**TOPIC 1: NUMBERS ~ MATHEMATICS FORM 1**

**NUMBERS**

All these types of numbers are categorized in different groups like counting numbers, integers, real numbers, whole numbers and rational and irrational numbers according to their properties. all this have been covered in this chapter

**Base Ten Numeration**

Numbers are represented by symbols called numerals. For example, numeral for the number ten is 10. Numeral for the number hundred is 110 and so on.

The symbols which represent numbers are called digits. For example the number 521 has three (3) digits which are 5, 2 and 1. There are only tendigits which are used to represent any number. These digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

**The Place Value in each Digit in Base Ten Numeration**

Identify the place value in each digit in base ten numeration

When we write a number, for example 521, each digit has a different value called place value. The 1 on the right means 1 ones which can be written as 1 × 1, the next number which is 2 means 2 tens which can be written as 2 × 10 and the last number which is 5 means 5 hundreds which can be written as 5 × 100. Therefore the number 521 was found by adding the numbers 5 × 100 + 2 × 10 + 1× 1 = 521.

Note that when writing numbers in words, if there is zero between numbers we use word ‘and’

Example 1

Write the following numbers in words:

**Operations with Whole Numbers**

We have four operations which are: addition (+), subtraction (-), multiplication ( X) and division (÷).

**Addition of Whole Numbers**

Add whole numbers

When adding numbers we add the corresponding digits in their corresponding place values and we start adding from the right side i.e. from the place value of ones to the next.

We can add numbers horizontally or vertically.

**Horizontal addition**

- 972 + 18=
- 23 750 + 250 =

**Solution**

- 972 + 18 = 990
- 23 750 + 250 = 24 000

**Vertical addition**

Example 6

**Subtraction of Whole Numbers**

Subtract whole numbers

Subtraction is denoted by the sign (-). It is sometimes called **minus**. Subtraction is the opposite of addition. Subtraction also means reduce a number from certain number and the answer that is obtained is called **difference.**.

Subtraction is done in similar way like addition. We subtract the corresponding digits in their corresponding place value. For example; 505 – 13.

We first subtract ones, which are 5 and 3. Subtract3 from 5 gives 2. Followed by tens which are 0 and 1.

Subtract 1 from 0 is not possible. In order to make it easy, take 1 from 5 (hundreds). When 1 is added to 0 it has to be changed to be tens since it is added to a place of tens.

So, when 1 comes into a place of tens it becomes 10. So add 10 to 0. We get 10. Now, subtract 1 from 10. We get 9. We are left with 4 in a place of hundreds since we took 1. There for our answer will be 492.

**Note tha**t similar manner will be used when subtracting.

**Multiplication of Whole Numbers**

Multiply whole numbers

Multiplication means adding repeatedly depending on the times number given. For example; 25 6 means add 25, repeat adding 6 times i.e. 25 + 25 + 25 + 25 +25 + 25 = 150. The answer obtained after multiplying two or more numbers is called **product**.

The number being multiplied is called a **multiplicand **while the number used in multiplying is called a **multiplier**. Referring our example, 25 is multiplicand and 6 is multiplier.

**Division of Whole Numbers**

Divide whole numbers

Division is the same as subtraction. You subtract**divisor**(the number used to divide another number) from **dividend **(the number which is to be divided), we repeat subtracting divisor to the answer obtained until we get zero. The answer is how many times you repeat subtraction.

For example; 27 ÷9, we take 27 we subtract 9, we get 18. Again we take 18 we subtract 9, we get 9. We take 9 we subtract 9 we get 0. We repeat subtraction three times. Therefore the answer is 3.

The answer obtained is called **quotient**.

Referring to our example; 27 is dividend, 9 is divisor and 3 is quotient. If a number can’t be divided exactly, what remains or left over is called**remainder**.

The Four Operations in Solving Word Problems

Use the four operations in solving word problems

Sometimes you may be given a question with mixed operations +, -, xand ÷ . We do multiplication and division first then addition and subtraction.

- 12 ÷ 4 + 3 x 5
- 14 x2 ÷ 7 – 3 + 6

**Solution**

- 12 x 4 + 3 x 5 =3 + 15 (do division and multiplication fist) =18
- 14 x 2 ÷ 7 – 3 + 6 =28 ÷ 7 – 3 + 6 (multiply first) =4 – 3 + 6 (then divide) =10 – 3 (add then subtract) =7

We may use brackets to separate x,÷ , + and – if they are mixed in the same problem and use what is called BODMAS . BODMAS is the short form of the following:

B for Brackets O for Open D for Division M for Multiplication A for Addition and S for Subtraction

Therefore, with mixed operations, we first do the operation inside the brackets; we say that we open the brackets. Then we do division followed by multiplication, addition and lastly subtraction.

**Word problems on whole numbers**

Example 12

In a school library there are 6 shelves each with 30 books. How many books are there?

Solution

Each shelf has 30 books

6 shelves have 30 × 6 = 180 books.

Therefore, there are 180 books.

Example 13

Juma’s mother has a garden with Tomatoes, Cabbages and Water Lemons. There are 4 rows of Tomato each with 30 in it. 6 rows of Cabbages with 25 in each and 3 rows of Water Lemo

Solution

Therefore in Juma’s mother garden there are 315 plants.

Example 14

A school shop collects sh 90 000 from customers each day. If sh 380 000 from the collection of 6 days was used to buy books. How much money was left?

**Exercise 1**

1. For each of the following numbers write the place value of a digit in a bracket.

- 899 482 (4)
- 1 940 (0)
- 9 123 476

2. Write the numerals for each of the following problems.

i. Ten thousand and fifty one.

ii. Nine hundred thirty millio

**Factors And Multiples Of Numbers**

Factors of a Number

Find factors of a number

Consider two numbers 5 and 6, when we multiply these numbers i.e. 5 6 the answer is 30. The numbers 5 and 6 are called factors or divisors of 30 and number 30 is called a multiple of 5 and 6. Therefore factors are the divisors of a number.

Example 15

**Find all factors of 12**

Note that, when listing the factors we don’t repeat any of it.

The numbers from 0 to the right are called positive numbers and the numbers from 0 to the left with minus (-) sign are called negative numbers. Therefore all numbers with positive (+) or negative (-) sign are called integers and they are denoted by Ζ. Numbers with positive sign are written without showing the positive sign.

For example +1, +2, +3, … they are written simply as 1, 2, 3, … . But negative numbers must carry negative sign (-). Therefore integers are all positive and negative numbers including zero (0). Zero is neither positive nor negative number. It is neutral.

The numbers from zero to the right increases their values as the increase. While the numbers from zero to the left decrease their values as they increase. Consider a number line below.

If you take the numbers 2 and 3, 3 is to the right of 2, so 3 is greater than 2. We use the symbol ‘>’ to show that the number is greater than i. e. 3 >2(three is greater than two). And since 2 is to the left of 3, we say that 2 is smaller than 3 i.e. 2<3. The symbol ‘<’ is use to show that the number is less than.

Consider numbers to the left of 0. For example if you take -5 and -3. -5 is to the left of -3, therefore -5 is smaller than -3. -3 is to the right of -5, therefore -3 is greater than -5.

**Generally**, the number which is to the right of the other number is greater than the number which is to the left of it.

If two numbers are not equal to each to each other, we use the symbol ‘≠’ to show that the two numbers are not equal. The not equal to ‘≠’ is the opposite of is equal to ‘=’.

Example 21

- 0 is greater than Ζ and Ζ is greater than -4
- -2 is less than Ζ and Ζ is less than or equal to 1.

**a**. 0 is greater than Ζ means the integers to the left of zero and Ζ is greater than -4 means integers to the left of -4. These numbers are -1, -2 and -3. Consider number line below

**b.**-2 is less than Ζ means integers to the right of -2 and Ζ is less than or equal to 1 means integers to the left of 1 including 1. These integers are -1, 0 and 1. Consider the number line below

Put the signs ‘is greater than’ (>), ‘is less than’ (<), ‘is equal to’ (=) to make a true statement.

#### Addition of Integers

When drawing integers on a number line, the arrows for the positive numbers goes to the right while the arrows for the negative numbers goes to the left. Consider an illustration bellow.

The distance from 0 to 3 is the same as the distance from 0 to -3, only the directions of their arrows differ. The arrow for positive 3 goes to the right while the arrow for the negative 3 goes to the left.

**Subtraction of Integers**

Since subtraction is the opposite of addition, if for example you are given 5-4 is the same as 5 + (-4). So if we have to subtract 4 from 5 we can use a number line in the same way as we did in addition. Therefore 5-4 on a number line will be:

Take five steps from 0 to the right and then four steps to the left from 5. The result is 1.

**Multiplication of Integers**

Example 25

Multiplication of a negative integer by a negative integer cannot be shown on a number line but the product of these two negative integers is a positive integer.

From the above examples we note that multiplication of two positive integers is a positive integer. And multiplication of a positive integer by a negative integer is a negative integer. In summary:

- (+)×,(+) = (+)
- (-)×,(-) = (+)
- (+)×,(-) = (-)
- (-)×,(+) = (-)

**Division of Integers**

Example 26

6÷3 is the same as saying that, which number when you multiply it by 3 you will get 6, that number is 2, so, 6÷3 = 2.

Therefore division is the opposite of multiplication. From our example 2×3 = 6 and 6÷3 = 2. Thus multiplication and division are opposite to each other.

Dividing two integers which are both positive the quotient (answer) is a positive integer. If they are both negative also the quotient is positive. If one of the integer is positive and the other is negative then the quotient is negative. In summary:

- (+)÷(+) = (+)
- (-)÷(-) = (+)
- (+)÷(-) = (-)
- (-)÷(+) = (-)

**Mixed Operations on Integers**

Perform mixed operations on integers

You may be given more than one operation on the same problem. Do multiplication and division first and then the rest of the signs. If there are brackets, we first open the brackets and then we do division followed by multiplication, addition and lastly subtraction. In short we call it BODMAS. The same as the one we did on operations on whole numbers.

Example 27

Example 28

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