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# Year 9 Maths: Real Numbers and Scientific Notation
- URL: https://same-page.ghost.io/year-9-maths-real-numbers-scientific-notation/
- Published: 2026-11-02T09:43:34.000Z
- Updated: 2026-11-03T07:20:05.000Z
- Description: 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.
- Author: Same Page
- Tags: Year 9, Maths, Free sample

Same Page · Year 9 Maths

# 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).

[Start here](#start)[The real number system](#real)[Exact and approximate values](#exact)[Real numbers on the number line](#line)[Scientific notation](#sci)[Calculating in scientific notation](#calc)[SI prefixes](#prefixes)[Significant figures](#sigfig)[Very large and very small](#scale)[Flashcards](#cards)[Exam-style practice](#practice)[Checklist](#check)

## 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.

For parents 
1. **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?
2. **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.
3. **Let the checkers mark.** The answer boxes tell your student straight away whether an answer is right.

How to use this guide

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 |

Which square roots are irrational?

√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).

Watch out

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.

Why exact values matter

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 ≥.

Find the area of a circle of radius 5 cm, correct to 2 decimal places.

Solve 3x − 4 > 11\. Type the boundary value of x.

## 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.

01234√2 ≈ 1.414√5 ≈ 2.23611√2√5Square 1 × 1: diagonal² = 1² + 1² = 2Rectangle 2 × 1: diagonal² = 2² + 1² = 5Swing each diagonal down with a compass.

The diagonal of a 1 × 1 square is √2; the diagonal of a 2 × 1 rectangle is √5\. A compass arc carries each length down to the number line. Original Same Page diagram.

1. Draw a rectangle a units long and b units high, with one corner at 0 on the number line.
2. Its diagonal has length √(a² + b²) by Pythagoras' theorem.
3. 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.

A rectangle 3 units by 2 units is drawn on the number line. Its diagonal is √n. What is n?

## 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          |

Negative powers

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.

Common mistakes
- 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.

Write 0.000 36 in scientific notation. Type the power of 10.

## 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.

Worked examples 

(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.

Calculate (6 × 10⁵) × (4 × 10⁻²) and write it in scientific notation as a × 10ⁿ. Type the value of a.

## 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                                 |

Converting between prefixes

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.

How many nanometres are in one millimetre?

## 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.

Exam tip

Give a final answer to no more significant figures than the least precise measurement you were given, unless the question says otherwise.

Round 7.0496 to 3 significant figures.

## 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                  |

Worked example: how far does light go in a nanosecond? 

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.

Worked example: a year in seconds 

365 × 24 × 60 × 60 = 31 536 000 s ≈ **3.15 × 10⁷ s** (3 significant figures).

Using the exact values in the table, how long does light take to travel one astronomical unit? Answer to the nearest second.

## 09Flashcards

## 10Exam-style practice

35 marks. Calculator allowed. Show working. Show / hide all answers

1\. Classify each number as rational or irrational: √49, √50, 0.12, π ÷ 2, −7/3\. (3)

√49 = 7 rational; √50 irrational (50 is not a perfect square); 0.12 \= 12/99 = 4/33 rational; π ÷ 2 irrational; −7/3 rational.

2\. A circle has diameter 10 cm. Find its circumference and area (a) exactly (b) correct to 2 decimal places. (3)

(a) C = 10π cm, A = π × 5² = 25π cm². (b) C ≈ **31.42 cm**, A ≈ **78.54 cm²**.

3\. Solve 2x + 7 < 0 and describe how to show the solution on a number line. (2)

2x < −7, so **x < −3.5**. Draw an open circle at −3.5 (not included) and an arrow to the left.

4\. Explain how to locate √10 exactly on a number line using a ruler and compass. (2)

Draw a 3 × 1 rectangle with one corner at 0 and its long side on the number line. Its diagonal is √(3² + 1²) = √10\. With the compass point at 0, swing the diagonal down to the line: the arc meets it at √10 (≈ 3.162).

5\. Write in scientific notation: (a) 38 000 000 (b) 0.000 006 05 (c) 0.5 (3)

(a) 3.8 × 10⁷ (b) 6.05 × 10⁻⁶ (c) 5 × 10⁻¹

6\. Write as ordinary numbers: (a) 2.07 × 10⁵ (b) 9.1 × 10⁻³ (2)

(a) 207 000 (b) 0.0091

7\. Calculate, giving answers in scientific notation: (a) (2.5 × 10⁶) × (4 × 10³) (b) (9.6 × 10⁻²) ÷ (3.2 × 10⁴) (4)

(a) 2.5 × 4 = 10, and 10⁶ × 10³ = 10⁹, so 10 × 10⁹ = **1 × 10¹⁰**. (b) 9.6 ÷ 3.2 = 3 and 10⁻² ÷ 10⁴ = 10⁻⁶, so **3 × 10⁻⁶**.

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)

(2 × 10¹²) ÷ (4 × 10⁹) = 0.5 × 10³ = **500 films**.

9\. Round 0.060 749 to (a) 2 significant figures (b) 3 significant figures. (2)

(a) 0.061 (b) 0.0607

10\. A bacterium is 2 µm long. How many would fit end to end across 1 cm? (2)

1 cm = 10⁻² m and 2 µm = 2 × 10⁻⁶ m. (10⁻²) ÷ (2 × 10⁻⁶) = 0.5 × 10⁴ = **5000**.

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)

time = distance ÷ speed = 1.496 × 10¹¹ ÷ 3.00 × 10⁸ ≈ 498.7 s ≈ **499 s**, which is about **8 min 19 s**.

12\. How many seconds are in a 365-day year? Give your answer in scientific notation correct to 3 significant figures. (2)

365 × 24 × 60 × 60 = 31 536 000 s ≈ **3.15 × 10⁷ s**.

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)

3.14 × 144 = 452.16 cm²; 144π ≈ 452.39 cm². Difference ≈ 0.23 cm². Rounding early builds in an error that is carried, and can grow, through every later step. Keep exact values (144π) until the final answer.

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)

5.97 × 10²⁴ ÷ 70 ≈ **8.53 × 10²²** times.

## 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?

**Sources and credits.** Speed of light and SI prefixes: BIPM, *The International System of Units (SI Brochure)*, 9th edition. Astronomical unit: International Astronomical Union Resolution B2 (2012). Mass of Earth: NASA Goddard Space Flight Center, *Earth Fact Sheet*. Curriculum alignment: content descriptions from the Australian Curriculum Version 9.0 © Australian Curriculum, Assessment and Reporting Authority (ACARA), licensed under CC BY 4.0 (creativecommons.org/licenses/by/4.0). Topic codes from the Victorian Curriculum F–10 © VCAA are cited for alignment only. ACARA and VCAA do not endorse this guide. All explanations, diagrams and questions are original Same Page material. Same Page is a study aid: check your school's course outline. Spotted an error? Email atlasknoxbuilds@gmail.com and we'll fix it.