Lesson 3 of 4

Scientific notation.

About 20 minutes · 5 practice questions

How to write very large and very small numbers with powers of ten: what the power counts, what a negative power means, and how to tidy the number in front.

Step 1 of 5 · Powers of ten

Step 1 of 5

Powers of ten

In the last Unit you saw that $10^2 = 100$ ⁠ and $10^3 = 1000$ ⁠: the power counts how many times you multiply by $10$ ⁠, which is also the number of zeros after the $1$ ⁠. On the board’s line each tick is $10$ ⁠ times the one to its left, so an equal step along the line is a multiplication, not an addition.

Move Power of ten until the number reaches the ring at a million, then press Check.

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Step 2 of 5

A large number in scientific notation

In scientific notation you write every number as a number between $1$ ⁠ and $10$ ⁠ times a power of $10$ ⁠. The Sun is 1,400,000 km across, that is $1.4$ ⁠ times a power of $10$ ⁠. The power says how many places the decimal point moves on the way from $1.4$ ⁠ to $1{,}400{,}000$ ⁠.

Answer, then choose Written both ways under Look inside to check.

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Step 3 of 5

A negative power

Each tick to the left on the line divides by $10$ ⁠: $10^0 = 1$ ⁠, $10^{-1} = \frac{1}{10} = 0.1$ ⁠, $10^{-2} = \frac{1}{100} = 0.01$ ⁠. That is how the line goes on below $1$ ⁠ without ever reaching zero.

Predict what $10^{-3}$ ⁠ is.

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Step 4 of 5

A small number in scientific notation

A grain of fine sand is about 0.00025 m across. To write it in scientific notation you move the point to the right until $2.5$ ⁠ is left, and count the steps. Each step is a multiplication by $10$ ⁠, so the original number is $2.5$ ⁠ divided by $10$ ⁠ to the power of the number of steps.

Answer, then choose Written both ways under Look inside to check.

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Step 5 of 5

Tidying the number in front

$32 \times 10^4$ ⁠ equals $320{,}000$ ⁠, but it is not in scientific notation, because $32$ ⁠ is not between $1$ ⁠ and $10$ ⁠. To fix it, write $32$ ⁠ as $3.2 \times 10$ ⁠: the number in front gets $10$ ⁠ times smaller, and the power of ten takes one more multiplication by $10$ ⁠, so the value does not change.

Answer, then choose Written both ways under Look inside to check.

To the question

Interactive scene

Loading the interactive version…

Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

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Practice · Reach the target

Number in front is locked at $1$ ⁠. Move Power of ten until the number reaches a million, $1{,}000{,}000$ ⁠, at the ring, then press Check.

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Practice · Calculate

The Sun is 1,400,000 km across. Write that as $1.4 \times 10^n$ ⁠. What is $n$ ⁠?

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Practice · Predict, then look

What is $10^{-3}$ ⁠?

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Practice · Calculate

A grain of fine sand is about 0.00025 m across. Write the number as $2.5 \times 10^n$ ⁠. What is $n$ ⁠?

To the experiment

Practice · Calculate

$32 \times 10^4$ ⁠ is not in scientific notation. Write it as $3.2 \times 10^n$ ⁠. What is $n$ ⁠?