Writing Numbers in Scientific Notation
Convert any number — big or tiny — into $a \times 10^n$. Count places, pick the sign of $n$, check the coefficient.
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To write $93\,000\,000$ in scientific notation, where does the decimal point start? Where should it land? How many places did it move?
To convert a number to scientific notation, move the decimal so it sits after the first non-zero digit. The number of places you moved becomes the power $n$ — negative if you moved right (small number), positive if you moved left (big number).
Big number: move decimal left $\to$ positive $n$. Small number: move decimal right $\to$ negative $n$. The count of places is the size of $n$.
Know
- Big numbers $\Rightarrow$ positive $n$
- Small numbers ($< 1$) $\Rightarrow$ negative $n$
- Numbers in $[1, 10) \Rightarrow n = 0$
Understand
- Why each place-shift corresponds to one factor of $10$
- Why the sign of $n$ is opposite to the direction the decimal moved
- How leading zeros set the negative power
Can Do
- Convert $93\,000\,000$ to $9.3 \times 10^7$
- Convert $0.000\,007$ to $7 \times 10^{-6}$
- Choose the right power for any size of number
Wrong: “$0.000\,7 = 7 \times 10^{4}$” — sign of the power is back-to-front.
Right: $0.000\,7 = 7 \times 10^{-4}$ — the number is small, so the power is negative.
Wrong: “$45\,000 = 45 \times 10^3$” — the coefficient is not between $1$ and $10$.
Right: $45\,000 = 4.5 \times 10^4$ — shift one more place left, bump the power up by one.
Find the decimal (it sits at the right end for whole numbers). Slide it left until exactly one non-zero digit is in front. Count places; that's $n$.
$93\,000\,000.$ → slide left to $9.3$, $7$ places. So $93\,000\,000 = 9.3 \times 10^7$. The Sun is about $1.5 \times 10^{11}$ m away.
The decimal is already in front. Slide it right until it sits after the first non-zero digit. Count places; $n$ is negative.
$0.000\,007 \to 7.0$, slid $6$ places right, so $0.000\,007 = 7 \times 10^{-6}$. A hydrogen atom is roughly $5 \times 10^{-11}$ m across — the negative power tells you instantly that it's tiny.
Watch Me Solve It · 3 examples
- 1Find the decimal$93\,000\,000.$ — sitting at the right.
- 2Slide left to the first non-zero digit$9.3\,000\,000$ — moved $7$ places.
- 3Write as $a \times 10^n$$9.3 \times 10^{7}$
- 1Identify the first non-zero digitIt is $7$, in the $10^{-6}$ place.
- 2Slide the decimal right until it sits after the $7$$0.000\,007 \to 7.0$ — moved $6$ places right.
- 3Use a negative power because it's a small number$7 \times 10^{-6}$
- 1First non-zero digit$4$, in the $10^{-4}$ place.
- 2Slide the decimal right past the $4$$0.000\,486 \to 4.86$ — moved $4$ places right.
- 3Write with a negative power$4.86 \times 10^{-4}$
Common Pitfalls
Method
- Find first non-zero digit
- Slide decimal until it's right after that digit
- Count places, set sign of $n$
Big examples
- $93\,000\,000 = 9.3 \times 10^7$
- $45\,000 = 4.5 \times 10^4$
Small examples
- $0.000\,007 = 7 \times 10^{-6}$
- $0.000\,486 = 4.86 \times 10^{-4}$
Sign reminder
- Big $\to +n$ (decimal moves left)
- Small $\to -n$ (decimal moves right)
How are you completing this lesson?
Brain Trainer · 4 problems
Mix of big and small. Watch the sign.
1 Write $45\,000$ in scientific notation.
Slide $4$ places left from $45\,000.$ to $4.5$.$4.5 \times 10^4$2 Write $0.000\,082$ in scientific notation.
Slide $5$ places right to $8.2$.$8.2 \times 10^{-5}$3 Write $1\,200\,000$ in scientific notation.
Slide $6$ places left to $1.2$.$1.2 \times 10^6$4 A particular bacterium is $0.000\,002$ m long. Write that length in scientific notation, in metres.
Slide $6$ places right to $2$.$2 \times 10^{-6}$ m
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Write each in scientific notation: (a) $5\,200$, (b) $1\,200\,000$, (c) $607\,000\,000$.
Q7. Write each in scientific notation: (a) $0.000\,082$, (b) $0.000\,000\,3$, (c) $0.005\,06$.
Q8. The diameter of a human hair is about $0.000\,07$ m and the radius of Earth is about $6\,370\,000$ m. (a) Write each in scientific notation. (b) State the order of magnitude difference between them. (c) Roughly how many hair-widths fit across Earth's radius?
Quick Check
1. C — $9.3 \times 10^7$.
2. A — $7 \times 10^{-6}$.
3. B — $4.5 \times 10^4$.
4. D — $4.86 \times 10^{-4}$.
5. C — $1.5 \times 10^{11}$.
Show Your Working Model Answers
Q6 (3 marks): (a) $5\,200 = 5.2 \times 10^3$ [1]. (b) $1\,200\,000 = 1.2 \times 10^6$ [1]. (c) $607\,000\,000 = 6.07 \times 10^8$ [1].
Q7 (3 marks): (a) Slide $5$ right: $8.2 \times 10^{-5}$ [1]. (b) Slide $7$ right: $3 \times 10^{-7}$ [1]. (c) Slide $3$ right: $5.06 \times 10^{-3}$ [1].
Q8 (3 marks): (a) Hair $7 \times 10^{-5}$ m; Earth radius $6.37 \times 10^6$ m [1]. (b) Powers $6 - (-5) = 11$ orders of magnitude apart [1]. (c) $\dfrac{6.37 \times 10^6}{7 \times 10^{-5}} = \dfrac{6.37}{7} \times 10^{11} \approx 0.91 \times 10^{11} \approx 9.1 \times 10^{10}$ hair-widths [1].
Fixing Broken Notation
Each of these is NOT correctly in scientific notation. For each, explain what's wrong and rewrite it correctly: (a) $124 \times 10^5$, (b) $0.7 \times 10^{-3}$, (c) $10 \times 10^4$, (d) $3.2 \times 10^{2.5}$.
Reveal solution
(a) Coefficient $124$ is not in $[1, 10)$; shift $2$ left and bump power by $2$: $1.24 \times 10^7$. (b) Coefficient $0.7 < 1$; shift one right and lower power by $1$: $7 \times 10^{-4}$. (c) Coefficient must be $< 10$; shift one left, bump power by $1$: $1 \times 10^5$. (d) The exponent must be an integer; $10^{2.5}$ is not scientific notation at all.
Method
Decimal $\to$ first non-zero digit
Big $\to$ left
$93\,000\,000 = 9.3 \times 10^7$
Small $\to$ right
$0.000\,007 = 7 \times 10^{-6}$
Coefficient
$1 \leq a < 10$
Exponent
Must be an integer
Sign check
Sign opposite to decimal direction
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