Learning Objectives
Chapter Overview
What you will master by the end of Chapter 1.
Use negative, rational and real numbers in real-world contexts
Represent real numbers on a number line and order them
Perform all four operations on real numbers including using a calculator
Round off to required decimal places and significant figures
Explain rounding errors and truncation (chopping) errors
Estimate results of computations and apply estimation in real-world contexts
Mathematics: Real Numbers and Approximation
Complete chapter notes with all worked examples and solved exercises: PDF format
Please note: The content on this page is a concise preview of the chapter. The complete notes, including all detailed explanations, diagrams, worked examples, and exam-style questions, are available in the PDF. We strongly encourage you to download the PDF for the full learning material.
Section 1.1
Rational Numbers, Real Numbers and Decimal Fractions
Complete definitions, number classification tree, and all types of decimals and fractions.
Types of Numbers: Definitions
1. What are Natural Numbers?
• Define N• Give examples of Natural Numbers
Natural Numbers (N) are the counting numbers starting from 1. They do not include zero or negative numbers.
N = {1, 2, 3, 4, 5, …}
📝 All natural numbers are whole numbers, but zero is a whole number that is NOT a natural number. Therefore N ⊂ W (N is a proper subset of W).
2. What are Whole Numbers?
• Define W• What is the difference between N and W?
Whole Numbers (W) are the natural numbers together with zero. Whole numbers extend natural numbers by including 0.
W = {0, 1, 2, 3, 4, …}
3. What are Integers?
• Define Z• Give examples of Integers• What is the difference between Integers and Whole Numbers?
Integers (Z) include all whole numbers and their negatives. They extend whole numbers to include all negative numbers.
Z = {…, −3, −2, −1, 0, 1, 2, 3, …}
📝 All integers are rational numbers because any integer n can be written as n/1. Therefore Z ⊂ Q.
4. What are Rational Numbers?
• Define Q• What makes a number rational?• Give examples of rational numbers
A Rational Number (Q) is any number that can be expressed as the ratio of two integers a and b, written in the form a/b, where b ≠ 0.
Examples: 3/4, −2, 0, 1.5, −1/3, 5¼
Examples: 3/4, −2, 0, 1.5, −1/3, 5¼
📝 All integers are rational numbers. All terminating decimals and recurring decimals are also rational.
5. What are Irrational Numbers? Give examples.
• Define irrational numbers• Are π and √2 rational or irrational?• What type of decimal is an irrational number?
Irrational numbers cannot be expressed as a ratio of two integers. Their decimal form is non-terminating and non-recurring.
Examples: π = 3.14159265…, √2 = 1.41421356…, √35 = 1.70997594…
Examples: π = 3.14159265…, √2 = 1.41421356…, √35 = 1.70997594…
📝 Key Rule: A non-terminating, non-recurring decimal is always an IRRATIONAL number.
Classification Tree of Real Numbers
REAL NUMBERS
├── RATIONAL NUMBERS (Q)
│ ├── INTEGERS (Z)
│ │ ├── WHOLE NUMBERS (W) = {0, 1, 2, 3, …}
│ │ │ └── NATURAL NUMBERS (N) = {1, 2, 3, …}
│ │ └── NEGATIVE INTEGERS = {…, −3, −2, −1}
│ └── FRACTIONS (non-integer rationals e.g. 1/3, −2⅛)
└── IRRATIONAL NUMBERS (e.g. π, √7, √35)
├── RATIONAL NUMBERS (Q)
│ ├── INTEGERS (Z)
│ │ ├── WHOLE NUMBERS (W) = {0, 1, 2, 3, …}
│ │ │ └── NATURAL NUMBERS (N) = {1, 2, 3, …}
│ │ └── NEGATIVE INTEGERS = {…, −3, −2, −1}
│ └── FRACTIONS (non-integer rationals e.g. 1/3, −2⅛)
└── IRRATIONAL NUMBERS (e.g. π, √7, √35)
Decimal Fractions: Three Types
6. What are the three types of decimal fractions?
• Define terminating decimal• What is a recurring decimal?• Which decimals are rational and which are irrational?
Type 1: Terminating Decimals: The decimal ends after a finite number of digits.
Examples: 0.5, −0.75, −3.125 → These are RATIONAL numbers.
Type 2: Non-Terminating Recurring Decimals: The decimal never ends but one digit or a group of digits repeats indefinitely.
Examples: 1/3 = 0.333…, 22/7 = 3.142857142857… → These are RATIONAL numbers.
Type 3: Non-Terminating Non-Recurring Decimals: The decimal never ends and no digit pattern repeats.
Examples: π = 3.14159265…, √2 = 1.41421356… → These are IRRATIONAL numbers.
Examples: 0.5, −0.75, −3.125 → These are RATIONAL numbers.
Type 2: Non-Terminating Recurring Decimals: The decimal never ends but one digit or a group of digits repeats indefinitely.
Examples: 1/3 = 0.333…, 22/7 = 3.142857142857… → These are RATIONAL numbers.
Type 3: Non-Terminating Non-Recurring Decimals: The decimal never ends and no digit pattern repeats.
Examples: π = 3.14159265…, √2 = 1.41421356… → These are IRRATIONAL numbers.
📝 Key Rule: Terminating → Rational. Recurring → Rational. Non-recurring non-terminating → IRRATIONAL.
7. What are the four types of fractions?
• Define proper fraction• What is an improper fraction?• What is a mixed number?• What is a unit fraction?
Proper Fraction: The numerator is less than the denominator. Its value lies between 0 and 1.
Examples: 3/4, 1/2, −2/7
Improper Fraction: The numerator is greater than or equal to the denominator. Its value is 1 or greater.
Examples: 5/3, 8/2, −9/4
Mixed Number: A whole number combined with a proper fraction. It represents the same value as an improper fraction but is written differently.
Examples: 5¼, −2⅓, 3⅝
Unit Fraction: A proper fraction whose numerator is 1.
Examples: 1/2, 1/3, 1/7, 1/100
Examples: 3/4, 1/2, −2/7
Improper Fraction: The numerator is greater than or equal to the denominator. Its value is 1 or greater.
Examples: 5/3, 8/2, −9/4
Mixed Number: A whole number combined with a proper fraction. It represents the same value as an improper fraction but is written differently.
Examples: 5¼, −2⅓, 3⅝
Unit Fraction: A proper fraction whose numerator is 1.
Examples: 1/2, 1/3, 1/7, 1/100
📝 Always write final answers as mixed numbers, not improper fractions, unless the question specifies otherwise.
Operations on Fractions: Key Rules
8. What are the rules for the four operations on fractions?
• How do you add and subtract fractions?• How do you multiply fractions?• How do you divide fractions?
Addition and Subtraction: Convert to like fractions (same denominator using LCM), then add or subtract the numerators.
Multiplication: Multiply the numerators together and the denominators together. Simplify by dividing by common factors before multiplying.
Division: Multiply the first fraction by the reciprocal of the second. The reciprocal of a/b is b/a.
Worked Example (Practise Now 1):
Evaluate: −2¾ + (−5/6) − (−2/3)
Step 1: Remove brackets → −2¾ − 5/6 + 2/3
Step 2: Convert to improper fractions → −11/4 − 5/6 + 2/3
Step 3: LCM of 4, 6, 3 = 12 → −33/12 − 10/12 + 8/12
Step 4: = −35/12 = −2 11/12
Multiplication: Multiply the numerators together and the denominators together. Simplify by dividing by common factors before multiplying.
Division: Multiply the first fraction by the reciprocal of the second. The reciprocal of a/b is b/a.
Worked Example (Practise Now 1):
Evaluate: −2¾ + (−5/6) − (−2/3)
Step 1: Remove brackets → −2¾ − 5/6 + 2/3
Step 2: Convert to improper fractions → −11/4 − 5/6 + 2/3
Step 3: LCM of 4, 6, 3 = 12 → −33/12 − 10/12 + 8/12
Step 4: = −35/12 = −2 11/12
Section 1.2
Approximation: Rounding and Truncation
Rules for rounding off, truncation (chopping), and the key differences between them.
When and Why We Use Approximation
1. When is approximation used?
• Define approximation• When do we use approximate values in real life?
Approximation is used in the following situations:
First, when the actual value is known but not necessary: for example, saying a journey takes about 30 minutes.
Second, when an approximate value is easier to remember.
Third, when a non-exact number has too many digits, such as π = 3.14159265…
Fourth, when measuring instruments limit the accuracy that can be achieved.
Fifth, when the actual value is impossible or impractical to obtain: in this case we use estimation.
First, when the actual value is known but not necessary: for example, saying a journey takes about 30 minutes.
Second, when an approximate value is easier to remember.
Third, when a non-exact number has too many digits, such as π = 3.14159265…
Fourth, when measuring instruments limit the accuracy that can be achieved.
Fifth, when the actual value is impossible or impractical to obtain: in this case we use estimation.
Rounding Off: Rule and Steps
2. What is the rule for rounding off?
• State the rounding off rule• What are the three steps for rounding off?• How do you round a number to a given place?
The Rule: Look at the digit immediately to the RIGHT of the position being rounded to.
If it is 5 or more, round UP by adding 1 to the digit being kept.
If it is less than 5, round DOWN by keeping the digit the same and dropping the rest.
Three Steps for Rounding Off:
Step 1: Underline the digit in the required place.
Step 2: Circle the digit immediately to its right.
Step 3a: If the circled digit is 5 or more, add 1 to the underlined digit.
Step 3b: If the circled digit is less than 5, keep the underlined digit unchanged.
If it is 5 or more, round UP by adding 1 to the digit being kept.
If it is less than 5, round DOWN by keeping the digit the same and dropping the rest.
Three Steps for Rounding Off:
Step 1: Underline the digit in the required place.
Step 2: Circle the digit immediately to its right.
Step 3a: If the circled digit is 5 or more, add 1 to the underlined digit.
Step 3b: If the circled digit is less than 5, keep the underlined digit unchanged.
📝 Do NOT round twice. Always work from the original number directly to the required accuracy. For example, to round 96.482 to the nearest whole number, look at the original digit 4, not at a rounded version of the number.
3. What is truncation (chopping)? How is it different from rounding?
• Define truncation• What is the difference between rounding and truncation?• Give an example comparing both
Truncation (Chopping) means to simply cut off all digits after the required position without any rounding. There is no looking at the next digit: everything beyond the cut-off point is removed.
Example: √162 = 12.727922…
Rounded to 3 decimal places: 12.728 (because the next digit 7 is ≥ 5, so round up)
Truncated to 3 decimal places: 12.727 (simply cut off after the third decimal place)
Key Difference:
Rounding considers the next digit and may round up.
Truncation always cuts off regardless of the next digit, so it always rounds towards zero.
Example: √162 = 12.727922…
Rounded to 3 decimal places: 12.728 (because the next digit 7 is ≥ 5, so round up)
Truncated to 3 decimal places: 12.727 (simply cut off after the third decimal place)
Key Difference:
Rounding considers the next digit and may round up.
Truncation always cuts off regardless of the next digit, so it always rounds towards zero.
📝 Truncation error is generally less than or equal to rounding error in most cases.
Section 1.3
Significant Figures
Definition, five rules, and rounding to significant figures.
Rules for Significant Figures
1. What are significant figures? State all five rules.
• Define significant figures• How do you count significant figures?• What are the five rules for significant figures?
Significant figures (s.f.) are the digits that carry meaning contributing to the precision of a number. A number is more accurate when it is given to a greater number of significant figures.
Rule 1: All non-zero digits are significant.
Example: 3748 has 4 significant figures. 2.15 has 3 significant figures.
Rule 2: All zeros between non-zero digits are significant.
Example: 3068 has 4 significant figures. 0.006019 has 4 significant figures.
Rule 3: Zeros after a non-zero digit in the decimal part are significant.
Example: 3.50 has 3 significant figures. 0.00620 has 3 significant figures.
Rule 4: Leading zeros (before the first non-zero digit) are NOT significant. They are place-holders only.
Example: 0.0082 has 2 significant figures. 0.0060195 has 5 significant figures.
Rule 5: Trailing zeros in a whole number may or may not be significant depending on the degree of accuracy.
Example: 8982 rounded to 2 s.f. = 9000, where the trailing zeros are NOT significant.
Rule 1: All non-zero digits are significant.
Example: 3748 has 4 significant figures. 2.15 has 3 significant figures.
Rule 2: All zeros between non-zero digits are significant.
Example: 3068 has 4 significant figures. 0.006019 has 4 significant figures.
Rule 3: Zeros after a non-zero digit in the decimal part are significant.
Example: 3.50 has 3 significant figures. 0.00620 has 3 significant figures.
Rule 4: Leading zeros (before the first non-zero digit) are NOT significant. They are place-holders only.
Example: 0.0082 has 2 significant figures. 0.0060195 has 5 significant figures.
Rule 5: Trailing zeros in a whole number may or may not be significant depending on the degree of accuracy.
Example: 8982 rounded to 2 s.f. = 9000, where the trailing zeros are NOT significant.
2. What is the difference between rounding to decimal places and rounding to significant figures?
• Compare decimal places and significant figures• How do you round 0.004709 to 4 significant figures?
Decimal places are counted from the decimal point, regardless of leading zeros.
Significant figures are counted from the first non-zero digit.
Example: Round 0.004709 to 4 significant figures.
The first significant figure is 4 (the leading zeros are not counted).
Counting from 4: the four significant figures are 4, 7, 0, 9.
Answer: 0.004709 (already 4 s.f.)
Rounding the same number to 4 decimal places would give 0.0047: a completely different result.
Significant figures are counted from the first non-zero digit.
Example: Round 0.004709 to 4 significant figures.
The first significant figure is 4 (the leading zeros are not counted).
Counting from 4: the four significant figures are 4, 7, 0, 9.
Answer: 0.004709 (already 4 s.f.)
Rounding the same number to 4 decimal places would give 0.0047: a completely different result.
📝 Always identify the first non-zero digit before counting significant figures.
Section 1.4
Rounding and Truncation Errors
How intermediate rounding causes errors in final answers.
Understanding Rounding and Truncation Errors
1. What is a rounding error?
• Define rounding error• How does rounding cause errors in calculations?• What is the key rule for intermediate steps?
A rounding error is introduced when numbers are rounded off at intermediate steps of a calculation. If intermediate working is rounded to fewer significant figures than the final answer requires, the final answer becomes inaccurate.
Key Rule: To ensure a final answer accurate to n significant figures, keep at least (n + 1) significant figures in all intermediate calculations.
Example: The area of a square is 131 cm². Find the perimeter.
Length = √131. Do NOT write √131 ≈ 11.45 and then round again.
Correct method: Length = √131 = 11.4455… ≈ 11.4 cm (3 s.f.)
Perimeter = 4 × 11.4455… ≈ 45.8 cm (3 s.f.): use the calculator value, not the rounded intermediate value.
Key Rule: To ensure a final answer accurate to n significant figures, keep at least (n + 1) significant figures in all intermediate calculations.
Example: The area of a square is 131 cm². Find the perimeter.
Length = √131. Do NOT write √131 ≈ 11.45 and then round again.
Correct method: Length = √131 = 11.4455… ≈ 11.4 cm (3 s.f.)
Perimeter = 4 × 11.4455… ≈ 45.8 cm (3 s.f.): use the calculator value, not the rounded intermediate value.
2. What is a truncation error?
• Define truncation error• Compare rounding error and truncation error
A truncation error occurs when a number is cut off (truncated) rather than rounded. Truncation always removes digits without rounding up, which means the approximation is slightly smaller than the true value for positive numbers.
Comparison:
Rounding error: may be positive or negative depending on whether the number was rounded up or down.
Truncation error: always makes the result smaller than the true value (for positive numbers), so it is always in one direction.
In most practical cases, truncation error is less than or equal to rounding error.
Comparison:
Rounding error: may be positive or negative depending on whether the number was rounded up or down.
Truncation error: always makes the result smaller than the true value (for positive numbers), so it is always in one direction.
In most practical cases, truncation error is less than or equal to rounding error.
Section 1.5
Estimation: Real-World Applications
How to estimate computations and compare value for money.
Estimation Methods
1. What is estimation and when is it used?
• Define estimation• How is estimation different from approximation?• When do we use estimation?
Estimation is the process of finding an approximate value when the exact value is unknown or impractical to obtain. It is a special case of approximation.
Estimation is used to:
Check whether a calculator answer is reasonable.
Find an approximate answer quickly without a calculator.
Compare value for money between two options.
Scale up from a known smaller quantity to estimate a larger one.
Method: Round each number to 1 or 2 significant figures (or convenient round numbers), then calculate mentally.
Estimation is used to:
Check whether a calculator answer is reasonable.
Find an approximate answer quickly without a calculator.
Compare value for money between two options.
Scale up from a known smaller quantity to estimate a larger one.
Method: Round each number to 1 or 2 significant figures (or convenient round numbers), then calculate mentally.
2. How do you use estimation to check a calculator answer?
• How do you verify a calculation using estimation?• Give an example of checking a calculator answer with estimation
Example (Practise Now 10):
Nora evaluates 798 × 195 using a calculator and says the answer is 15,561.
Estimation: 800 × 200 = 160,000.
Nora's answer of 15,561 is clearly far too small: it is not reasonable.
The actual correct answer is 155,610.
Nora evaluates 798 × 195 using a calculator and says the answer is 15,561.
Estimation: 800 × 200 = 160,000.
Nora's answer of 15,561 is clearly far too small: it is not reasonable.
The actual correct answer is 155,610.
📝 Always estimate first, then verify with the calculator. If your estimate and calculator answer differ significantly, check your calculator input.
3. How do you compare value for money using estimation?
• How do you find better value for money without a calculator?• Give an example comparing two products
To compare value for money, find the cost per unit (cost per gram, cost per mL, etc.) for each option.
Example (Practise Now 12):
Option A: 300 mL of Olive Oil at $8.80
Option B: 350 mL of Olive Oil at $10.40
Option A: $8.80 ÷ 300 = approximately $8.80 ÷ 300 ≈ $2.93 per 100 mL
Option B: $10.40 ÷ 350 = approximately $10.40 ÷ 350 ≈ $2.97 per 100 mL
Since Option A costs less per mL, Option A is better value for money.
Example (Practise Now 12):
Option A: 300 mL of Olive Oil at $8.80
Option B: 350 mL of Olive Oil at $10.40
Option A: $8.80 ÷ 300 = approximately $8.80 ÷ 300 ≈ $2.93 per 100 mL
Option B: $10.40 ÷ 350 = approximately $10.40 ÷ 350 ≈ $2.97 per 100 mL
Since Option A costs less per mL, Option A is better value for money.
Key Vocabulary
Chapter 1 Vocabulary
Important terms you must know: learn these definitions for exam questions.
| Term | Definition |
|---|---|
| Natural Numbers (N) | Counting numbers starting from 1. N = {1, 2, 3, 4, …} |
| Whole Numbers (W) | Natural numbers plus zero. W = {0, 1, 2, 3, …} |
| Integers (Z) | All whole numbers and their negatives. Z = {…, −2, −1, 0, 1, 2, …} |
| Rational Numbers (Q) | Numbers expressible as a/b where b ≠ 0. Includes terminating and recurring decimals. |
| Irrational Numbers | Non-terminating, non-recurring decimals that cannot be written as a fraction. Examples: π, √2 |
| Proper Fraction | A fraction where the numerator is less than the denominator. Value is between 0 and 1. |
| Improper Fraction | A fraction where the numerator is greater than or equal to the denominator. |
| Mixed Number | A whole number combined with a proper fraction, such as 5¼. |
| Unit Fraction | A proper fraction where the numerator is 1, such as 1/3 or 1/7. |
| Approximation | Using an approximate value instead of the exact value when precision is not required. |
| Rounding Off | Adjusting a number to a required degree of accuracy using the rule of 5 or more rounds up. |
| Truncation | Cutting off digits after a required position without rounding: always rounds towards zero. |
| Significant Figures | Digits that carry meaning contributing to the precision of a number. |
| Rounding Error | Error introduced when intermediate steps are rounded to fewer figures than required. |
| Truncation Error | Error introduced by cutting off digits rather than rounding them. |
| Estimation | Finding an approximate value without the exact value: used to check reasonableness of answers. |
Test Yourself: Chapter 1 Quiz
Chapter-wise MCQs aligned with APSACS and FBISE syllabus. No login required. Instant results.
Start Quiz NowWant Live Explanation of Real Numbers and Approximation?
Join W.H. Academy: Live tutoring, recorded lectures, and full chapter solutions for APSACS & FBISE students in Rawalpindi & online Pakistan.