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NUMBERS

NUMBERS

What is numbers?
Numbers are symbols or words which represent quality of something for example in form 1B there are forty four students.

i.e. 44 students

The numbers are represented by symbols called numerals.

Each symbol in a numeral is called a digit.

E.g. in 256 there are three digits which are 2, 5, and 6.

There are ten digits which are 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 6x1, 1is called tens and can be written 1x10 and 3 is called hundreds and can be written 3x100.

Therefore 316 can be written as 3x100+1x10+6x1 this form is called expanded form.

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

i. 9x100+0x10+1x1

ii. 5x10000+5x1000+5x100+5x10+5x1

iii. 6x100000+8x10000+0x1000+1x100+7x10+0x1

iv. 5x100+0x10+1x1



NATURAL AND WHOLE NUMBERS


Natural numbers

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

They are counting number, they denoted by N

Number line of natural numbers.


Whole numbers

Are the ones that begin by 0, 1, 2, 3, to infinity? They are denoted by W, in a number line.


Even, odd and prime numbers;-

Even;- are natural numbers which are divisible by 2 without remainder.

Example; - 2, 4, 6, 8, 10, 12.

Odd;- are natural numbers which are not divisible by 2, when they are divided by 2 they give remainder, so the answer is not a natural numbers.

Example; - 1, 3, 5, 7, 9, 11……………………….

Prime numbers;-

Prime numbers are natural numbers which are divisible by one and by themselves. They are the ones with only two factors.

Example;- 2, 3, 5, 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, 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.

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. 57392 – Five hundred and seventy three thousand nine hundred and twenty one

f. 952675 – Nine hundred and fifty two thousand six hundred and seventy five

g. 105,451,225 – One hundred and five million four hundred and fifty one thousand two hundred and 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 =9876

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 = 98764

b. The smallest possible number = 46789

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 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 and fifty thousand

c. 106,000 – One hundred and six thousand

d. 100,006 – One hundred thousand and six

e. 205,020 – Two hundred and five thousand and twenty

f. 2,415,982,728 – Two billion, four hundred and fifteen million, nine hundred and eighty two thousand, seven hundred and 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 =137214

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;-

These are four operations which are addition, subtraction, multiplication and division.

Addition (plus)(+)


Horizontal addition

E.G. 254+369=623

14796+230+14=15030

Vertical addition


The answer obtained after adding is called sum.

Subtraction (-) (minus);-

Horizontal subtraction

E.g. 2349610 – 1396789 = 952821

129 – 98 =31

Vertical subtraction;-

The answer obtained after subtracting numbers is called Difference.

Multiplication/Times (x)

a x b = c

a = Multiplicant

b = Multiplier

c = Product



There are two types of Multiplication

Horizontal multiplication

E.g. 2486 x 7 = 17402

126 x 8 = 1008



Vertical multiplication

The number that multiples is called multiplier

The number to be multiplied is called multiplicand

The answer obtained after multiplication is called product.


Division (÷)

a ÷ b = c

a = dividend

b = divisor

c = quotient

Short division



Long or vertical division



The number that divides is called divisor

The number to be divided is called dividend

The answer obtained after division is called Quotient

The left over upon division is called Remainder





Word problems on whole numbers

1. E.g. 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;-

1row = 12cabbage

4rows = x


X = 12 cabbages x 4 rows

X = 48 cabbages


1row = 8 tomatoes

6rows = x


X = 8 tomatoes x 6rows

X = 48 tomatoes


There are 48 cabbages plants and

48 tomatoes plants



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

Solution;-

2400 shs = 1 month

1 year = 12 months

2 years = 24 months

2400 shs x 24 months

  576000/=


More examples

1. 324 + 1984

Solution;-

    324

+1984

  2308



2. 28782-2784+294

Solution;-

28782

-2784

25998

+ 294

26292

3. 2479249872

    -944129872

   1535120000


Will the sums of the following be even or odd;-

4. Two numbers which are Odd.

The answer must be even.

5. Two numbers which are even.

Even numbers

6. An odd and even number

Odd numbers

7. Any two odd numbers

Even numbers

8. Use horizontal way to find products in questions;-

1. 128 x 5 =640

2. 195 x 9 =1755

3. 17289 x 2 = 34578

9. Use long multiplication method to evaluate;-

249752 x 8921

Solution;-





10. Find the product and quotient of the following

a) 247 X 100

Solution;-

                247

              x100

                000

              000

        + 247

          24700

b) 4000 ÷ 100

Solution;-



Revision exercise

1. In a school garden there are 4 rows of carrots with 10 in each row.

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 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 magazine 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 these 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. 75000 from the customers. If the collection was made for eight days and sh. 275000 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 80shillings. How much money per day does the school earn in 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 respective, how much of each material of cloth will the school need so as to give uniforms to all students?

SOLUTIONS

Qn 1.

4 rows @ 10 plants

4 x10 = Total plants

40

There are 40 carrot plants.

Qn 2.

6 students = 96000 shillings

96000 ÷ 6 =16000

Each student got 16000shs

Qn 3.

1kg = 700shs

Juma buys 8 kgs

700 x 8= 5600

It costed him 5600/=

Qn 4.

240 x 300 =72000

198 x 200 =39600

6 x 500 =3000

72000 + 39600 + 3000 =114600

He got 114,600/=

Qn 5.

17 streams x 35 pupils

17 x 35 = 595

Therefore the total number of pupils in the school is 595.

Qn 6.

36 lines x 14 words

36 x 14 = 504 x 250 pages

126,000 words

Qn 7.

1 day = 26 km

196 days = ? km

196 x 26 =?

5096 kms

He travel 5096 km

Qn 8.

8 days = 75000 x 8

600,000/=

600,000

-275,000

325,000

325,000/= was left.

Qn 9.

Each egg costs = 80 shillings

Each day 140 eggs

We multiply 140 x 80

=112000

The school earns 112000 each day

Qn 10.

900 x 2 m = 1800m

900 x 1.5 = 1350m

The school will need 1800m and 1350m respectively.

ORDER OF OPERATIONS

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

B rackets

O rders

M ultiplication

A ddition

S ubtraction

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 x 2 = 12 + (4 x 2) = 12 + 8 = 20

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

c. 4 x 3 + 7 x 2 = (4 x 3) + (7 x 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 a second number.

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

Example; 16 ÷ 8 = 2 so 2 is the factor of 16 and 16 is the 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 the given number.

- Factors are finite, and divisors of the given number.

- Multiple are number greater than the given number.

- They are infinite and dividend of a particular number.

E.g. factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 and 36

Multiple of 3 = 3, 6, 9, 12, 15, 18, 21, 243, 729………………



PRIME FACTORIZATION

Is the process of writing numbers using their prime factors we use the short division in writing/factorization.

E.g. write the following as the product of their prime factors.



THE LOWEST COMMON MULTIPLES (LCM)

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

E.g. 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 …………

3 = 3, 6, 9, 12, 15, 18, 21, 24, 27 ……………………..

Common multiples are = (6, 12, 18 and 24)

LCM = 6

E.g. find the lowest common multiples of 9, 18, 24

Soln

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………………

27 = 27, 54, 81, 198, 135, 162, 189…………………………………

Common multiples are = {54, 108….}

The least common multiples = 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 numbers.

That is if a and b are whole numbers. Then a + b = c. c is also a whole number. a x b = d d is also a whole number.

E.g. 4 + 6 = 10

3 + 5 = 15

2. Commutative property

If a and b are whole numbers.

Then (i) a + b = b + a

(ii) a x b = b x a

E.g. 3 + 4 = 7

4 + 3 = 7

Implies: 3 + 4 = 4 + 3

a + b = b + a



9 x 6 = 54

6 x 9 = 54

a x b = b x a

3. Association property

If a + b & c are whole number.

Then (i) (a + b) + c = a + (b + c)

(ii) (a x b) c = a (b x c)

e.g. (2 + 3) + 4 = 5 + 4 = 9

2 + (3 + 4) = 2 + 7 =9

Implies: (2 + 3) + 4 = 2 + (3 + 4)

(a + b) + c = a + (b + c)



e.g. (4 x 2) x 5 = 8 x 5 = 40

4 x (2 x 5) = 4 x 10 = 40

Implies: (4 x 2) x 5 = 4 x (2 x 5)

(a x b) c = a (b x c)

4. Distributive property

If a, b and c are whole numbers, then

a x (b + c) = (a x b) + (a x c)

E.g. 4 x (6 + 9) = 4 x 15 = 60

4 x (6 + 9) = (4 x 6) + (4 x 9)

= 24 + 36

= 60

5. Identity property

If a is a whole number,

Then (i) a + 0 = 0 + a = a

0 is called an identity of addition

(ii) a x 1 = 1 x a = a

1 is called and identity of multiplication.



Exercise 1:5

1. Match the property in group A and group B then name the property

Group A

(a) 2 + (3 + 5)

(b) 1 + (2 + 3)

(c) 2 + (2 + 1)

(d) 4 + (3 + 5)

(e) 1 + (1 + 3)

(f) 8 + (4 + 4)

(g) (1 + 3) + 1



Group B

(h) (2 + 3) + 5

(i) (2 + 2) + 1

(j) (8 + 4) + 4

(k) (1 + 2) + 3

(l) (3 + 5) + 4

(m) (1 + 1) + 3

(n) 1 + (1 + 3)



Solution

a and h have associative property

b and k have commutative property

c and l have associative property

e and m have commutative property

f and j have associative property

g and n have commutative property



2. By using properties of whole numbers given reasons why each of the following two expressions are equal

(a) 5 + (6 + 7) = (7 + 6) +5

- The answer to the two expressions will be the same because the numbers have commuted.

(b) (8 + 3) + 4 = 8 + (4 + 3)

- When you solve this you will get the same answer because of the exchange of the numbers.

(c) 5 + (2 + 1) = 1 + (3 + 2)

- In this expression, we have seen that they are two different expressions so you cannot get the same answer from the two.

(d) (5 + 3) + 2 = 3 + (2 + 5)

- In this two expression you can get the same answer because only the position of numbers and brackets as the one that has changed.



3. Telling / writing equal expressions;

(a) 2 x (5 x 3) = (2 x 5) x 3 – associative property

(b) 4 x (3 x 5) = (4 x 3) x 5 – associative property

(c) 2 x (4 x 7) = 7 x (2 x 4) – associative property

(d) 3 x (2 x 2) = 3 x (2 x 2) x 3 – commutative property

(e) (3 x 2) x 6 = 16 x (2 x 2) – commutative property

(f) (3 x 2) x 3 = (2 x 3) x 3 – commutative property

4. Why is it permitted to write 3 x 2 x 5 x 9 without any brackets?

Because even if they are written without brackets they don’t destroy the meaning of the question / statement.



5. Express each of the following in the form of (a x b) + (a x c)

(a) 2 x (3 + 5)

= (2 x 3) + (2 x 5)



(b) 4 x (3 + 10)

= (4 x 3) + (4 x 10)



(c) 2 x (3 + 7)

= (2 x 3) + (2 x 7)



(d) 4 x (3 + 1)

= (4 x 3) + (4 x 1)



6. Express each of the following in the form a x (b + c)

(a) (2 x 5) + (2 x 3)

= 2 x (5 + 3)



(b) (5 x 3) + (5 x 7)

= 5 x (3 + 7)

(c) (2 x 1) + (2 x 3)

= 2 x (1 + 3)



(d) (7 x 3) + (7 x 1)

= 7 x (3 + 1)



7. Calculate the following;

(a) 14 x 5 + 16 – 4 =82

(b) 36 x 72 ÷ 4 = 648

(c) (144 + 20) x 48 + 4 ÷ 2 = 7872

(d) 24 x (10 + 54) + 8 = 192



1:5 INTEGERS

The integers are whole numbers from negative infinity to positive infinity.

They are denoted by Z.

The order of the size of the number 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 have to consider which one is to the right or left?

Consider the following




E.g. write down all the integers between – 5 and 4



Solution








Operation with integers

Integers can be operated using number lines except for division

Addition of integers

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

Example

-2 + -5 = -7

Start at -2 move 5 steps to the left or star at 0 move 2 steps to the left then -5 steps to the left



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

Example





Subtraction of integers

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

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







Multiplication

The rules for multiplying 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
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