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National 5 Maths
Scientific Notation

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Course content

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

Textbook page references

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.

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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.

N5 Maths practice papers

Non-calculator papers and solutions 
Calculator papers and solutions 

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.

Example 10 (calculator)

SQA National 5 Maths 2024 P2 Q2

An ant colony occupies an area of 250 hectares. There is an average of 1.22 \(\times\) 106 ants per hectare. Calculate the number of ants in the colony. Give your answer in scientific notation.

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Maths.scot worksheet

• Scientific notation worksheet
• Answer sheet
• See all National 5 Maths worksheets

Past paper questions

• All past paper questions by topic
• 2015 Paper 2 Q6 (with rounding)
• 2016 Paper 2 Q2
• 2017 Specimen Paper 2 Q2
• 2018 P2 Q11 (with percentages)
• 2019 Paper 2 Q4
• 2021 Paper 2 Q2
• 2023 Paper 2 Q2
• 2024 Paper 2 Q2

Other great resources

Videos - Maths180.com
Scroll down, below the indices videos.
Video - Math Antics
American, but relevant.
Video - YouKenMaths
Standard Grade 2009 Paper 2 Q1
Video - Larbert High School
Notes - Maths4Scotland
Lesson notes - Maths 777
1. Converting both ways
2. Calculator use
Practice questions - Maths Hunter
Worksheet - D R Turnbull
Exercises - Larkhall Academy
Pages 17-20 Ex 1-3 (no answers)

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