Year 9 Maths: Real Numbers and Scientific Notation
The real number system, exact and approximate values, locating square roots on the number line, scientific notation, SI prefixes from pico to tera, significant figures and very large and very small measurements. Worked examples, answer-checkers and exam-style practice.
Real Numbers and Scientific Notation
Rational and irrational numbers, exact answers, and the scientific notation used for the very large and the very small.
Aligned to the Australian Curriculum v9 Year 9 Mathematics: Number (AC9M9N01) and Measurement (AC9M9M02), and the Victorian Curriculum Mathematics 2.0 Level 9 (VC2M9N01, VC2M9M02).
00Start here
Year 9 number work is about two big ideas. First, the real numbers: every point on the number line is either rational or irrational, and you need to know when to keep an answer exact. Second, scientific notation: a compact way to write and calculate with the very large and very small numbers used in science, technology and measurement.
- Spot big and small numbers together. Phone storage (GB, TB), news figures in billions, the size of a virus. Ask: how would you write that in scientific notation?
- Check the calculator. Make sure your student can enter 3.2 × 10⁻⁵ using the ×10ˣ or EXP key. Many marks are lost to keying errors, not to maths.
- Let the checkers mark. The answer boxes tell your student straight away whether an answer is right.
Work through the examples with a pencil, try every Check box, then do the practice set in section 10 under exam conditions.
01The real number system
Each set of numbers sits inside the next one, like nested boxes.
| Set | What it contains | Examples |
|---|---|---|
| Natural numbers | Counting numbers | 1, 2, 3, 40 |
| Integers | Natural numbers, their negatives and zero | −12, 0, 7 |
| Rational numbers | Anything that can be written as a fraction of two integers. Their decimals terminate or recur. | −⅝, 2.75, 0.3, 9 |
| Irrational numbers | Real numbers that are not rational. Their decimals never end and never repeat. | √2, √20, π |
| Real numbers | All rational and irrational numbers together | Every point on the number line |
√n is rational only when n is a perfect square (1, 4, 9, 16, 25, …). So √36 = 6 is rational, but √37 is irrational. It lies between √36 = 6 and √49 = 7, very close to 6 (√37 ≈ 6.083).
A long calculator decimal does not prove a number is irrational: 1/17 = 0.0588235294117647… looks random, but its block of 16 digits does repeat, so it is rational. What matters is whether the number can be written as a fraction of integers.
02Exact and approximate values
An exact answer has no rounding in it. An approximate answer has been rounded. In Year 9 you are expected to choose the right one.
- A circle of radius 3 cm has circumference 2π × 3 = 6π cm (exact) ≈ 18.85 cm (2 decimal places).
- Its area is π × 3² = 9π cm² (exact) ≈ 28.27 cm².
- A right-angled triangle with shorter sides 2 and 3 has hypotenuse √(4 + 9) = √13 (exact) ≈ 3.61.
For a circle of radius 20 cm, 3.14 × 20² = 1256 cm², but 400π = 1256.64 cm² (2 d.p.). The early rounding has already lost about 0.64 cm², and that error is carried into every later step that uses it. Keep π, surds and full calculator values until the last line, then round once.
Inequalities on the number line. Solve like an equation, but reverse the sign if you multiply or divide by a negative. 2x + 7 < 0 gives x < −3.5: an open circle at −3.5 with an arrow to the left. Use a closed circle for ≤ or ≥.
03Real numbers on the number line
Every real number has an exact position on the number line. You can place some irrational square roots exactly using Pythagoras' theorem and a compass.
- Draw a rectangle a units long and b units high, with one corner at 0 on the number line.
- Its diagonal has length √(a² + b²) by Pythagoras' theorem.
- Put the compass point on 0, open it to the end of the diagonal, and draw an arc down to the line.
So a 3 × 1 rectangle locates √10, and a 2 × 2 square locates √8. Fractions can also be placed exactly: to mark ⅗, divide the interval from 0 to 1 into five equal parts and count three.
04Scientific notation
A number in scientific notation has the form a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
| Ordinary number | Move the decimal point | Scientific notation |
|---|---|---|
| 4 500 000 | 6 places left | 4.5 × 10⁶ |
| 602 000 | 5 places left | 6.02 × 10⁵ |
| 7.3 | no move | 7.3 × 10⁰ |
| 0.081 | 2 places right | 8.1 × 10⁻² |
| 0.000 072 | 5 places right | 7.2 × 10⁻⁵ |
10⁻¹ = 0.1, 10⁻² = 0.01, 10⁻³ = 0.001. A negative power of ten means a number smaller than 1, not a negative number: 7.2 × 10⁻⁵ is a small positive number.
- 45 × 10⁵ is not in scientific notation, because 45 is not between 1 and 10. Rewrite it as 4.5 × 10⁶.
- On a calculator, type 3.2 × 10⁻⁵ as 3.2 [×10ˣ] (−) 5. Don't type × 10 as well.
05Calculating in scientific notation
Use the index laws: multiply the front numbers and add the powers; divide the front numbers and subtract the powers. Then tidy up so the front number is between 1 and 10.
(3 × 10⁴) × (5 × 10⁶) = 15 × 10¹⁰ = 1.5 × 10¹¹
(8.4 × 10⁹) ÷ (2.1 × 10³) = 4 × 10⁶
(2 × 10⁻³) × (7 × 10⁻⁴) = 14 × 10⁻⁷ = 1.4 × 10⁻⁶
(3 × 10⁵)² = 9 × 10¹⁰
Adding and subtracting need the same power of 10 first: 4.2 × 10⁵ + 3 × 10⁴ = 4.2 × 10⁵ + 0.3 × 10⁵ = 4.5 × 10⁵.
06SI prefixes
SI prefixes are shorthand for powers of 10. They work with any unit: metres, grams, seconds, bytes, watts.
| Prefix | Symbol | Power of 10 | Example |
|---|---|---|---|
| tera | T | 10¹² | 1 TB = 10¹² bytes |
| giga | G | 10⁹ | a 3 GHz processor ticks 3 × 10⁹ times a second |
| mega | M | 10⁶ | 5 MW = 5 × 10⁶ watts |
| kilo | k | 10³ | 4 km = 4 × 10³ m |
| milli | m | 10⁻³ | 250 mL = 2.5 × 10⁻¹ L |
| micro | µ | 10⁻⁶ | 8 µm = 8 × 10⁻⁶ m |
| nano | n | 10⁻⁹ | 1 ns = 10⁻⁹ s |
| pico | p | 10⁻¹² | 1 pg = 10⁻¹² g |
Find the gap between the powers. From millimetres (10⁻³) to nanometres (10⁻⁹) the gap is 6, so 1 mm = 10⁶ nm. Going to a smaller unit gives a bigger number.
07Significant figures
Significant figures show how precise a value is. Count from the first non-zero digit.
| Number | Significant figures | Why |
|---|---|---|
| 4 087 | 4 | Zeros between non-zero digits count. |
| 0.003 20 | 3 | Leading zeros don't count; the final zero after the decimal point does. |
| 5.0 × 10³ | 2 | Scientific notation makes the precision clear. |
| 52 000 | 2, 3, 4 or 5 | Ambiguous: write 5.2 × 10⁴ if you mean 2 significant figures. |
Rounding: 0.004 082 7 to 3 significant figures is 0.004 08; 52 718 to 2 significant figures is 53 000 = 5.3 × 10⁴.
Give a final answer to no more significant figures than the least precise measurement you were given, unless the question says otherwise.
08Very large and very small
Scientific notation makes problems with huge or tiny measurements, timescales and intervals manageable.
| Quantity | Value | Source |
|---|---|---|
| Speed of light in a vacuum | 299 792 458 m/s exactly ≈ 3.00 × 10⁸ m/s | SI definition of the metre (BIPM) |
| Astronomical unit (about the average Sun–Earth distance) | 149 597 870 700 m exactly ≈ 1.496 × 10¹¹ m | International Astronomical Union, 2012 |
| Mass of Earth | about 5.97 × 10²⁴ kg | NASA Earth Fact Sheet |
distance = speed × time = 2.997 924 58 × 10⁸ m/s × 1 × 10⁻⁹ s ≈ 0.300 m. Light travels only about 30 cm in a nanosecond, which is why the speed of signals matters inside fast computers.
365 × 24 × 60 × 60 = 31 536 000 s ≈ 3.15 × 10⁷ s (3 significant figures).
09Flashcards
10Exam-style practice
35 marks. Calculator allowed. Show working.
1. Classify each number as rational or irrational: √49, √50, 0.12, π ÷ 2, −7/3. (3)
2. A circle has diameter 10 cm. Find its circumference and area (a) exactly (b) correct to 2 decimal places. (3)
3. Solve 2x + 7 < 0 and describe how to show the solution on a number line. (2)
4. Explain how to locate √10 exactly on a number line using a ruler and compass. (2)
5. Write in scientific notation: (a) 38 000 000 (b) 0.000 006 05 (c) 0.5 (3)
6. Write as ordinary numbers: (a) 2.07 × 10⁵ (b) 9.1 × 10⁻³ (2)
7. Calculate, giving answers in scientific notation: (a) (2.5 × 10⁶) × (4 × 10³) (b) (9.6 × 10⁻²) ÷ (3.2 × 10⁴) (4)
8. A 2 TB hard drive is used to store films of 4 GB each. How many films fit? (1 TB = 10¹² bytes, 1 GB = 10⁹ bytes) (2)
9. Round 0.060 749 to (a) 2 significant figures (b) 3 significant figures. (2)
10. A bacterium is 2 µm long. How many would fit end to end across 1 cm? (2)
11. Taking the Sun–Earth distance as 1.496 × 10¹¹ m and the speed of light as 3.00 × 10⁸ m/s, how long does sunlight take to reach Earth? Give the answer in seconds (3 significant figures) and in minutes and seconds. (3)
12. How many seconds are in a 365-day year? Give your answer in scientific notation correct to 3 significant figures. (2)
13. Calculate the area of a circle of radius 12 cm using π ≈ 3.14 and using the π key. What is the difference, and why does it matter if the result is used in further calculations? (3)
14. Earth's mass is about 5.97 × 10²⁴ kg. About how many times heavier is Earth than a 70 kg person? Answer in scientific notation (3 significant figures). (2)
11Checklist
- sort numbers into natural numbers, integers, rational and irrational numbers?
- explain why √n is irrational unless n is a perfect square?
- give exact answers with π and surds, and round only at the end?
- solve a linear inequality and show it on a number line?
- locate a square root such as √5 on a number line with Pythagoras and a compass?
- convert between ordinary numbers and scientific notation, including negative powers?
- multiply and divide numbers in scientific notation, by hand and on a calculator?
- use SI prefixes from pico to tera and convert between them?
- count and round to significant figures?
- solve problems with very large and very small measurements and times?