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 768frs≈ 770푓푟푠. Why 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
ퟏ)
3
4
ퟐ)
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)
ퟏ)
1
2
표푓 2
1
2
− 1
1
6
÷ 2
1
3
ퟐ) 7
1
2
− 2
1
7
× 1
2
5
+ 7 ÷ 3
3)
4
3
(2
1
2
) − 1
1
6
÷ 2
1
3
4) 1.2
0.012
5) 9.02 × 100
6) 2 ÷ 100
7) 9.8+1.07+69
8) 5-4.667
9) 44.36 x 32
10) a)
0.3
0.2
b)
0.3
1000
b)
2
100
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 ≥
5 , 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

Remark: 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
100
10
Unit 10th
(thousand)
(hundred)
(ten)
(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
Neare
st
10frs
Nearest
100th
Nearest
Degree
444.525
5059.996
267.537

Answer: Check for correctness
Number 1dp 2dp Whole
number
or unit
Neare
st
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× ퟏퟎ풏 퐰퐡퐞퐫퐞 ퟏ ≤ 풂 < ퟏퟎ, 풏흐풁
a× 10+푛 ⇒ 푑. 푝 푚표푣푒푑 푡표 푡ℎ푒 푙푒푓푡
a× 10−푛 ⇒ 푑. 푝 푚표푣푒푑 푡표 푡ℎ푒 푟푖푔ℎ푡
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)
(푎 × 10푛)( b× 10푚) = (a × b)(10푛 × 10푚)
푎×10푛
푎
10푛
=
×
푎×10푚푏
10푚
Examples: Evaluate
1. (2 × 105 )(4 × 10−3)
2. 3.6×10−5
1.2×10−1
3. (2 × 103)5
4. Convert to decimals:
a) 2.5 x 10-2
b) 3.40 x 10-3
c) 0.4 x 104
Activity 1: Simplify:
1. (4 × 10−2)( 20× 108)
2. (3 × 10−3) ÷ ( 30× 10−10)
3. 4×10−3
2.5×104
4. (7 × 10−5) ÷ ( 1.4× 10−8)
5. 5×10−4
2×100
6. (3 × 100) × ( 4.004× 103)
7. 4 × ( 25 × 1012)
8. 4×104 2
2.5×102 3

Review Exercise
1. Write the number
1
400
in a) Standard form b) two significant figures
2. Simplify 3
1
3
− 2
1
4
÷4
1
2
+ 1
1
6
3. Evaluate
12.78×10−3
9 × 10−1 Express your answer in
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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