Scientific Notation
From the mass of an electron to the distance between galaxies — scientific notation lets us work with numbers of any magnitude.
Printable Worksheets
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Worksheet
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Q1: The mass of Earth is about 6 000 000 000 000 000 000 000 kg. The mass of a hydrogen atom is about 0.000 000 000 000 000 000 000 001 67 kg. Write both numbers as ordinary decimals and describe the problem with comparing them directly.
Q2: A scientist measures two bacteria colonies: Colony A has $3.2 \times 10^7$ cells and Colony B has $8.5 \times 10^6$ cells. Without converting to ordinary numbers, how can you tell which colony is larger? Explain your reasoning.
Learning Intentions
Know
- Scientific notation: $a \times 10^n$ where $1 \leq a < 10$.
- How to convert between scientific notation and decimal form.
Understand
- Why scientific notation is essential for working with extreme scales in science.
- How the exponent indicates the magnitude and direction of the number.
Can Do
- Convert any decimal number to and from scientific notation.
- Compare and order numbers in scientific notation.
- Perform multiplication and division with numbers in scientific notation.
Success Criteria
- I can write any large or small number in scientific notation correctly.
- I can convert from scientific notation back to decimal form.
- I can compare two numbers in scientific notation and determine which is larger.
- I can multiply and divide numbers in scientific notation without a calculator.
Key Terms
Common Mistakes to Avoid
Wrong: “$5.2 \times 10^4 = 52{,}000$”. Multiplying by $10^4$ means moving the decimal point 4 places to the right, not adding a zero.
Right: $5.2 \times 10^4 = 52{,}000$. The decimal moves 4 places right: $5.2 \rightarrow 52 \rightarrow 520 \rightarrow 5{,}200 \rightarrow 52{,}000$.
Wrong: “$340 = 34 \times 10^1$ is scientific notation.” The mantissa must satisfy $1 \leq a < 10$.
Right: $340 = 3.4 \times 10^2$. The decimal goes after the first non-zero digit.
Scientific notation solves a real problem: how do we write and compare numbers that are impossibly large or vanishingly small?
A number is in scientific notation when written as:
For large numbers (positive exponent): move the decimal point to the left until it sits after the first digit.
$6{,}500{,}000 = 6.5 \times 10^6$ (decimal moved 6 places left)
For small numbers (negative exponent): move the decimal point to the right until it sits after the first non-zero digit.
$0.000{,}042 = 4.2 \times 10^{-5}$ (decimal moved 5 places right)
Real-World Anchor: The diameter of a red blood cell is about $7 \times 10^{-6}$ metres. Australia's population is approximately $2.6 \times 10^7$ people. The speed of light is $3 \times 10^8$ m/s. Without scientific notation, comparing these scales would be nearly impossible.
What to write in your book
- Scientific notation has the form $a \times 10^n$ where $1 \leq a < 10$
- Large numbers get positive exponents; small numbers get negative exponents
- Count how many places the decimal moves to get the exponent
Correct! In scientific notation, the mantissa must satisfy $1 \leq a < 10$ and the exponent must be an integer. Only Option A meets both conditions.
Not quite. In scientific notation, the mantissa must satisfy $1 \leq a < 10$ and the exponent must be an integer. Only Option A meets both conditions.
Because scientific notation uses powers of 10, multiplication and division become surprisingly simple using index laws.
Multiplication: Multiply the mantissas, add the exponents.
$(3 \times 10^4) \times (2 \times 10^3) = (3 \times 2) \times 10^{4+3} = 6 \times 10^7$
Division: Divide the mantissas, subtract the exponents.
$(8 \times 10^6) \div (4 \times 10^2) = (8 \div 4) \times 10^{6-2} = 2 \times 10^4$
Remember: After multiplying or dividing, check that your answer is still in proper scientific notation ($1 \leq a < 10$). If the mantissa is 10 or greater, adjust by moving the decimal and changing the exponent.
What to write in your book
- Multiply the mantissas and add the exponents: $(a \times 10^m)(b \times 10^n) = (ab) \times 10^{m+n}$
- Divide the mantissas and subtract the exponents
- Check the final answer is still in proper scientific notation ($1 \leq a < 10$)
Correct! Multiply mantissas: $3 \times 2 = 6$. Add exponents: $10^5 \times 10^3 = 10^8$. Answer: $6 \times 10^8$.
Not quite. Multiply mantissas: $3 \times 2 = 6$. Add exponents: $10^5 \times 10^3 = 10^8$. Answer: $6 \times 10^8$.
Interactive: Scientific Notation Converter
Your Turn
Question 1: Write $45{,}600{,}000$ in scientific notation.
Question 2: Write $0.000{,}000{,}89$ in scientific notation.
Question 3: Calculate $(4 \times 10^5) \times (3 \times 10^3)$, giving your answer in scientific notation.
Revisit Your Thinking
Look back at your Think First answer about the mass of Earth and a hydrogen atom. Write both masses in scientific notation. How many orders of magnitude separate them? (Hint: subtract the exponents.)
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
Which of the following is written correctly in scientific notation?
Write $0.000{,}072$ in scientific notation.
Calculate $(2 \times 10^3) \times (5 \times 10^4)$, giving your answer in scientific notation.
Which is larger: $3.5 \times 10^6$ or $7.2 \times 10^5$?
Write $6.02 \times 10^{-3}$ as a decimal.
Write the following numbers in scientific notation.
(a) $234{,}000$ (1 mark)
(b) $0.000{,}005{,}6$ (1 mark)
(c) $9{,}100{,}000{,}000$ (1 mark)
(a) Calculate $(6 \times 10^4) \times (3 \times 10^7)$, giving your answer in scientific notation. (2 marks)
(b) Calculate $(8 \times 10^9) \div (2 \times 10^5)$, giving your answer in scientific notation. (2 marks)
The mass of the Sun is approximately $1.989 \times 10^{30}$ kg. The mass of Earth is approximately $5.972 \times 10^{24}$ kg.
(a) How many times more massive is the Sun than Earth? Give your answer in scientific notation. (3 marks)
(b) If a new planet were discovered with mass $2.4 \times 10^{25}$ kg, would it be more or less massive than Earth? Justify your answer. (2 marks)
Scientific notation
$a \times 10^n$, $1 \leq a < 10$
Large numbers
Positive exponent — decimal moves left
Small numbers
Negative exponent — decimal moves right
Multiply
Multiply $a$ values, add exponents
Divide
Divide $a$ values, subtract exponents
Compare
Larger exponent means larger number
Real-Life Link
Australian scientists use scientific notation daily. CSIRO researchers measuring nanoparticles work at scales of $10^{-9}$ metres. Astronomers at the Australian Square Kilometre Array Pathfinder telescope detect radio waves from objects billions of light-years away — distances around $10^{25}$ metres. Without scientific notation, writing these numbers would require lines of zeros, and comparing them would be almost impossible. Every calculator and computer uses scientific notation internally to handle these extreme ranges.
Game Time!
Test your scientific notation skills in an interactive challenge.
Play Scientific Notation Challenge