Grade 8 · Grade 8: Real Numbers, Exponents and Scientific Notation · Real Numbers, Exponents and Scale
Scientific Notation and Orders of Magnitude
Compress very large and very small numbers into powers of ten and compute with them.
Learning objectives
- Write numbers in scientific notation.
- Compare quantities using orders of magnitude.
- Multiply and divide numbers in scientific notation.
- Choose units of appropriate size for reporting results.
AERO Mathematics alignment
AERO.M8.NS.5
Use numbers expressed as a single digit times an integer power of ten to estimate very large or very small quantities and to express how many times as much one is than the other.
AERO.M8.NS.6
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used, and choose units of appropriate size.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
The Sun is about 150 000 000 km away. A virus is about 0.00000012 m across.
Think about it
How could you compare these two numbers without counting zeros?
Hint: Write each as a single digit times a power of ten.
EXPLAIN
Powers of ten
Scientific notation writes a number as a × 10ⁿ where 1 ≤ a < 10 and n is an integer. So 150 000 000 = 1.5 × 10⁸ and 0.00000012 = 1.2 × 10⁻⁷.
| Ordinary form | Scientific notation | Meaning |
|---|---|---|
| 6 300 000 | 6.3 × 10⁶ | Millions |
| 0.00045 | 4.5 × 10⁻⁴ | Ten-thousandths |
| 93 | 9.3 × 10¹ | Tens |
Computing
Multiply the digits and add the exponents: (3 × 10⁵)(2 × 10⁴) = 6 × 10⁹. Divide the digits and subtract the exponents. If the digit part falls outside 1 to 10, adjust the exponent: 12 × 10³ becomes 1.2 × 10⁴.
Orders of magnitude
To find how many times larger one number is than another, divide: 4 × 10⁸ ÷ 8 × 10⁵ = 0.5 × 10³ = 500 times larger.
Where this is used
Astronomers, microbiologists, data engineers and economists all work at scales where ordinary notation becomes unreadable.
INVESTIGATION
Try it yourself
Find the populations of five countries and write each in scientific notation. Rank them, then state how many times larger the largest is than the smallest using a single power of ten.
Watch
Scientific Notation
Math Antics · 10:00
Shows how powers of ten tame both huge and tiny numbers.
Before you watch: How would you write 0.00042 using a power of ten?
- Write 6 300 000 in scientific notation.
- How many times larger is 5 × 10⁹ than 5 × 10⁴?
Interactive simulation · PhET
Number Play
Build large numbers, then rewrite each one as a single digit times a power of ten.
While you explore
- How many times larger is 4 × 10⁸ than 8 × 10⁵?
- Why is 12 × 10³ not correct scientific notation?
Key vocabulary
- Order of magnitude
- A rough size in powers of ten.
- Scientific notation
- Writing a number as one digit, a decimal part and a power of ten.
Practice questions
0/1 correct
Level 1 · Criterion A
Write 0.00082 in scientific notation.
MYP criterion tasks
Level 2 · Criterion D
How many times larger is 6 × 10⁹ than 3 × 10⁶? Show your working.
Reflect & track
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