Scientific Notation

The Sun is about $150\,000\,000\,000$ m from Earth. A red blood cell is about $0.000\,007\,5$ m wide. Numbers like these are hard to read and easy to mistype, because of all the zeros. Scientific notation writes them in a short, standard way using powers of $10$, so scientists, engineers and calculators can work with them easily. It all comes from a simple pattern in the powers of $10$.

Key ideas

Patterns in powers of 10

Start with $10^3 = 1000$ and keep dividing by $10$. Each time, the exponent goes down by $1$:

Power	$10^3$	$10^2$	$10^1$	$10^0$	$10^{-1}$	$10^{-2}$	$10^{-3}$
Value	$1000$	$100$	$10$	$1$	$0.1$	$0.01$	$0.001$
As a fraction					$\dfrac{1}{10}$	$\dfrac{1}{100}$	$\dfrac{1}{1000}$

Following the pattern, $10^0 = 1$, and a negative exponent gives a decimal between $0$ and $1$:

$$10^{-n} = \frac{1}{10^n}$$

A neat shortcut: for a positive exponent, $10^n$ is a $1$ followed by $n$ zeros. For a negative exponent, $10^{-n}$ has its $1$ in the $n$th place after the decimal point. For example, $10^{-4} = 0.0001$.

How the sign and size of the exponent matter

For a base bigger than $1$, like $10$:

• A positive exponent gives a number bigger than $1$. The bigger the exponent, the bigger the number: $10^6$ is much bigger than $10^2$.
• An exponent of zero gives exactly $1$.
• A negative exponent gives a number between $0$ and $1$. The more negative the exponent, the smaller the number (the closer to $0$): $10^{-6} = 0.000001$ is smaller than $10^{-2} = 0.01$.

A negative exponent never makes the number negative. $10^{-3} = 0.001$ is still positive.

The same pattern works for other bases. With base $2$: $2^3 = 8$, $2^0 = 1$, $2^{-3} = \dfrac{1}{8}$. You’ll explore the rules for multiplying and dividing powers in exponent laws.

Scientific notation

A number is in scientific notation when it’s written as

$$a \times 10^n \qquad \text{where } 1 \le a \lt 10 \text{ and } n \text{ is an integer}$$

The coefficient $a$ has exactly one non-zero digit before the decimal point. So $4.56 \times 10^6$ is in scientific notation, but $45.6 \times 10^5$ and $0.456 \times 10^7$ are not (even though they’re equal to it).

Converting

Standard form to scientific notation. Move the decimal point until the number is between $1$ and $10$, and count the places you moved it.

• A large number (10 or more): you move the point left, and the exponent is positive. $\quad 4\,560\,000 = 4.56 \times 10^6$ (moved $6$ places).
• A small number (between $0$ and $1$): you move the point right, and the exponent is negative. $\quad 0.000\,032 = 3.2 \times 10^{-5}$ (moved $5$ places).

Scientific notation to standard form. Do the reverse. A positive exponent moves the point right (the number gets bigger); a negative exponent moves it left (the number gets smaller).

Multiplying and dividing

Group the coefficients together and the powers of $10$ together. When you multiply powers of $10$, add the exponents; when you divide, subtract them:

$$(3 \times 10^4)(5 \times 10^6) = (3 \times 5) \times 10^{4 + 6} = 15 \times 10^{10}$$

If the coefficient ends up outside $1$ to $10$, fix it: $15 \times 10^{10} = 1.5 \times 10^1 \times 10^{10} = 1.5 \times 10^{11}$.

Calculator notation

Calculators and spreadsheets often show scientific notation with an E: the display 3.2E8 means $3.2 \times 10^8$, and 2.5E-4 means $2.5 \times 10^{-4}$. The E stands for “exponent”. It does not mean subtraction.

To enter $3.2 \times 10^8$, type 3.2, press the key marked EE, EXP or ×10ˣ, then type 8. That key already includes the "$\times 10$". If you type 3.2 × 10 EXP 8, the calculator reads it as $3.2 \times 10 \times 10^8 = 3.2 \times 10^9$, which is ten times too big.

Very big and very small things

Scientific notation makes it easy to compare sizes across the universe:

Each tick on this scale is $10$ times the one to its left. Moving three ticks right multiplies the length by $1000$.

Worked examples

Example 1: Writing numbers in scientific notation

Write each number in scientific notation.

• (a) $4\,560\,000$
• (b) $0.000\,032$
• (c) $38 \times 10^4$

Solution.

(a) Move the decimal point $6$ places left to get $4.56$. The number is large, so the exponent is positive: $4.56 \times 10^6$.

(b) Move the decimal point $5$ places right to get $3.2$. The number is small, so the exponent is negative: $3.2 \times 10^{-5}$.

(c) $38$ isn’t between $1$ and $10$. Write $38 = 3.8 \times 10^1$, then add the exponents:

$$38 \times 10^4 = 3.8 \times 10^1 \times 10^4 = 3.8 \times 10^5$$

Check: $38 \times 10^4 = 380\,000$ and $3.8 \times 10^5 = 380\,000$. ✓

Example 2: Back to standard form

Write each number in standard form.

• (a) $7.09 \times 10^5$
• (b) $2.5 \times 10^{-4}$

Solution.

(a) Positive exponent: move the decimal point $5$ places right, filling in zeros: $709\,000$.

(b) Negative exponent: move the decimal point $4$ places left: $0.000\,25$.

Check (b): $0.000\,25$ is between $0$ and $1$, as a negative exponent should give. ✓

Example 3: Multiplying and dividing

Evaluate. Give each answer in scientific notation.

• (a) $(3 \times 10^4)(5 \times 10^6)$
• (b) $\dfrac{8.4 \times 10^{-3}}{2.1 \times 10^5}$

Solution.

(a)

$$\begin{aligned} (3 \times 10^4)(5 \times 10^6) &= (3 \times 5) \times 10^{4 + 6} \\ &= 15 \times 10^{10} \\ &= 1.5 \times 10^{11} && 15 = 1.5 \times 10^1 \end{aligned}$$

(b) Divide the coefficients and subtract the exponents. Be careful with the negative:

$$\frac{8.4 \times 10^{-3}}{2.1 \times 10^5} = \frac{8.4}{2.1} \times 10^{-3 - 5} = 4 \times 10^{-8}$$

Example 4: Sunlight’s trip to Earth

The Sun is about $1.5 \times 10^{11}$ m from Earth, and light travels at about $3.0 \times 10^8$ m/s. How long does sunlight take to reach Earth? Give your answer in seconds and in minutes.

Solution. Time is distance divided by speed:

$$\begin{aligned} t &= \frac{1.5 \times 10^{11}}{3.0 \times 10^8} \\ &= 0.5 \times 10^{11 - 8} \\ &= 0.5 \times 10^3 \\ &= 5 \times 10^2 && 0.5 = 5 \times 10^{-1} \end{aligned}$$

That’s $500$ s. In minutes: $500 \div 60 \approx 8.3$ min. The sunlight you see left the Sun about $8$ minutes ago!

On a calculator, enter 1.5 EE 11 ÷ 3 EE 8. The display shows 500.

Common mistakes

Leaving the coefficient outside 1 to 10. $45 \times 10^3$ and $0.6 \times 10^4$ are equal to numbers in scientific notation, but they aren’t in scientific notation. Fix them: $4.5 \times 10^4$ and $6 \times 10^3$.

Getting the sign of the exponent backwards. Small numbers (between $0$ and $1$) have negative exponents; large numbers have positive exponents. Check by asking: is my original number big or small?

Thinking a negative exponent makes a negative number. $10^{-3} = 0.001$, which is positive. A negative number would be written with a negative coefficient, like $-3 \times 10^4$.

Multiplying the exponents instead of adding them. $10^4 \times 10^6 = 10^{10}$, not $10^{24}$. You’re counting how many factors of $10$ there are: four plus six.

Adding exponents when you add numbers. $3 \times 10^4 + 2 \times 10^3$ is not $5 \times 10^7$. The add-the-exponents rule is for multiplying. Write both in standard form to add: $30\,000 + 2000 = 32\,000 = 3.2 \times 10^4$.

Typing ”× 10” before the EE key. The EE (or EXP) key already means ”$\times 10$ to the power of”. Typing × 10 EE 8 makes your answer ten times too big. Also, read 2.3E-4 as $2.3 \times 10^{-4}$, not $2.3 - 4$.

Practice

1. (Warm-up)

• (a) Evaluate $10^4$, $10^0$ and $10^{-2}$.
• (b) Put these in order from smallest to largest: $10^{2}$, $10^{-1}$, $10^{0}$, $10^{-3}$.

Solution

(a) $10^4 = 10\,000$, $\ 10^0 = 1$, $\ 10^{-2} = \dfrac{1}{100} = 0.01$.

(b) The more negative the exponent, the smaller the number: $10^{-3}$, $10^{-1}$, $10^0$, $10^2$ (that is, $0.001$, $0.1$, $1$, $100$).

2. (Warm-up) Write in scientific notation.

• (a) $93\,000\,000$
• (b) $0.0061$
• (c) $500$
• (d) $0.4$

Solution

(a) $9.3 \times 10^7$ (point moved $7$ places left)

(b) $6.1 \times 10^{-3}$ (point moved $3$ places right)

(c) $5 \times 10^2$

(d) $4 \times 10^{-1}$

3. (Warm-up) Write in standard form.

• (a) $1.2 \times 10^3$
• (b) $6.02 \times 10^{-2}$
• (c) $9 \times 10^{-6}$

Solution

(a) $1200$

(b) $0.0602$

(c) $0.000\,009$

4. (Core) A calculator shows each result below. Write it in scientific notation and in standard form.

• (a) 4.7E-6
• (b) 1.08E9

Solution

(a) $4.7 \times 10^{-6} = 0.000\,004\,7$

(b) $1.08 \times 10^9 = 1\,080\,000\,000$

5. (Core) Evaluate. Give answers in scientific notation.

• (a) $(2.5 \times 10^3)(4 \times 10^{-7})$
• (b) $\dfrac{6 \times 10^8}{1.5 \times 10^{-2}}$
• (c) $(9 \times 10^5)^2$

Solution

(a) $(2.5 \times 4) \times 10^{3 + (-7)} = 10 \times 10^{-4} = 1 \times 10^{-3}$

(b) $\dfrac{6}{1.5} \times 10^{8 - (-2)} = 4 \times 10^{10}$

(c) $(9 \times 10^5)(9 \times 10^5) = 81 \times 10^{10} = 8.1 \times 10^{11}$

6. (Core) Put these numbers in order from smallest to largest: $3.2 \times 10^{-4}$, $\ 5 \times 10^{-5}$, $\ 1.1 \times 10^{-3}$, $\ 0.0007$.

Solution

Write $0.0007 = 7 \times 10^{-4}$. Compare exponents first (more negative is smaller), then coefficients:

$$5 \times 10^{-5}, \quad 3.2 \times 10^{-4}, \quad 0.0007, \quad 1.1 \times 10^{-3}$$

In standard form: $0.000\,05 \lt 0.000\,32 \lt 0.0007 \lt 0.0011$. ✓

7. (Core) A red blood cell is about $7.5 \times 10^{-6}$ m wide. About how many red blood cells would fit side by side across $1$ cm ($1 \times 10^{-2}$ m)?

Solution

$$\frac{1 \times 10^{-2}}{7.5 \times 10^{-6}} = \frac{1}{7.5} \times 10^{-2 - (-6)} \approx 0.1333 \times 10^4 = 1333$$

About $1300$ red blood cells would fit across $1$ cm.

8. (Challenge) Earth’s mass is about $5.97 \times 10^{24}$ kg, and the Moon’s mass is about $7.35 \times 10^{22}$ kg. About how many times more massive is Earth than the Moon?

Solution

$$\frac{5.97 \times 10^{24}}{7.35 \times 10^{22}} = \frac{5.97}{7.35} \times 10^{24 - 22} \approx 0.812 \times 10^2 = 81.2$$

Earth is about $81$ times as massive as the Moon.

9. (Challenge) A light-year is the distance light travels in one year. Light travels about $3.0 \times 10^8$ m/s.

• (a) How many seconds are in a year of $365$ days? Write your answer in scientific notation.
• (b) How far is a light-year, in metres and in kilometres?

Solution

(a) $365 \times 24 \times 60 \times 60 = 31\,536\,000 \approx 3.15 \times 10^7$ s.

(b) Distance is speed times time. Use the unrounded number of seconds, $3.1536 \times 10^7$:

$$(3.0 \times 10^8)(3.1536 \times 10^7) = 9.4608 \times 10^{15} \approx 9.46 \times 10^{15} \text{ m}$$

There are $1000 = 10^3$ m in a kilometre, so divide by $10^3$:

$$\frac{9.46 \times 10^{15}}{10^3} = 9.46 \times 10^{12} \text{ km}$$

A light-year is about $9.46 \times 10^{15}$ m, or $9.46$ trillion kilometres.