Course content

• Convert to and from scientific notation (positive or negative powers)
• Calculations using scientific notation.

Key ideas

• "Scientific notation" is also called "standard form".
• We use powers of 10 to express very large or very small numbers in scientific notation.
• Multiplying by a positive power of 10 moves the decimal point to the right.
• Dividing by a positive power of 10 moves the decimal point to the left.

Textbook page references

• TeeJay Maths N5 Book 4+ pp.66-71
• Zeta National 5+ Maths pp.23-26
• Leckie National 5 Maths pp.22-24

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Powers of 10

10¹ = 10
10² = 100
10³ = 1000
10⁴ = 10 000
10⁵ = 100 000
etc.

Large numbers

Multiplication by a positive power of 10 moves the decimal point to the right. For example: 1.23 \(\times\) 10⁴ = 12 300.

Small numbers

Division by a positive power of 10 moves the decimal point to the left. For example: 9.87 ÷ 10³ = 0.00987.

Dividing by 10³ is the same as multiplying by 10^–3. Please study the index laws if this isn't obvious to you.

So we can also write 0.00987 as 9.87 \(\times\) 10^–3. Scientific notation always uses a \(\times\) sign, not \(\div.\)

Note

In \(a\times 10^n\), the number \(a\) is called the "mantissa" and \(n\) is the "exponent" (or index, or power).

We must always make sure that 1 ≤ \(a\) < 10, so that there is a single non-zero digit before the decimal point.

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Example 1 (non-calculator)

Express 51 600 000 in scientific notation.

Example 2 (non-calculator)

Express 0.00007081 in standard form.

Example 3 (non-calculator)

Evaluate (4 \(\times\) 10¹⁴) ÷ (8 \(\times\) 10⁶). Express your answer in scientific notation.

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Example 4 (calculator)

In the year 2015, there were estimated to be 7.34 \(\times\) 10⁹ people and 3.04 \(\times\) 10¹² trees in the world. Calculate the number of trees per person in 2015. Express your answer correct to 2 significant figures.

Example 5 (calculator)

The average adult has approximately 7.6 \(\times\) 10⁶ white blood cells per millilitre of blood. On average, an adult has 5.2 litres of blood. Calculate the average number of white blood cells in an adult body. Express your answer in scientific notation, correct to 1 significant figure.

Example 6 (calculator)

SQA National 5 Maths 2016 P2 Q2

A pollen sample weighs 12 grams and contains 1.5 \(\times\) 10^–9 pollen grains. Calculate the weight of one pollen grain in grams. Give your answer in scientific notation.

Books for Maths teachers

Example 7 (calculator)

SQA National 5 Maths 2019 P2 Q4

A sesame seed weighs 3.6 \(\times\) 10^–6 kilograms. The weight of a poppy seed is 8% of the weight of a sesame seed. Calculate the weight of a poppy seed in kilograms. Give your answer in scientific notation.

Example 8 (calculator)

SQA National 5 Maths 2021 P2 Q2

Light travels at 3 \(\times\) 10⁸ metres per second. A star is 4.2 \(\times\) 10¹⁷ metres away from Earth. Calculate the number of seconds it takes for light from this star to reach Earth. Give your answer in scientific notation.

Example 9 (calculator)

SQA National 5 Maths 2023 P2 Q2

The mass of a helium atom is 6.64 \(\times\) 10^–24 grams. A flask contains 300 grams of helium. Calculate the number of helium atoms in this flask. Give your answer in scientific notation, correct to 3 significant figures.

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