Chapter 1 Number Calculations Notes Form 2 2013

Form 2 [CHAPTER 1: NUMBER CALCULATIONS]

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Chapter 1: Number Calculations It is important to revise Chapter 1 – Number Calculations done in form 1. You can revise from the notes found on the website in the form 1 section. 1.1 – Place value and rounding numbers to the nearest whole, ten, hundred, thousand, etc. Each digit in a number has a place value:

1234.567

1000 200 30 4 0.5 0.06 0.007

thousands hundreds tens units tenths hundredths thousandths 1000 s 100s 10 s 1 s 1/10 s 1/100s 1/1000s

Example 1: Round 2645 to the:

(i) Nearest 10 (ii) Nearest 100 (iii) Nearest 1000

(i)

• Underline the tens: 2645 • Look at the number behind it • If it is 5 or bigger add 1 to the underlined value • If it is strictly smaller than 5 leave it as it is

In this case we add 1 to the 4 and the answer is 2650

(ii)

• Underline the hundreds: 2645 • Look at the number behind it • If it is 5 or bigger add 1 to the underlined value • If it is strictly smaller than 5 leave it as it is

In this case we leave it as it is: 2600



(iii)

• Underline the thousands: 2645 • Look at the number behind it • If it is 5 or bigger add 1 to the underlined value • If it is strictly smaller than 5 leave it as it is

In this case add 1 to 2 and the answer is 3000 Example 2: Complete the following table:

Number To the nearest 10 To the nearest 100 To the nearest 1000

1675

678

9397

230

2675

Example 3: There are 1300 supporters to the nearest 100 at Ta’ Qali National Stadium. List all the possible number of supporters that could have been at the stadium.



Example 4: Round the following numbers to the nearest whole number:

4.6 -‐ _________ 56.3 -‐ _________ 743.4999 -‐ _________

89.5 -‐ _________ 12.3 -‐ __________ 1.1 -‐ ________

1.2 − Rounding Numbers: Decimal places Parts of a unit can be written either as fractions or as decimals by placing a point after the units column and continuing to the right e.g. 1.234 The first column after the decimal point represents tenths and is called the first decimal place. The second column after the point represents hundredths and is called the second decimal place. The third column after the point represents the thousandths and is called the third decimal place. Example 1: Correct 3.451 to:

(i) one decimal place (ii) two decimal places

(i)

• Underline the first decimal place: 3.451 • Look at the number behind it • If it is 5 or bigger add 1 to the underlined value • If it is strictly smaller than 5 leave it as it is

In this case we add 1 to the 5 and the answer is 3.5

(ii) • Underline the second decimal place: 3.451 • Look at the number behind it • If it is 5 or bigger add 1 to the underlined value • If it is strictly smaller than 5 leave it as it is

In this case we leave it as it is and the answer is 3.45



Example 2: Complete the following table:

Number 1 d.p. 2 d.p. 3 d.p.

56.1682

1.6431

3.287

6.7176

3.456723

1.3 – Rounding Numbers: Significant figures

Ø All digits that are not zero are significant.

Example: 16.2 (3 significant figures)

Ø Zeros between non zero digits are significant.

Example: 103 (3 significant figures)

Ø Zeros to the left of the first non zero digit are not significant. They indicate placement

of the decimal point only.

Example: 0.0002 (1 significant figure)

Ø Zeros to the right of a decimal point are significant if they are at the end of a number.

Example: 16.000 (5 significant figures)



How do we correct numbers to a given number of significant figures?

Example 1: Correct 5.744562647 correct to four significant figures.

• Underline the fourth significant figure: 5.744562647

• If the number behind it is 5 or bigger, add one

• If the number behind it is strictly smaller than 5, leave it as it is.

The number behind the fourth significant figure is bigger thus we add one to 4.

This leaves us with 5.745

Example 2: Correct the following numbers to the number of significant figure

shown in the brackets.

(i) 345 (2)

(ii) 0.0023 (1)

(iii) 506 (2)

(iv) 5.673 (3)

(v) 9743 (1)

(vi) 6.097 (2)

(vii) 64864 (4)



1.4 − Ordering positive and negative decimals

Each digit in a number means something different depending on its position. For example the number 4531 means 4 thousands 5 hundreds 3 tens 1 unit

Thousands Hundreds Tens Units

4 5 3 1

Thousands are bigger than Hundreds which are bigger than Tens which are bigger than Units.

So 4531 is larger than 999 because 4531 has a digit in the thousands column.

Example Which is larger 89 or 809? 809 since it has a digit in the hundreds column

Extending to Decimals

Let us build the same table for 1234.567

Th H T U . t h th

1 2 3 4 . 5 6 7

Units are larger than tenths which are larger than hundredths which are larger than thousandths.



Example 1: Which is the larger 0.23 or 0.023?

Units . Tenths Hundredths Thousandths

0 . 2 3

0 . 0 2 3

The larger is 0.23 because it has a figure in the tenth column.

Example 2: Which is the larger from 0.203 and 0.23?

Units . Tenths Hundredths Thousandths

0 . 2 0 3

0 . 2 3

0.23 is the larger because they both have 2 tenths but the 3 in the 0.23 is larger than the 0 in

0.203 since there are 3 hundredths in 0.23.

Example 3: Arrange the following numbers in order of size:

0.909, 0.99, 0.099, 9.09

Looking at the units

9 is larger than 0

9.09 is the largest

Looking at the tenths

9 is larger than 0

0.909 and 0.99 are larger than 0.099

Compare the hundredths of these two

0.99 is larger than 0.909



Next after is 0.909

The smaller one is 0.099

Ordered

9.09, 0.99, 0.909, 0.099

Example 4: Arrange 5.26, 0.526, 0.502, 0.00526 in ascending order.

Working with negative numbers and decimals

Numbers further on the right of the number line are the larger. Positives are always larger

than negatives.

⇒ Which number is larger -‐11 or 2?

2 because negative is always smaller than positive.

This is called descending order

since it starts from the largest to the

smallest. (devil)

Ascending order: from the smallest to the largest. (angel)



Example 5: Arrange the following decimals in ascending order: −0.25, 0.25,

−1.265, 0.568

• Start looking at the signs

• There are 2 negatives which must be smaller than the positives

• We look at the units first

• −1 is smaller than -‐0

• The smaller one is −1.265

• Immediately after −0.25

• We look at the positives now

• Let us compare the tenths

• 5 is larger than 2

• The next is 0.25

• The largest one is 0.568

Drawing a number line can also help.

−1.265, −0.25, 0.25, 0.568

Example 6: Arrange the following decimals in descending order: 0.32, 0.56, 17.2, -‐

0.63, -‐0.65

(Hint: Use a number line to help you out)



1.5 – Multiplication of decimals Example 1: Work out: 105.32 × 4.56

Step 1: Remove the points

105.32 to10532 (multiply by 100)

4.56 to 456 (multiply by 100)

In total we have moved four point to the right (multiplied by 10 000)

Step 2: Multiply the numbers obtained

10532 x 456

63192 526600 + 4212800

4802592

Step 3: Move back the points

Previously we had moved four points to the right ( multiplication by 10 000)

Now, move them back (division by 10 000)

Answer = 480.2592 Example 2: Evaluate the following

(i) 13.5 × 56.5

Revise Long multiplication!



(ii) 0.76 × 0.43

(iii) 0.045 × 3.1

(iv) 0.04 × 0.092

(v) 0.3 × 0.5 × 0.7



1.6 – Division of decimals (short division) Example 1: Work out the following

(i) 4.8 ÷ 2

(ii) 6.9 ÷ 3

(iii) 1.2 ÷ 3

(iv) 1.25 ÷ 5

(v) 4.8 ÷ 8

(vi) 0.2 ÷ 8

(vii) 0.1 ÷ 5



Example 2: Evaluate

(i) 0.4 ÷ 0.2

(ii) 15 ÷ 0.3

(iii) 16 ÷ 0.4

(iv) 3.6 ÷ 0.6

1.7 – Mental multiplication and division by powers of 10 Example 1: 7.4 x 10

One needs not use any mental or standard method of multiplication. It is easy to calculate the

answer straight away by ‘moving the point to the right’

= 74

Example 2: 93 x 1000



Example 3: 960 ÷ 10

This time we ‘subtract zeros’ or ‘move the point to the left’

=96

Example 4: 14600 ÷ 100

Example 5: 16.7 x 10

Example 6: 25.137 x 100

Example 7: 74.58 x 1000

Example 8: 6800 ÷ 100

Example 9: 1986 ÷ 1000

Example 10: 345 ÷ 10000



In this section we will also be multiplying and dividing by numbers such as 0.1, 0.01, 0.001, etc. Example 11: 45 × 0.1

0.01 is the same as 110

Thus, 45 × 110

= 4510

=4.5 In general, ×0.1 means ÷ 10 × 0.01 means ÷ 100 × 0.001 means ÷ 1000 Also, ÷ 0.1 means × 10 ÷ 0.01 means × 100 ÷ 0.001 means × 1000 Example 12: Work out the following:

(i) 342 × 0.01

(ii) 5 × 0.1

(iii) 6.7 × 0.001

(iv) 9.9 ÷ 0.01

(v) 5.6 ÷ 0.1

(vi) 6.89 ÷ 0.001



Example 13: Work out as a decimal. i) 7 × 10

ii) 0.7 × 100

iii) !!.!

iv) !.!"!.!!"

v) !"#!"""

vi) 1.9 × !!""



1.8 – Operations with negative numbers Positive and negative numbers are collectively known as directed numbers. They can be used to describe any measurement that goes above or below an obvious zero level e.g. temperature, lift (above and below ground level), etc. Positive and negative numbers can be presented on a number line.

On this number line 5 is to the right of 3 so 5 is greater than 3 and we can write it as : 5 > 3

Any number to the right of another number is greater than the second number e.g. -‐1 > − 4

and 3 > − 5

Remember that last year we followed two rules whilst adding and subtracting negative

numbers.

Rule 1: Adding a negative number is the same as subtracting a positive number.

For example: + −4 = −4 thus, 7 + − 4 = 7 − 4 = 3 Rule 2: Subtracting a negative number is the same as adding a positive number.

For example : − − 7 = 7 thus, 9 − − 7= 9 + 7 =16 Example 1: Work out the following:

(a) – 2 + 3 (b) 4 + − 5

(c) -‐1 + − 5 (d) 6 – 9 (e) − 6 − 3 (f) − 5 + − 3



We also had three rules about division and multiplication of negative numbers:

Rule 3: When you multiply a negative number by a positive number you get a negative answer. For example, -‐4 × 4 = -‐16

Rule 4: When you divide a negative number by a positive number you get a negative answer. For example, -‐45 ÷ 5 = −9 or 28 ÷ − 4 = −7 Rule 5: When you divide or multiply a negative number by another negative number you get a positive answer. For example, −28 ÷ −7 = 4 and −4 × −5 = 20 Example 2: Evaluate the following:

(i) 32 × − 2

(ii) 45 ÷ − 9

(iii) − 7 × − 5

(iv) − 36 ÷ 6

(v) − 8 × 9



(vi) 156 × − 0.1

Example 3: Use an appropriate method to calculate these:

(i) − 1.5 + 4.7

(ii) 49.6 + 56.12 + 12.34

(iii) 3.7 − −1.8 + 16.7

(iv) 45.67 + − 12.45



1.9 – Further +/− using calculator Example 1: Work out these using a calculator:

(i) 45 + − 115

(ii) −5.0 − − 27 + 3.5

(iii) 6.7 − 8.9 + − 9

(iv) − 8.2 − 9.0 − − 13 1.10 – Use of calculator for powers and roots Remember:

A power or index is an operator, like +, −, x and ÷.

32 means 3 × 3, which is equal to 9.

3 is the base 2 is the power, or index

32 is not the same as 3 x 2, which is 6.



Further examples:

42 = 4 x 4 = 16

33 = 3 x 3 x 3 = 27

52 = 5 x 5 = 25

We can use a calculator to work out powers. Not all calculators are the same and some will have different keys to work powers.

In general, the keys of the calculator are the following:

x2 : To work the square of a number (power 2)

x3 : To work the cube of a number (power 3)

x□ or xy : To work any power

⇒ Look at your calculator and try to work out several powers.

⇒

Example 1: Use a calculator to find:

(i) 42 + 53

(ii) 64 − 82

(iii) 112 ÷ 73 (correct to 2 d.p)

(iv) 712 × 63

(v) 0.52 + 0.459



A root is the reverse of a power. For example: 23 = 8 3√8 = 2 The calculator can also me used to find roots. In general one finds the following keys on the

calculator:

√□ : For square roots □√□ : For other roots

Example 2: Work out the following:

(i) √565

(ii) 4√498

(iii) 3√34

(iv) 5√156

1.11 – BIDMAS (Order of Operations) Work out:

i) (82 + 54) – (43 + 62) ii) 22 × 3 iii) (2 + 3)2 + 62 × 2

From where do we start? The word BIDMAS gives you the order with which you work such sums. Brackets Indices Division Multiplication



Addition Subtraction

• This means that if there are brackets first, they are worked out.

• Powers are worked out in the same stage as that of the Indices.

• If no brackets are present in the question the multiplication and division are worked out prior to the addition and subtraction

When we have fractions we work out the numerator and denominator using BIDMAS, then we work out the fraction. Example 1: Work the following in the correct order. Correct your answer to 1 decimal place.

(i) (5.43 + 6.64)

(ii) 3

2

4.5 183.4 10

−

−

(iii) 4 + ( 6 − 4)3



(iv) 34 − 4 × 5

(v) 4 × 3 ÷ (6 − 2)2

(vi) 2

2

(6 8) 6 97 8

+− − ×−

−

Documents

Chapter 1 Number Calculations Notes Form 2 2013