = 9,999.
4. Write down the largest four-digit number when the digits are not repeating = 9,876.
5. Write down the smallest three-digit number without using a zero = 111.
6. Change the order of the digits in 47986 to make:
   • (a) The largest possible number = 98,764
   • (b) The smallest possible number = 46,789
7. Write down the number with 6 in the hundred place, 9 in the tens place, 0 in the thousands place, 4 in the units place, and 3 in the ten thousands place: 30,694.
8. Write down the next three counting numbers after 6999: 7000, 7001, 7002.
9. Write the numbers in words:
   • (a) 6054 – Six thousand and fifty-four
   • (b) 3,250,000 – Three million two hundred fifty thousand
   • (c) 106,000 – One hundred six thousand
   • (d) 100,600 – One hundred thousand six hundred
   • (e) 100,006 – One hundred thousand six
   • (f) 205,020 – Two hundred five thousand and twenty
   • (g) 2,415,982,728 – Two billion four hundred fifteen million nine hundred eighty-two thousand seven hundred twenty-eight
10. Write in numerals:
    • (a) Six hundred seventy-five – 675
    • (b) Four hundred and five – 405
    • (c) Three thousand and sixteen – 316
    • (d) Eight thousand and sixteen – 8016
    • (e) One hundred thirty-seven thousand two hundred and fourteen – 137,214
    • (f) One million five hundred thousand – 1,500,000
    • (g) Two million twenty-three – 2,000,023
    • (h) Nineteen thousand and forty-five – 19,045
    • (i) One billion fourteen million two hundred and fifteen thousand – 1,014,215,000

Operations with whole numbers

There are four operations: addition, subtraction, multiplication, and division.

Addition (plus) (+):

Horizontal addition

Example: 254 + 369 = 623

14,796 + 230 + 14 = 15,030

Vertical addition:

The answer obtained after adding is called the sum.

Subtraction (-) (minus):

Horizontal subtraction

Example: 2,349,610 – 1,396,789 = 952,821

129 – 98 = 31

The answer obtained after subtracting numbers is called the difference.

Vertical subtraction:

Multiplication / Times (×)

a × b = c

a = Multiplicand

b = Multiplier

c = Product

There are two types of multiplication:

Horizontal multiplication

Example: 2486 × 7 = 17,402

Example: 126 × 8 = 1,008

Vertical multiplication:

The number that multiplies is called the multiplier.

The number to be multiplied is called the multiplicand.

The answer obtained after multiplication is called the product.

Division (÷)

a ÷ b = c

a = dividend

b = divisor

c = quotient

Short division:

Long or vertical division:

The number that divides is called the divisor.

The number to be divided is called the dividend.

The answer obtained after division is called the quotient.

The leftover upon division is called the remainder.

Example: 29 ÷ 6 =

Word problems on whole numbers

1. In a school garden there are four rows of cabbage with 12 in each row, six rows of tomatoes with eight in each row. How many plants of each kind are there?

Solution:

1 row = 12 cabbages

4 rows = x

x = 12 cabbages × 4 rows

x = 48 cabbages

1 row = 8 tomatoes

6 rows = x

x = 8 tomatoes × 6 rows

x = 48 tomatoes

There are 48 cabbage plants and 48 tomato plants.

2. Two thousand four hundred shillings are deposited by a teacher each month. How much is this after two years?

Solution:

2400 shs = 1 month

1 year = 12 months

2 years = 24 months

2400 shs × 24 months = 57,600 shs

3. 28782 – 2784 + 294.

Solution:



4. Will the sums of the following be even or odd:
   • Two numbers which are odd – The answer must be even.
   • Two numbers which are even – Even numbers.
   • An odd and even number – Odd numbers.
   • Any two odd numbers – Even numbers.
5. Use horizontal way to find products in questions:
   1. 128 × 5 = 640
   2. 195 × 9 = 1755
   3. 17289 × 2 = 34578
6. Use long multiplication method to evaluate:

   249752 × 8921

   Solution:

   

7. Find the product and quotient of the following:

   (a) 247 × 100

   Solution:

   

Revision exercise

1. In a school garden there are 4 rows of carrots with 10 in each row. How many plants are there?
2. Six students were given 96,000 shillings to share equally. How many shillings did each student get?
3. The cost of one kilogram of sugar is 700 shillings and Juma buys 8 kgs of sugar. How much did it cost him?
4. Richard sold 240 copies of the daily newspaper for 300 shillings each, 198 pieces of the Uhuru newspaper for 200 shillings, and 6 sports magazines for 500 shillings each. How much did he collect in all?
5. There are 17 streams in a school, each stream has 35 pupils. How many pupils are there in the school?
6. A page of a book has 36 lines. If each line contains 14 words, how many words are there if the book has 250 pages?
7. Jim’s school is 13 km from his home. If he goes to school daily, how many kilometers does he travel in 196 days?
8. Each day a school shop collects sh. 75,000 from customers. If the collection was made for eight days and sh. 275,000 of the collected money was used to make a fence for the school, how much money was left?
9. A school collects 140 eggs from its poultry farm each day. Each egg costs 80 shillings. How much money per day does the school earn from selling the eggs?
10. A school of 900 students has decided to make uniforms for each student. If a shirt and a pair of trousers need 2 m and 1.5 m respectively, how much of each material of cloth will the school need to give uniforms to all students?

Solutions

Qn 1.

Qn 2.

Qn 3.

Qn 4.

Qn 5.

Qn 6.

Qn 7.

Qn 8.

Qn 9.

Qn 10.

ORDER OF OPERATIONS

The order of priority is often given by the word BODMAS to mean:

• Brackets
• Orders (powers and roots)
• Division
• Multiplication
• Addition
• Subtraction

So first perform any operation in brackets, then any use of orders, division or multiplication, followed by addition or subtraction.

Example: evaluate the following

(a) 12 + 4 × 2 = 12 + (4 × 2) = 12 + 8 = 20

(b) 20 ÷ 2 + 3 = (20 ÷ 2) + 3 = 10 + 3 = 13

(c) 4 × 3 + 7 × 2 = (4 × 3) + (7 × 2) = 12 + 14 = 26

1:4 FACTORS AND MULTIPLES OF NUMBERS

When a number divides exactly into another number, it is called a factor of the second number.

The second number is called a multiple of the first number.

Example: 16 ÷ 8 = 2, so 2 is a factor of 16 and 16 is a multiple of 2. Other factors of 16 are 8, 4, 1, and 16.

In general, one (1) is a factor of every number and each number is a factor of itself.

• Factors of a number are numbers less than or equal to the given number.
• Factors are finite and divisors of the given number.
• Multiples are numbers greater than or equal to the given number.
• Multiples are infinite and dividends of a particular number.

Example: Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, and 36

Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …

PRIME FACTORIZATION

Prime factorization is the process of writing numbers using their prime factors. We use short division in writing/factorization.

Example: Write the following as the product of their prime factors:

∴ 24 = 2 × 2 × 2 × 3

∴ 48 = 2 × 2 × 2 × 2 × 3

∴ 420 = 2 × 2 × 3 × 5 × 7

∴ 360 = 2 × 2 × 2 × 3 × 3 × 5

THE LOWEST COMMON MULTIPLES (LCM)

LCM is the short form of lowest or least common multiple.

Example: Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, …

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, …

Common multiples are: 6, 12, 18, 24, …

LCM = 6

Example: Find the lowest common multiple of 9, 18, 24

Solution:

9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, …

18 = 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, …

24 = 24, 48, 72, 96, 120, 144, 168, 192, …

Common multiples are: 54, 108, …

The least common multiple = 54

PROPERTIES OF WHOLE NUMBERS

There are five properties of whole numbers.

1. Closure property (law): When two or more whole numbers are operated by adding or multiplying, the answer is also a whole number.

That is, if a and b are whole numbers, then a + b = c, where c is also a whole number; a × b = d, where d is also a whole number.

Example: 4 + 6 = 10

3 × 5 = 15

2. Commutative property: If a and b are whole numbers, then
   • (i) a + b = b + a
   • (ii) a × b = b × a

   Example: 3 + 4 = 7 and 4 + 3 = 7, implies a + b = b + a

   3 × 6 = 18 and 6 × 3 = 18, implies a × b = b × a

3. Associative property: If a, b, and c are whole numbers, then
   • (i) (a + b) + c = a + (b + c)
   • (ii) (a × b) × c = a × (b × c)

   Example: (2 + 3) + 4 = 5 + 4 = 9 and 2 + (3 + 4) = 2 + 7 = 9

   (4 × 2) × 5 = 8 × 5 = 40 and 4 × (2 × 5) = 4 × 10 = 40

4. Distributive property: If a, b, and c are whole numbers, then

   a × (b + c) = (a × b) + (a × c)

   Example: 4 × (6 + 9) = 4 × 15 = 60

   4 × (6 + 9) = (4 × 6) + (4 × 9) = 24 + 36 = 60

5. Identity property: If a is a whole number, then
   • (i) a + 0 = 0 + a = a (0 is called the identity of addition)
   • (ii) a × 1 = 1 × a = a (1 is called the identity of multiplication)

Exercise 1:5

1. Match the property in the left column with the right column and name the property:

   (a) 2 + (3 + 5)	(h) (2 + 3) + 5
   (b) 1 + (2 + 3)	(i) (2 + 2) + 1
   (c) 2 + (2 + 1)	(j) (8 + 4) + 4
   (d) 4 + (3 + 5)	(k) (1 + 2) + 3
   (e) 1 + (1 + 3)	(l) (3 + 5) + 4
   (f) 8 + (4 + 4)	(m) (1 + 1) + 3
   (g) (1 + 3) + 1	(n) 1 + (1 + 3)

Solution:

a and h have Associative property

b and k have Commutative property

c and i have Associative property

e and m have Commutative property

f and j have Associative property

g and n have Commutative property

2. Using properties of whole numbers, give reasons why each of the following two expressions are equal:
   • (a) 5 + (6 + 7) = (7 + 6) + 5 – The answer is the same because the numbers have commuted.
   • (b) (8 + 3) + 4 = 8 + (4 + 3) – The answer is the same because of the exchange of the numbers.
   • (c) 5 + (2 + 1) = 1 + (3 + 2) – These are two different expressions, so the answers are not the same.
   • (d) (5 + 3) + 2 = 3 + (2 + 5) – The answers are the same because only the position of numbers and brackets has changed.
3. Telling/writing equal expressions:
   • (a) 2 × (5 × 3) = (2 × 5) × 3 – Associative property
   • (b) 4 × (3 × 5) = (4 × 3) × 5 – Associative property
   • (c) 2 × (4 × 7) = 7 × (2 × 4) – Associative property
   • (d) 3 × (2 × 2) = 3 × (2 × 2) – Commutative property
   • (e) (3 × 2) × 6 = 16 × (2 × 2) – Commutative property
   • (f) (3 × 2) × 3 = (2 × 3) × 3 – Commutative property
4. Why is it permitted to write 3 × 2 × 5 × 9 without any brackets?

Because even if they are written without brackets, they don’t change the meaning of the expression.

5. Express each of the following in the form of (a × b) + (a × c):
   • (a) 2 × (3 + 5) = (2 × 3) + (2 × 5)
   • (b) 4 × (3 + 10) = (4 × 3) + (4 × 10)
   • (c) 2 × (3 + 7) = (2 × 3) + (2 × 7)
   • (d) 4 × (3 + 1) = (4 × 3) + (4 × 1)
6. Express each of the following in the form a × (b + c):
   • (a) (2 × 5) + 2 × 3 = 2 × (5 + 3)
   • (b) (5 × 3) + 5 × 7 = 5 × (3 + 7)
   • (c) (2 × 3) + 2 × 7 = 2 × (3 + 7)
   • (d) (4 × 3) + 4 × 1 = 4 × (3 + 1)
7. Calculate the following:
   • (a) 14 × 5 + 16 – 4 = 82
   • (b) 36 × 72 ÷ 4 = 648
   • (c) (144 + 20) × 48 + 4 ÷ 2 = 7872
   • (d) 24 × (10 + 54) + 8 = 192

1:5 INTEGERS

Integers are whole numbers from negative infinity to positive infinity. They are denoted by Z.

The order of the size of the numbers 1, 2, 3, 4, … can be represented as points on a number line.

Starting from 0 to the right and other numbers from 0 to the left.

Number line for Z:



If we want to know which number is larger than the other, we consider which one is to the right or left.

Consider the following:



Example: Write down all the integers between -5 and 4

Solution:



Example: present on a number line

-3 5



-3 -2 -1 0 1 2 3 4

Operation with integers

Integers can be operated using number lines except for division.

Addition

RULE 1: To add a negative integer, move to the left on the number line.

Example: -2 + -5 = -7

Start at -2, move 5 steps to the left or start at 0, move 2 steps to the left then 5 steps to the left.



OR



RULE 2: To add a positive integer, move to the right on the number line.

Example: -4 + (+6) = +2

Start at -4, move 6 steps to the right.



OR



-4 + -6 = -10

Start at -4, move 6 steps to the left.



Subtraction of integers

Subtraction of positive integers gives the same result as addition of integers.

RULE 1: To subtract a positive integer, move to the left on the number line.

Example: -4 – (+3) = -7

Start at -4, move 3 steps to the left.



RULE 2: To subtract a negative integer, move to the right on the number line.

Example: -7 – (-3) = -4

Start at -7, move 3 steps to the right.



Multiplication

The rules for multiplying are as follows:



Division

The rules for division are as follows:



Note: The results are similar to the rules of multiplication.

Example:

-36 ÷ 6 = -6

24 ÷ 6 = +4

24 ÷ -6 = -4

-24 ÷ 6 = -4

-24 ÷ -6 = 4