Estimation, Approximation and Standard form

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IB Mathematics

Approximation and Estimation



Estimation:Estimation is an Intelligent guess made about something base on some information. Approximation is guess that is nearly exact. Estimate in science help scientist to guess before doing actual investigation. Estimate gives an idea about some quantities off hand before proper findings or measurement. For example, if you were asked to pay electricity bill given that you consumed 12units and the cost of a unit being 64frs. The cost of 12 units is 768frsWhy would SONEL Company prefer to approximate this value t0 770frs?. What if the value was 764? Will they still approximate it to 770 or 775. “It’s better to be roughly right than precisely wrong”(Allan Greenspan, U.S. Federal Reserve Chairman -retired)

Introduction Activity:

1. Guess intelligently the length and width of the classroom and calculate its area

2. Estimate the amount of water you drink on daily basis. (1.5litres per day)

3. Guess the weight of your exercise book (250g)

4. Approximate the height and weight (mass) of your mate (1.7m, 60kg or 500N)

5. Estimate the weight of a football (396 to 463g)

6. Estimate the angle between the wall the floor


Post Mathematics for Mathematics HL and SL


Review: Operations with Fractions and Decimals

Activity 1: Answers

Remark: To divide a decimal by a power of 10n, move n-places to the left. To multiply a decimal number by a power of 10n, move n-places to the right.

Examples:

a) 450 1000

b) 780 1000

Activity 1: Simplify as much as possible (Hint: Use BODMAS)

34) 5)

6)

7) 9.8+1.07+69

8) 5-4.667

9) 44.36 x 32

10) a) b) b)

11) 0.2 x 1000



NB: Some numbers are too big to be rounded up or too small to be rounded down.

To round up or down a decimal number (number with decimal point), We study the number of decimal places and we approximate accordingly.

Examples:

1) Round 2475 to the nearest 10frs.

(Ans: 2480frs )

2) Put 2.963 to 1decimal place

(Ans: 3.1)

3)Round 344frs to the nearest 5frs.

Rounding Up and Down:

To round up and down a number or decimal, Draw a vertical line (stroke)

in front of the desired digit or unit

If the digit after the line is , round it up to one and add to the digit immediately before the line; the rest of the digits are considered zeros

If the digit after the line is <5, round it down so that the digits before the line remain unchanged and those after the line are assumed zeros


Approximation and EstimationRemark: The value of any digit depends on its position. The value of 2 in the number 1267 is 100 and in 0.324 is 100th or 2dp. Given 2768.043, what is the value of each digit?

The value of 2=thousand, 3=hundred, 6=ten, 8=unit, .0=tenth (1dp), 4=hundredth (2dp), 3=thousandth (3dp)

1000(thousand)

100(hundred)

10(ten)

Unit 10th

(tenth or 1dp)

100th (hundredth or 2dp)

1000th (thousandth or 3dp)

(1000) Or 103 100 or 102 10 or 101

1 or 100

0.1 or 10-

1

0.01 or 10-2 0.001 or 10-3

Example: Copy and complete the table below

Number 1dp 2dp Whole number or unit

Nearest 10frs

Nearest 100th

Nearest Degree

444.525

5059.996

267.537


Approximation and EstimationAnswer: Check for correctness

Number 1dp 2dp Whole number or unit

Nearest 10frs

Nearest 100th

Nearest Degree

444.525 444.5 444.53 445 450 444.53 4450

5059.996 5060.0 5060.00 5060 5060 5060.00 50600

267.537 267.5 267.54 268 270 268.54 2680

Significant Figures

In the number 4380, 4=first significant figure or digit, 3=second significant figure, 8=third significant figure. and 0 is not significant although 0 has a value. However, in 4308, 0 is significant.

Remark: 0 between two whole numbers or digits is significant 0 at the beginning or end of a number is not significant If the digit after the required significant figure >=5, round up and add to

the previous digit



Examples: Write the following numbers to 2 significant

a) 0403670=

b) 0.052407=

c) 34.08945=

d) 040567 (to 3 s.f.)

Number 1dp 2dp 1sf 2sf 3sf Nearest whole number

42.546

0.9974

1.9995

295.6891



Examples: Write the following numbers to 2 significant

a) 0403670=40

b) 0.052407=0.052

c) 34.08945=34

d) 040567=34.1 to 3 sf

Number 1dp 2dp 1sf 2sf 3sf Nearest whole number

42.546 42.5 42.55 4 40 43.547 43

0.9974 1.0 1.00 1.0 1.00 0.997 1

1.9995 2.0 2.00 2 2.0 2.00 2

295.6891 295.7 295.69 3 30 296 296


Introduction

Activity: Write the biggest amount of money you dream to ever

have in a life time Write a number that starts with 1 and then followed by

30 zeros. Do not write it in any other form? Write another number that starts with a decimal point,

continues with 20 zeros and 2 at the end. Comment on this number What is the most convenient way of writing such a

number with many figures? Write it down. What is the name given to this format of writing this

number


Standard Form

Mathematicians or scientist use a convenient and economical manner or way to express very small numbers or very large numbers called the Standard Form. This form was adopted and made standard to be used all over the world.

a

a

a

Examples: Express the following numbers in standard form

1. 4500000

2. 0.0000045

3. 20

4. 5

5. 405 000 000

Activity 1: Express the following numbers in standard form

1. 10

2. 0.000104

3. 450000000000.0

4. 0.01

5. 9


Multiplication and Division of Numbers in Standard Form

To multiply (or divide) numbers in standard form, multiply (or divide) the numbers (a’s) separately as well as the powers of 10 (102)

b()

=

Examples: Evaluate

1. (2 )

2. (2

3. Convert to decimals:

a) 2.5 x 10-2

b) 3.40 x 10-3

c) 0.4 x 104

Activity 1: Simplify:

1. 20

2. 30

3. 1.4

4. 4.004

5. 25


Review Exercise

1. Write the number in a) Standard form b) two significant figures

2. Simplify 3 2 4 + 1

3. Evaluate

a) in standard form

b) correct to 2 significant figures

c) correct to 3 decimal places


END

Estimation, Approximation and Standard form

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