Lesson 15 ~25 min Unit 1 · Index Laws +85 XP

Introduction to Scientific Notation

Write huge and tiny numbers as $a \times 10^n$. Recognise the form, read the power, and see why index laws are the engine behind it.

Today's hook: The Sun is about $150\,000\,000\,000$ m away from Earth. Writing all those zeros gets messy — what if we could say the same thing as $1.5 \times 10^{11}$?

0/5QUESTS

Printable Worksheets

Print or save as PDF — or build a custom worksheet from any module's questions.

Build Apply Master Build custom

Think First

warm-up

How would you describe the number $300\,000$ using a power of $10$? What about $0.0004$? Try splitting each into “a number between $1$ and $10$” multiplied by some $10^n$.

Record in your workbook.

The Big Idea

+5 XP

Scientific notation writes any number as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer. The $a$ part captures the digits; the $10^n$ part captures the size.

The coefficient $a$ sits between $1$ and $10$ (never less than $1$, never $10$ or more). The power $n$ is a positive integer for big numbers, a negative integer for small numbers, and $0$ for numbers between $1$ and $10$.

$150\,000\,000\,000 = 1.5 \times 10^{11}$

Coefficient rule

$1 \leq a < 10$ — one non-zero digit before the point.

Big $\to$ positive $n$

$10^3 = 1000$, $10^6 = $ a million.

Small $\to$ negative $n$

$10^{-3} = 0.001$, $10^{-6} = $ a millionth.

What You'll Master

objectives

Know

Form $a \times 10^n$ with $1 \leq a < 10$
$10^1, 10^2, \ldots$ count zeros for big numbers
$10^{-1}, 10^{-2}, \ldots$ count place-shifts for small numbers

Understand

Why scientific notation makes huge/tiny numbers usable
How the power tells you the size (order of magnitude)
Why index laws are the maths behind the notation

Can Do

Read $3.2 \times 10^5 = 320\,000$
Read $4.7 \times 10^{-4} = 0.000\,47$
Decide if a number is in valid scientific notation

Words You Need

vocabulary

Scientific notationForm $a \times 10^n$, $1 \leq a < 10$, $n$ an integer.

Coefficient $a$The digits at the front, with one non-zero digit before the decimal point.

Power of ten $10^n$Tells you the order of magnitude (how big or small).

Order of magnitudeWhich power of $10$ a number sits near.

Standard formAnother name for scientific notation.

Significant figuresThe meaningful digits in $a$ — everything except the trailing zeros that just place the decimal.

Spot the Trap

heads-up

Wrong: “$15 \times 10^{10}$ is scientific notation.” The coefficient $15$ is not between $1$ and $10$.

Right: $1.5 \times 10^{11}$ — one digit before the decimal, $1.5$ is between $1$ and $10$.

Wrong: “$0.3 \times 10^4$ is scientific notation.” The coefficient is less than $1$.

Right: $3 \times 10^3$ — coefficient is $3$, which is in the valid range.

Why we need it

+5 XP

Numbers in science span an enormous range. Scientific notation keeps them readable, comparable, and easy to operate on with index laws.

Mass of the Earth $\approx 6 \times 10^{24}$ kg. Mass of a hydrogen atom $\approx 1.67 \times 10^{-27}$ kg. The notation lets you compare them at a glance — the difference is about $51$ orders of magnitude.

$6 \times 10^{24}$ vs $1.67 \times 10^{-27}$

Reading the power

+5 XP

The integer $n$ in $10^n$ is just an index law in disguise. Each $\times 10$ shifts the decimal one place to the right; each $\div 10 = 10^{-1}$ shifts it left.

$3.2 \times 10^5$ means “move the decimal $5$ places right” $\to 320\,000$. $4.7 \times 10^{-4}$ means “move it $4$ places left” $\to 0.000\,47$. Index laws like $10^5 \cdot 10^{-4} = 10^1$ are why it all works.

$3.2 \times 10^5 = 320\,000$

Watch Me Solve It · 3 examples

Watch Me Solve It · Read a positive power

+15 XP per step

PROBLEM

Write $3.2 \times 10^5$ as an ordinary number.

1
Note the power
$n = 5$, so move the decimal $5$ places right.
2
Shift the decimal
$3.2 \to 32 \to 320 \to 3200 \to 32\,000 \to 320\,000$
3
State the answer
$320\,000$

Answer$320\,000$

Watch Me Solve It · Read a negative power

+15 XP per step

PROBLEM

Write $4.7 \times 10^{-4}$ as an ordinary number.

1
Note the power
$n = -4$, so move the decimal $4$ places left.
2
Shift the decimal
$4.7 \to 0.47 \to 0.047 \to 0.0047 \to 0.000\,47$
3
State the answer
$0.000\,47$

Answer$0.000\,47$

Watch Me Solve It · Decide if it's in standard form

+15 XP per step

PROBLEM

Which of $42 \times 10^3$, $0.6 \times 10^7$, $4.2 \times 10^4$, $6 \times 10^6$ are correctly written in scientific notation?

1
Check each coefficient against $1 \leq a < 10$
$42 \times 10^3$: $a = 42$ — too big. $0.6 \times 10^7$: $a = 0.6$ — too small.
2
Check the valid ones
$4.2 \times 10^4$: $1 \leq 4.2 < 10$ ✓. $6 \times 10^6$: $1 \leq 6 < 10$ ✓.
3
List the valid ones
$4.2 \times 10^4$ and $6 \times 10^6$.

Answer$4.2 \times 10^4$ and $6 \times 10^6$

Common Pitfalls

heads-up

Coefficient outside $1 \leq a < 10$

$42 \times 10^3$ has the right value but the wrong form.

Fix: Shift the decimal in the coefficient and adjust the power: $4.2 \times 10^4$.

Confusing direction of shift

Positive $n$ shifts the decimal right (bigger); negative $n$ shifts it left (smaller).

Fix: Match the sign of $n$ to the direction.

Miscounting the power

$10^5$ is $100\,000$ (five zeros), not $1\,000\,000$.

Fix: $10^n$ has $n$ zeros for positive $n$.

Copy Into Your Books

The form

$a \times 10^n$, $1 \leq a < 10$, $n \in \mathbb{Z}$

Read it

$3.2 \times 10^5 = 320\,000$
$4.7 \times 10^{-4} = 0.000\,47$

Sign of $n$

$n > 0$ → big
$n < 0$ → small
$n = 0$ → between $1$ and $10$

Examples

Earth's mass $\approx 6 \times 10^{24}$ kg
Atom mass $\approx 1.67 \times 10^{-27}$ kg

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Recognise and read drills

4 problems

Read each, then decide what changes.

1 Write $2.5 \times 10^4$ as an ordinary number.
Shift $4$ places right.$25\,000$
2 Write $9 \times 10^{-3}$ as an ordinary number.
Shift $3$ places left.$0.009$
3 Is $0.5 \times 10^6$ in scientific notation? Why or why not?
No — the coefficient $0.5$ is less than $1$. Correct form: $5 \times 10^5$.No — rewrite as $5 \times 10^5$
4 Which is bigger: $7 \times 10^5$ or $3 \times 10^6$? Justify with the power.
The larger power wins: $10^6 > 10^5$ by a factor of $10$, and $3 \times 10^6 = 3\,000\,000$ vs $7 \times 10^5 = 700\,000$.$3 \times 10^6$ is bigger

Complete in your workbook.

Quick Check · 5 questions

Which is in scientific notation?

+10 XP

$3.2 \times 10^5$ equals:

+10 XP

$4.7 \times 10^{-4}$ equals:

+10 XP

Which is bigger: $7 \times 10^5$ or $3 \times 10^6$?

+10 XP

A number is in scientific notation if it is written as $a \times 10^n$ where:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyEasy3 MARKS

Q6. Write each as an ordinary number: (a) $2.5 \times 10^4$, (b) $9 \times 10^{-3}$, (c) $6.02 \times 10^7$.

Answer in your workbook.

UnderstandMedium3 MARKS

Q7. Each is NOT in scientific notation. Explain why, then rewrite each correctly. (a) $25 \times 10^4$, (b) $0.4 \times 10^6$, (c) $10 \times 10^3$.

Answer in your workbook.

ReasonHard3 MARKS

Q8. The distance from Earth to the Sun is about $1.5 \times 10^{11}$ m. The diameter of a hydrogen atom is about $1 \times 10^{-10}$ m. (a) How many orders of magnitude apart are these? (b) Why is scientific notation more useful than ordinary numbers here?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $4.2 \times 10^4$.

2. C — $320\,000$.

3. A — $0.000\,47$.

4. D — $3 \times 10^6$.

5. B — $1 \leq a < 10$ and $n$ any integer.

Show Your Working Model Answers

Q6 (3 marks): (a) $2.5 \times 10^4 = 25\,000$ [1]. (b) $9 \times 10^{-3} = 0.009$ [1]. (c) $6.02 \times 10^7 = 60\,200\,000$ [1].

Q7 (3 marks): (a) Coefficient $25$ is not in $[1, 10)$; shift one place left $\to 2.5 \times 10^5$ [1]. (b) Coefficient $0.4 < 1$; shift one place right $\to 4 \times 10^5$ [1]. (c) Coefficient $10$ is not less than $10$; shift one place left $\to 1 \times 10^4$ [1].

Q8 (3 marks): (a) Difference in powers: $11 - (-10) = 21$ orders of magnitude [1]. (b) Ordinary form would need $11$ digits for the Sun distance and $10$ leading zeros for the atom; scientific notation captures both with one digit and a power, and lets us compare and operate using index laws [2].

Stretch Challenge · +25 XP, +10 coins

Reading the Magnitude

The mass of the Earth is about $6 \times 10^{24}$ kg. The mass of the Moon is about $7.3 \times 10^{22}$ kg. (a) Which is bigger and by how many orders of magnitude? (b) Roughly how many times the Moon's mass is the Earth's? Justify using the powers.

Reveal solution

(a) Earth is bigger; the powers differ by $24 - 22 = 2$ orders of magnitude, so Earth is roughly $100$ times more massive in pure-power terms. (b) $\dfrac{6 \times 10^{24}}{7.3 \times 10^{22}} = \dfrac{6}{7.3} \times 10^{2} \approx 0.82 \times 100 \approx 82$ times the Moon's mass. The accepted value is about $81$, matching nicely.

Quick Review

The form

$a \times 10^n$, $1 \leq a < 10$

Big

$3.2 \times 10^5 = 320\,000$

Small

$4.7 \times 10^{-4} = 0.000\,47$

Sign of $n$

$+$ big, $-$ small, $0$ in $[1, 10)$

Order of magnitude

The power $n$ tells you the size

Why bother

Compare, compute, communicate

Your Badges

0 of 6

First Steps

3-Day Streak

3 in a Row

Lesson Ace

Stretch Seeker

Daily Warrior

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page