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Standard Form And Approximation

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Standard Form And Approximation
Week Five - Revision of Standard Form and Approximation

WEEK FIVE

Topic: Revision of Standard Form and Approximation

Content

  • Revision of Standard Form
  • Rounding Off Numbers
  • Decimal Places
  • Significant Figures

Learning Objectives

By the end of this lesson, students should be able to:

  1. Write large and small numbers in standard form.
  2. Change numbers from standard form to ordinary form.
  3. Round numbers to the nearest ten, hundred, and thousand.
  4. Round numbers to a given number of decimal places.
  5. Round numbers to a given number of significant figures.
  6. Distinguish between decimal places and significant figures.

1. Revision of Standard Form

Meaning of Standard Form

A number is said to be in standard form when it is written as:

A × 10n

where:

  • 1 ≤ A < 10
  • n is an integer (positive, negative, or zero)

Standard form is also called scientific notation. It is useful for writing very large or very small numbers in a short and neat way.

Examples

  • 2 × 106
  • 7 × 10-3
  • 2.5 × 104
  • 8.6 × 10-9

Important Idea

  • If the number is greater than 10, the power of 10 is usually positive.
  • If the number is between 0 and 1, the power of 10 is usually negative.

Illustration with Powers of 10

Power of 10 Ordinary Form
10110
102100
1031,000
10410,000
10-10.1
10-20.01
10-30.001
10-40.0001

How to Write a Number in Standard Form

  1. Move the decimal point until only one non-zero digit remains to the left of the decimal point.
  2. Count how many places the decimal point moves.
  3. If it moves to the left, the exponent is positive.
  4. If it moves to the right, the exponent is negative.

Worked Example 1: Express the following numbers in standard form

  1. 300,000
    300,000 = 3 × 105
    Decimal point moved 5 places to the left.
  2. 55
    55 = 5.5 × 101
  3. 2,300,000
    2,300,000 = 2.3 × 106
  4. 720,000,000
    720,000,000 = 7.2 × 108
  5. 9,400,000,000
    9,400,000,000 = 9.4 × 109

Worked Example 2: Change from standard form to ordinary form

Rule: For a positive power, move the decimal point to the right. For a negative power, move it to the left.

  1. 5.1 × 107 = 51,000,000
  2. 2.5 × 106 = 2,500,000
  3. 3.4 × 101 = 34
  4. 9.8 × 105 = 980,000
  5. 6 × 108 = 600,000,000

Worked Example 3: Express the following decimal fractions in standard form

  1. 0.0015 = 1.5 × 10-3
  2. 0.000026 = 2.6 × 10-5
  3. 0.000000067 = 6.7 × 10-8
  4. 0.3 = 3 × 10-1

Worked Example 4: Express the following as decimal fractions

  1. 9.4 × 10-5 = 0.000094
  2. 8.8 × 10-3 = 0.0088
  3. 1.8 × 10-1 = 0.18
  4. 2 × 10-7 = 0.0000002
Note: For numbers less than 1 written in standard form, the exponent is usually negative.
Examples:
  • 0.5 = 5 × 10-1
  • 0.04 = 4 × 10-2
  • 0.007 = 7 × 10-3

Quick Practice

  1. Change to ordinary form:
    • 9.18 × 105
    • 6.75 × 10-8
  2. Express in standard form:
    • 0.0000058
    • 458,000

2. Approximation

Approximation means giving a value that is close to the exact value.

We use approximation when:

  • the exact number is not necessary;
  • the number is too long;
  • we want easier calculations.

The symbol for approximation is ≈, which means approximately equal to.

3. Rounding Off Numbers

General Rule for Rounding

  1. Identify the place value you are rounding to.
  2. Look at the digit immediately to the right.
  3. If the digit is 0, 1, 2, 3, or 4, round down.
  4. If the digit is 5, 6, 7, 8, or 9, round up.

Example: Round to the nearest thousand, hundred, and ten

1. Number: 4,517

  • Nearest thousand: 4,517 ≈ 5,000
  • Nearest hundred: 4,517 ≈ 4,500
  • Nearest ten: 4,517 ≈ 4,520

2. Number: 30,637

  • Nearest thousand: 30,637 ≈ 31,000
  • Nearest hundred: 30,637 ≈ 30,600
  • Nearest ten: 30,637 ≈ 30,640

Place Value Illustration for 4,517

Thousands Hundreds Tens Units
4 5 1 7
  • Nearest thousand: check 5 → round up to 5,000
  • Nearest hundred: check 1 → round down to 4,500
  • Nearest ten: check 7 → round up to 4,520

4. Significant Figures

Meaning of Significant Figures

Significant figures begin from the first non-zero digit on the left. They show how precise a number is.

Important Notes on Significant Figures

  • Leading zeros are not significant. Example: 0.000925 has 3 significant figures.
  • Zeros between non-zero digits are significant. Example: 8.0296 has 5 significant figures.
  • Zeros after a decimal point may be significant. Example: 2.300 has 4 significant figures.
  • In whole numbers like 4500, the number of significant figures may be unclear unless standard form is used.

Examples:

  • 4.5 × 103 = 2 significant figures
  • 4.50 × 103 = 3 significant figures

How to Round to Significant Figures

  1. Start counting from the first non-zero digit.
  2. Keep the required number of digits.
  3. Look at the next digit to decide whether to round up or down.
  4. Keep the correct place value.

Worked Examples

1. Round 26,002 to:

  • 1 significant figure: 26,002 ≈ 30,000
  • 2 significant figures: 26,002 ≈ 26,000
  • 3 significant figures: 26,002 ≈ 26,000

2. Round 2.00567 to:

  • 1 significant figure: 2.00567 ≈ 2
  • 2 significant figures: 2.00567 ≈ 2.0
  • 3 significant figures: 2.00567 ≈ 2.01

3. Round 0.006307 to:

  • 1 significant figure: 0.006307 ≈ 0.006
  • 2 significant figures: 0.006307 ≈ 0.0063
  • 3 significant figures: 0.006307 ≈ 0.00631

5. Decimal Places

Meaning of Decimal Places

Decimal places are counted from the decimal point.

  • 3.4 has 1 decimal place
  • 5.67 has 2 decimal places
  • 0.908 has 3 decimal places

How to Round to Decimal Places

  1. Identify the number of decimal places required.
  2. Look at the next digit.
  3. Round up or down.
  4. Write the answer with the correct number of decimal places.

Worked Examples

1. Round 0.0089 to:

  • 1 decimal place: 0.0089 ≈ 0.0
  • 2 decimal places: 0.0089 ≈ 0.01
  • 3 decimal places: 0.0089 ≈ 0.009

2. Round 0.9002 to:

  • 1 decimal place: 0.9002 ≈ 0.9
  • 2 decimal places: 0.9002 ≈ 0.90
  • 3 decimal places: 0.9002 ≈ 0.900

3. Round 1.9875 to:

  • 1 decimal place: 1.9875 ≈ 2.0
  • 2 decimal places: 1.9875 ≈ 1.99
  • 3 decimal places: 1.9875 ≈ 1.988

6. Difference Between Significant Figures and Decimal Places

Number Rounded to 2 Decimal Places Rounded to 2 Significant Figures
4.376 4.38 4.4
0.006307 0.01 0.0063
1.9875 1.99 2.0

Explanation:
Decimal places are counted from the decimal point, while significant figures start from the first non-zero digit.

7. Common Mistakes to Avoid

  1. Writing a number in standard form with the first number greater than 10.
    Wrong: 25 × 103
    Correct: 2.5 × 104
  2. Forgetting that small numbers have negative powers.
    0.004 = 4 × 10-3, not 4 × 103
  3. Confusing decimal places with significant figures.
  4. Dropping important zeros.
    2.0 has 2 significant figures, while 2 has 1 significant figure.

8. Summary

  • Standard form is written as A × 10n, where 1 ≤ A < 10.
  • Large numbers usually have positive powers of 10.
  • Small numbers usually have negative powers of 10.
  • In rounding, digits 0 to 4 round down, while digits 5 to 9 round up.
  • Significant figures begin from the first non-zero digit.
  • Decimal places are counted from the decimal point.

9. Evaluation

A. Express the following in standard form

  1. 3,500,000
  2. 28
  3. 0.47
  4. 0.0000003

B. Change the following to ordinary numbers

  1. 9.18 × 105
  2. 6.75 × 10-8

C. Round each statement to two significant figures and write it in full

  1. It will cost ₦3.28 billion to renovate the state’s classrooms.
  2. The area of Ghana is 23.9 million hectares.
  3. It was estimated that the population of Lagos was about 9.44 million in 2000.

10. Answer Guide

A. Standard Form

  1. 3,500,000 = 3.5 × 106
  2. 28 = 2.8 × 101
  3. 0.47 = 4.7 × 10-1
  4. 0.0000003 = 3 × 10-7

B. Ordinary Form

  1. 9.18 × 105 = 918,000
  2. 6.75 × 10-8 = 0.0000000675

C. Two Significant Figures

  1. ₦3.28 billion ≈ ₦3.3 billion
    = ₦3,300,000,000
  2. 23.9 million hectares ≈ 24 million hectares
    = 24,000,000 hectares
  3. 9.44 million ≈ 9.4 million
    = 9,400,000

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