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Significant figures and scientific notation
Which digits are significant, rounding to significant figures, writing very large and very small numbers in scientific notation, and calculating with them on a calculator.
"The Sun is $149\,600\,000$ km away" and "the Sun is $150$ million km away" are both fine. They just claim different amounts of precision. Significant figures are how you say how precise a number is, and scientific notation is how you write the very big and very small numbers in science, medicine and computing without a page of zeros.
Which digits count?
A digit is significant if it carries real information about the measurement. Zeros that only hold the decimal point in place don't.
- Non-zero digits are always significant.
- Zeros between non-zero digits are significant: $305$ has $3$.
- Zeros at the front are never significant. They only show where the decimal point is: $0.0045$ has $2$.
- Zeros at the end of a decimal are significant, because someone chose to write them: $2.30$ has $3$.
- Zeros at the end of a whole number are unclear. $305\,000$ could have $3$, $4$, $5$ or $6$ significant figures, depending on how it was measured.
How many significant figures are in (a) $0.004050$ (b) $18.096$ (c) $2.30$?
- Try each example button. Which zeros turn grey, and which turn yellow? Explain the difference in one sentence.
- Type $0.0302$. How many significant figures? Now type $0.03020$. What changed and why?
- Type $149600000$ and round it to $2$, $3$ and $4$ significant figures. Watch the scientific notation.
Rounding to significant figures
Count from the first significant digit, look at the next digit, and round the usual way. In a whole number, fill the dropped places with zeros so the size of the number stays the same.
Round (a) $0.0067382$ to $2$ significant figures (b) $48\,963$ to $3$ significant figures (c) $3.14159$ to $4$ significant figures.
Scientific notation
A number in scientific notation is written as $a \times 10^n$, where $1 \le a \lt 10$ and $n$ is a whole number. The power of ten counts how many places the decimal point moves.
- Big numbers have a positive power: $149\,600\,000 = 1.496 \times 10^{8}$ (the point moves 8 places left).
- Small numbers have a negative power: $0.000\,007\,2 = 7.2 \times 10^{-6}$ (the point moves 6 places right).
Scientific notation also fixes the problem with trailing zeros. $3.05 \times 10^5$ has three significant figures and $3.050 \times 10^5$ has four. No guessing.
Write (a) $3.05 \times 10^4$ and (b) $6.1 \times 10^{-3}$ as decimals.
Calculating with very big and very small numbers
Light travels at about $3.0 \times 10^8$ m/s. The Sun is $1.496 \times 10^{11}$ m from Earth. How long does sunlight take to reach us? Give your answer in scientific notation to $2$ significant figures, then in minutes and seconds.
Your turn
A red blood cell is about $7.5 \times 10^{-6}$ m across. About how many would fit side by side across $1$ cm?
Show the working
$1$ cm $= 0.01$ m $= 1 \times 10^{-2}$ m. Then $\dfrac{1 \times 10^{-2}}{7.5 \times 10^{-6}} \approx 1333$. About $1300$ cells, to 2 significant figures.
How many significant figures are in $0.0302$?
The leading zeros don't count. The digits $3$, $0$ and $2$ do (the zero sits between non-zero digits): $3$.
Round $7.0451$ to $3$ significant figures.
Keep $7$, $0$, $4$. The next digit is $5$, so round up: $7.05$.
Round $26\,718$ to $2$ significant figures.
Keep $2$, $6$. The next digit is $7$, so round up to $27$, then fill with zeros: $27\,000$.
Write $4.5 \times 10^{-3}$ as a decimal.
Move the decimal point $3$ places left: $0.0045$.
$0.000\,81$ in scientific notation is:
Move the point 4 places right to get $8.1$, so the power is $-4$. The last two options are equal in value but $a$ must be between $1$ and $10$.
A country dam holds $52\,000\,000$ litres of water. Written as $5.2 \times 10^n$, what is $n$?
The point moves $7$ places left from $52\,000\,000$ to $5.2$, so $n = 7$.
Which number has exactly $4$ significant figures?
$0.4050$ has $4$, $0$, $5$ and the written trailing $0$. $0.0045$ has $2$, $40.5$ has $3$, and $4050$ is unclear (3 or 4).
Calculate $(2.4 \times 10^5) \times (3 \times 10^3)$ and write the answer as an ordinary number.
Multiply the numbers and add the powers: $2.4 \times 3 = 7.2$ and $10^{5 + 3} = 10^8$. $7.2 \times 10^8 = 720\,000\,000$.
Exam style: A $2$ TB drive holds $2 \times 10^{12}$ bytes. Each photo is $4.8$ MB, which is $4.8 \times 10^6$ bytes. How many whole photos can the drive hold?
$\dfrac{2 \times 10^{12}}{4.8 \times 10^6} \approx 416\,666.67$. Only whole photos fit, so round down: $416\,666$.
Exam style: Sound travels at $343$ m/s. A lightning strike is $1.7 \times 10^3$ m away. How many seconds until you hear the thunder, correct to $1$ significant figure?
$\dfrac{1.7 \times 10^3}{343} \approx 4.96$ s, which is $5$ s to 1 significant figure.