Mathematics · Topic 1.1
Rational Numbers & Scientific Notation
Extend from whole numbers to the full set of rational numbers — any number that can be written as a fraction. Learn to operate with negative numbers, express very large or very small quantities in scientific notation, and calculate percent change.
What You'll Learn
- Classify numbers as rational or irrational and explain why
- Perform addition, subtraction, multiplication, and division with negative numbers
- Find the absolute value of a number and use it to compare quantities
- Convert numbers to and from scientific notation (standard form)
- Calculate percent increase and percent decrease
- Apply these skills in real-world contexts
IB Assessment Focus
Criterion A: Apply number operations in familiar and unfamiliar situations.
Criterion B: Discover patterns in sign rules and verify by substitution.
Criterion C: Communicate working clearly with correct notation.
Criterion D: Apply scientific notation and percent change to real-world data.
Number Classification
Every number you encounter belongs to one or more of these categories. Understanding classification helps you choose the right methods for calculations.
| Type | Definition | Examples |
|---|---|---|
| Natural (ℕ) | Counting numbers from 1 onwards | 1, 2, 3, 47, 1000 |
| Integer (ℤ) | Whole numbers, including zero and negatives | −5, 0, 3, 12 |
| Rational (ℚ) | Any number expressible as p/q where p, q are integers and q ≠ 0 | 3/4, −2, 0.75, 0.333... |
| Irrational | Cannot be written as a fraction; decimal is non-terminating, non-repeating | √2, π, √5 |
| Real (ℝ) | All rational and irrational numbers combined | All of the above |
Key Distinction: Repeating decimals ARE rational. 0.333... = 1/3 (rational). A number like 0.101001000100001... with no repeating pattern is irrational. All integers are rational because they can be written as a fraction (e.g. 5 = 5/1).
Mathematics · Topic 1.1
Rational Numbers
A rational number is any number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0. This includes fractions, terminating decimals, repeating decimals, and all integers.
Absolute Value
Definition
|a| = the distance of a from zero on the number line (always positive or zero)
Examples:
- |5| = 5 (5 is 5 units from zero)
- |−5| = 5 (−5 is also 5 units from zero)
- |0| = 0
- |−3.7| = 3.7
Comparing Rational Numbers
To compare rational numbers, convert them to the same form (decimals or fractions with a common denominator), then compare.
Example: Which is greater, −2/3 or −3/4?
Convert to decimals: −2/3 ≈ −0.667 and −3/4 = −0.75
On a number line, −0.667 is to the right of −0.75
Therefore −2/3 > −3/4
Ordering on the Number Line
Numbers increase from left to right on the number line. For negative numbers, the one closer to zero is greater.
- −1 > −5 (because −1 is closer to zero)
- −10 < −2 < 0 < 3 < 7
- Any positive number is greater than any negative number
Converting Between Fractions, Decimals, and Percentages
| From → To | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide numerator by denominator | 3/8 = 3 ÷ 8 = 0.375 |
| Decimal → Fraction | Write over place value, then simplify | 0.6 = 6/10 = 3/5 |
| Decimal → Percentage | Multiply by 100 | 0.375 = 37.5% |
| Percentage → Decimal | Divide by 100 | 45% = 0.45 |
| Fraction → Percentage | Divide, then × 100 | 3/8 = 0.375 × 100 = 37.5% |
Mathematics · Topic 1.1
Operations with Negative Numbers
Negative numbers follow specific rules for each operation. Mastering these sign rules is essential for all of Grade 7 mathematics.
Addition and Subtraction Rules
- Adding a negative = subtracting: 5 + (−3) = 5 − 3 = 2
- Subtracting a negative = adding: 5 − (−3) = 5 + 3 = 8
- Adding two negatives: (−4) + (−6) = −10 (add the absolute values, keep the negative sign)
- Different signs: 7 + (−12) = −5 (subtract absolute values, take the sign of the larger)
Memory Tip: Think of two negatives next to each other as making a positive. When you see −(−), it becomes +. Same sign = positive, different signs = negative.
Multiplication and Division Sign Rules
The Sign Rule
Same signs → Positive result Different signs → Negative result
| Operation | Signs | Result Sign | Example |
|---|---|---|---|
| Positive × Positive | Same | Positive | 4 × 3 = +12 |
| Negative × Negative | Same | Positive | (−4) × (−3) = +12 |
| Positive × Negative | Different | Negative | 4 × (−3) = −12 |
| Negative × Positive | Different | Negative | (−4) × 3 = −12 |
The same rules apply to division:
- (−20) ÷ (−5) = +4 (same signs → positive)
- (−20) ÷ 5 = −4 (different signs → negative)
- 20 ÷ (−5) = −4 (different signs → negative)
Order of Operations with Negatives
Follow BEDMAS/BODMAS (Brackets, Exponents/Order, Division & Multiplication, Addition & Subtraction) even when working with negatives.
Example: (−3)² + 4 × (−2)
Exponent first: (−3)² = (−3) × (−3) = 9
Multiplication: 4 × (−2) = −8
Addition: 9 + (−8) = 1
Common Mistake: (−3)² = 9 but −3² = −9. With the brackets, the negative is squared. Without brackets, only the 3 is squared and the result stays negative.
Mathematics · Topic 1.1
Scientific Notation
Scientific notation (standard form) lets you express very large or very small numbers compactly using powers of 10.
Scientific Notation
a × 10n where 1 ≤ a < 10 and n is an integer
Converting to Scientific Notation
- Move the decimal point until you have a number between 1 and 10.
- Count how many places you moved it — that is the value of n.
- If you moved the decimal LEFT (large number), n is positive.
- If you moved the decimal RIGHT (small number), n is negative.
| Ordinary Number | Scientific Notation | Explanation |
|---|---|---|
| 3,400,000 | 3.4 × 10&sup6; | Decimal moves 6 places left |
| 72,000 | 7.2 × 10&sup4; | Decimal moves 4 places left |
| 0.000047 | 4.7 × 10−5 | Decimal moves 5 places right |
| 0.0081 | 8.1 × 10−3 | Decimal moves 3 places right |
Converting from Scientific Notation to Ordinary Number
- If n is positive, move the decimal point n places to the right (number gets bigger)
- If n is negative, move the decimal point |n| places to the left (number gets smaller)
Examples:
- 5.6 × 10³ = 5600
- 2.1 × 10−4 = 0.00021
Common Mistake: 34 × 10² is NOT valid scientific notation because 34 is not between 1 and 10. The correct form is 3.4 × 10³.
Comparing Numbers in Scientific Notation
Compare the exponents first. The larger the exponent, the larger the number (for positive numbers). If exponents are equal, compare the coefficients.
Example: Which is larger, 3.2 × 10&sup5; or 8.7 × 10&sup4;?
3.2 × 10&sup5; = 320,000 and 8.7 × 10&sup4; = 87,000. Since 10&sup5; > 10&sup4;, the first number is larger.
Mathematics · Topic 1.1
Percent Change
Percent change measures how much a quantity has increased or decreased relative to its original value.
Percent Change Formula
% change = new value − original valueoriginal value × 100
- A positive result means a percent increase
- A negative result means a percent decrease
- Always divide by the original value, not the new value
Detailed Examples
Percent Increase: A shirt was $40 and is now $52. Find the percent change.
% change = [(52 − 40) ÷ 40] × 100 = [12 ÷ 40] × 100 = 0.3 × 100 = 30% increase
Percent Decrease: A phone cost $800 and now costs $680. Find the percent change.
% change = [(680 − 800) ÷ 800] × 100 = [−120 ÷ 800] × 100 = −0.15 × 100 = −15% (a 15% decrease)
Finding the New Value
- After an increase: New value = original × (1 + rate)
- After a decrease: New value = original × (1 − rate)
Example: A population of 5000 increases by 12%.
New population = 5000 × 1.12 = 5600
Example: A $250 item is discounted by 20%.
Sale price = 250 × 0.80 = $200
Common Mistake: A 50% increase followed by a 50% decrease does NOT return to the original. If something goes from 100 → 150 (+50%) → 75 (−50%). The result is 75, not 100, because the second percentage is calculated from the new value (150).
Mathematics · Topic 1.1
Worked Examples
These examples show the multi-step reasoning expected at Grade 7. Notice how each step is clearly stated.
EXAMPLE 1Calculate (−3) × (−4) + (−2) × 5. Show all steps.
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Full Solution
Following order of operations, I calculate the multiplications first:
Step 1: (−3) × (−4) = +12 (negative × negative = positive)
Step 2: (−2) × 5 = −10 (negative × positive = negative)
Step 3: 12 + (−10) = 12 − 10 = 2
Step 1: (−3) × (−4) = +12 (negative × negative = positive)
Step 2: (−2) × 5 = −10 (negative × positive = negative)
Step 3: 12 + (−10) = 12 − 10 = 2
EXAMPLE 2A jacket costs $80 and goes on sale for $60. Calculate the percentage decrease.
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Full Solution
Step 1: Identify: original = $80, new = $60.
Step 2: Apply the formula: % change = [(new − original) ÷ original] × 100
Step 3: = [(60 − 80) ÷ 80] × 100 = [−20 ÷ 80] × 100
Step 4: = −0.25 × 100 = −25%
The price decreased by 25%.
Step 2: Apply the formula: % change = [(new − original) ÷ original] × 100
Step 3: = [(60 − 80) ÷ 80] × 100 = [−20 ÷ 80] × 100
Step 4: = −0.25 × 100 = −25%
The price decreased by 25%.
EXAMPLE 3Write 0.000047 in scientific notation. Then write 6.3 × 10&sup4; as an ordinary number.
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Full Solution
Part 1: 0.000047 → Move the decimal 5 places right to get 4.7.
Since I moved right (small number), the power is negative: 4.7 × 10−5
Part 2: 6.3 × 10&sup4; → Move the decimal 4 places right.
6.3 → 63 → 630 → 6300 → 63,000
Since I moved right (small number), the power is negative: 4.7 × 10−5
Part 2: 6.3 × 10&sup4; → Move the decimal 4 places right.
6.3 → 63 → 630 → 6300 → 63,000
EXAMPLE 4Evaluate −5 + 3 × (−2)² − (−4). Show order of operations.
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Full Solution
Step 1 (Exponent): (−2)² = (−2) × (−2) = 4
Step 2 (Multiply): 3 × 4 = 12
Step 3 (Subtraction of negative): −(−4) = +4
Step 4 (Combine): −5 + 12 + 4 = 11
Step 2 (Multiply): 3 × 4 = 12
Step 3 (Subtraction of negative): −(−4) = +4
Step 4 (Combine): −5 + 12 + 4 = 11
EXAMPLE 5Order these numbers from least to greatest: −3/4, −0.5, 1/3, −1, 0.8
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Full Solution
Step 1: Convert all to decimals:
−3/4 = −0.75, −0.5, 1/3 ≈ 0.333, −1, 0.8
Step 2: Place on number line: −1 is furthest left, then −0.75, then −0.5, then 0.333, then 0.8.
Answer: −1, −3/4, −0.5, 1/3, 0.8
−3/4 = −0.75, −0.5, 1/3 ≈ 0.333, −1, 0.8
Step 2: Place on number line: −1 is furthest left, then −0.75, then −0.5, then 0.333, then 0.8.
Answer: −1, −3/4, −0.5, 1/3, 0.8
EXAMPLE 6A shop increases prices by 15%, then later offers a 15% discount. Is the final price the same as the original? Explain.
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Full Solution
Let the original price be $100.
Step 1 (Increase): $100 × 1.15 = $115
Step 2 (Decrease): $115 × 0.85 = $97.75
The final price is NOT the same as the original. It is $2.25 less because the 15% discount is calculated on the higher price ($115), so the decrease is larger in absolute terms than the original increase.
Step 1 (Increase): $100 × 1.15 = $115
Step 2 (Decrease): $115 × 0.85 = $97.75
The final price is NOT the same as the original. It is $2.25 less because the 15% discount is calculated on the higher price ($115), so the decrease is larger in absolute terms than the original increase.
EXAMPLE 7The Earth is approximately 1.496 × 10&sup8; km from the Sun. Express this as an ordinary number.
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Full Solution
The exponent is +8, so I move the decimal 8 places to the right:
1.496 → 14.96 → 149.6 → 1496 → 14960 → 149600 → 1496000 → 14960000 → 149600000
Answer: 149,600,000 km (approximately 150 million kilometres)
1.496 → 14.96 → 149.6 → 1496 → 14960 → 149600 → 1496000 → 14960000 → 149600000
Answer: 149,600,000 km (approximately 150 million kilometres)
Mathematics · Topic 1.1
Practice Q&A
Attempt each question before revealing the model answer. Show your working clearly.
CALCULATEEvaluate (−8) ÷ (−2) + (−3) × 4
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Model Answer
Multiplication and division first (left to right):
(−8) ÷ (−2) = +4 (same signs → positive)
(−3) × 4 = −12 (different signs → negative)
Then add: 4 + (−12) = 4 − 12 = −8
(−8) ÷ (−2) = +4 (same signs → positive)
(−3) × 4 = −12 (different signs → negative)
Then add: 4 + (−12) = 4 − 12 = −8
CLASSIFYState whether each is rational or irrational and explain: (a) 0.7777... (b) √3 (c) −5 (d) π
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Model Answer
(a) Rational — 0.7777... = 7/9, a repeating decimal can always be written as a fraction.
(b) Irrational — √3 cannot be expressed as a fraction; its decimal is non-terminating and non-repeating.
(c) Rational — −5 = −5/1, any integer is rational.
(d) Irrational — π = 3.14159... is non-terminating and non-repeating.
(b) Irrational — √3 cannot be expressed as a fraction; its decimal is non-terminating and non-repeating.
(c) Rational — −5 = −5/1, any integer is rational.
(d) Irrational — π = 3.14159... is non-terminating and non-repeating.
CONVERTWrite 56,700,000 in scientific notation.
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Model Answer
Move the decimal 7 places left to get 5.67.
Since the number is large and I moved left, the power is positive.
5.67 × 10&sup7;
Since the number is large and I moved left, the power is positive.
5.67 × 10&sup7;
CALCULATEA school had 1200 students. Enrollment increased to 1380. Calculate the percent change.
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Model Answer
% change = [(1380 − 1200) ÷ 1200] × 100
= [180 ÷ 1200] × 100
= 0.15 × 100 = 15% increase
= [180 ÷ 1200] × 100
= 0.15 × 100 = 15% increase
COMPAREWhich is greater: |−7| or |4|? Explain using the number line.
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Model Answer
|−7| = 7 (distance from zero) and |4| = 4 (distance from zero).
Since 7 > 4, |−7| is greater.
This shows that −7 is farther from zero than 4, even though −7 is a smaller number than 4.
Since 7 > 4, |−7| is greater.
This shows that −7 is farther from zero than 4, even though −7 is a smaller number than 4.
EVALUATECalculate (−2)³ and explain why the result is negative.
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Model Answer
(−2)³ = (−2) × (−2) × (−2)
= (+4) × (−2) = −8
The result is negative because an odd power of a negative number is always negative. The first two negatives multiply to give positive 4, but the third negative makes the final result negative.
= (+4) × (−2) = −8
The result is negative because an odd power of a negative number is always negative. The first two negatives multiply to give positive 4, but the third negative makes the final result negative.
CONVERTExpress 8.05 × 10−4 as a decimal number.
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Model Answer
The exponent is −4, so move the decimal 4 places to the left:
8.05 → 0.805 → 0.0805 → 0.00805 → 0.000805
8.05 → 0.805 → 0.0805 → 0.00805 → 0.000805
APPLYA car was bought for $24,000 and sold two years later for $18,000. Calculate the percentage depreciation.
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Model Answer
% change = [(18000 − 24000) ÷ 24000] × 100
= [−6000 ÷ 24000] × 100
= −0.25 × 100 = −25%
The car depreciated by 25% over two years.
= [−6000 ÷ 24000] × 100
= −0.25 × 100 = −25%
The car depreciated by 25% over two years.
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
What is a rational number?
Any number that can be expressed as p/q where p and q are integers and q ≠ 0. Includes fractions, terminating decimals, repeating decimals, and all integers.
Tap to reveal
What is |−8|?
8. Absolute value is the distance from zero, always positive or zero.
Tap to reveal
Negative × Negative = ?
Positive. Same signs always give a positive result.
Tap to reveal
Positive × Negative = ?
Negative. Different signs always give a negative result.
Tap to reveal
What is scientific notation?
A way to write numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Used for very large or very small numbers.
Tap to reveal
Write 0.00036 in scientific notation
3.6 × 10−4 (move decimal 4 places right)
Tap to reveal
Write 4,200,000 in scientific notation
4.2 × 10&sup6; (move decimal 6 places left)
Tap to reveal
What is the percent change formula?
% change = [(new − original) ÷ original] × 100. Positive = increase, negative = decrease.
Tap to reveal
Subtracting a negative is the same as?
Adding. 5 − (−3) = 5 + 3 = 8. Two negatives make a positive.
Tap to reveal
Is 0.333... rational or irrational?
Rational. 0.333... = 1/3. Repeating decimals are always rational because they can be written as fractions.
Tap to reveal
What is (−2)³?
−8. An odd power of a negative number stays negative.
Tap to reveal
What is (−2)²?
+4. An even power of a negative number is positive.
Tap to reveal
Difference between (−3)² and −3²?
(−3)² = 9 (the negative is squared). −3² = −9 (only 3 is squared, then negated).
Tap to reveal
How do you compare negative numbers?
The one closer to zero is greater. −1 > −5 because −1 is closer to zero on the number line.
Tap to reveal
Is √2 rational or irrational?
Irrational. Its decimal (1.41421...) is non-terminating and non-repeating. It cannot be written as a fraction.
Tap to reveal
Mathematics · Practice Test
Practice Test — 20 Questions
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