Prep

Section 2.1 Prep

This preparatory section will equip you with the knowledge and skills to wield significant figures with confidence and precision. Through a combination of instructional video, explanatory text, and practice questions, you will become adept at all things significant figures.

Subsection 2.1.1 Prep Videos: Significant Figures

Watch the following video(s) to get an introduction to significant figures.

Subsection 2.1.2 Prep Text: Significant Figures

Read the following text to learn about significant figures. Don’t forget to try the checkpoint questions along the way!

Subsubsection 2.1.2.1 What Are Significant Figures?

Definition 2.1.1.

Significant figures

The digits in a measured or calculated value that convey meaningful precision. Significant figures include all certain digits and the first uncertain digit.

Significant figures (or sig figs) are the trustworthy digits in a measurement. They help ensure precision and honesty in your scientific spells and calculations.

Subsubsection 2.1.2.2 Counting Significant Figures

To wield them wisely, follow these rules:

Start with what you see. All non-zero digits are significant. If your magic crystal reads \(3.472\) volts, that’s four significant figures; each non-zero digit counts.
Zeroes have secrets.
- Zeroes between non-zero digits are always significant. Example: \(101\) has three sig figs.
- Leading zeroes (before any non-zero digits) are just placeholders; they don’t count. Example: \(0.0035\) has two sig figs.
- Trailing zeroes (after the decimal point) do count! Example: \(2.300\) has four sig figs.
- Trailing zeroes in a whole number without a decimal? Mysterious! They may not be significant unless written in scientific notation. Example: \(1500\) could have two, three, or four sig figs, but \(1.500 \times 10^3\) definitely has four. To help clarify our communication in this guide, we will treat \(1500\) as two sig figs and \(15\underline{0}0\) as three sig figs.

Checkpoint 2.1.2.

How many significant figures are in the number: \(42.02\text{?}\)

Hint.

All non-zero digits count. Also, zeros between non-zero digits count.

Answer.

\(42.02\) has four significant figures.

Solution.

\(42.02\) has three non-zero digits: \(4\text{,}\) \(2\text{,}\) and \(2\text{.}\) This counts as three significant figures so far. There is also a zero between two of the non-zero digits, which counts as a fourth significant figure.

So, \(42.02\) has four significant figures.

Checkpoint 2.1.3.

How many significant figures are in the number: \(0.007\text{?}\)

Hint.

All non-zero digits count. Leading zeros (zeros that occur to the left of the first non-zero digit) don’t count.

Answer.

\(0.007\) has one significant figure.

Solution.

\(0.007\) has one non-zero digit: \(7\text{.}\) This counts as one significant figures so far. There is also zeros, but all the zeros occur to the left of the \(7\) and are not between non-zero digits, so the zeros do not count.

So, \(0.007\) has one significant figure.

Subsubsection 2.1.2.3 Rounding with Significant Figures

To round properly, follow these enchanted guidelines:

If digit to the right of your last desired digit is \(6, 7, 8, 9\) then round the last desired digit up by one. Example: \(2.46 \rightarrow 2.5\) (if keeping two sig figs).
If digit to the right of your last desired digit is \(4, 3, 2, 1, or 0\) then keep the desired digit unchanged. Example: \(2.14 \rightarrow 2.1\) (if keeping two sig figs)
If digit to the right of your last desired digit is \(5\) and there are other non-zero digits to the right of that \(5\text{,}\) then round the last desired digit up by one. Example: \(2.2501 \rightarrow 2.3\) (if keeping two sig figs)
If digit to the right of your last desired digit is \(5\) and there is no other non-zero digits to the right of that \(5\text{,}\) then round to the nearest even digit. Example: \(2.250 \rightarrow 2.2\) (if keeping two sig figs). Example: \(2.35 \rightarrow 2.4\) (if keeping two sig figs).

Subsubsection 2.1.2.4 Using Sig Figs in Calculations

When combining numbers in your calculations, the significant figures must follow the rules of magical balance:

Multiplying or dividing? Your answer should have the same number of significant figures as the measurement with the fewest. For example, \(2.5 \times 3.42 = 8.55\text{,}\) but since \(2.5\) has only two sig figs, you round your answer to \(8.6\text{.}\)
Adding or subtracting? Line up your decimal points. Your answer should be rounded to the least precise decimal place from your inputs. For example, \(12.11 + 3.6 = 15.71\text{,}\) which gets rounded to \(15.7\) because \(3.6\) is only precise to the tenths place.
Multiple Operations? Cast each spell in the proper order. Follow the standard order of operations (parentheses, exponents, multiplication/division from left to right, then addition/subtraction from left to right). Only round at the very end after all the magic is done. Do not round after each step. Rounding too early can throw your final result out of balance, like mimeasuring a potion halfway through.

Subsubsection 2.1.2.5 Why Use Significant Figures?

Significant figures are like a secret code shared among scientists and magic folk alike. When you report a measurement of \(1.27\) meters, you’re really telling your fellow colleague: "I’m confident about the \(1\) and the \(2\text{,}\) but the last digit (the \(7\)) could vary a bit." That message is embedded in the number’s three-digit significant figure code, quietly revealing both precision and uncertainty.

Subsubsection 2.1.2.6 Sig Figs and Scientific Notation

Scientific notation and significant figures are best friends. Writing a number like \(6.02 \times 10^{23}\) clearly shows that you’ve got three sig figs, no more, no less.

Here’s how to combine them:

\(3.0 \times 10^2\) has \(2\) sig figs
\(3.00 \times 10^2\) has \(3\) sig figs
\(3 \times 10^2\) has \(1\) sig fig

Each extra zero you intentionally include after the decimal adds more confidence to your measurement.

Subsubsection 2.1.2.7 A Parting Word

With significant figures in your database, you can communicate your uncertainty across the vast realms of physics, chemistry, and cosmic measurement. Write your numbers with care, check your sig figs, and let no leading zero invade your notes.

Subsection 2.1.3 Prep Questions: Significant Figures

Count the number of significant figures.

Checkpoint 2.1.4.

How many significant figures are in the number: \(4202\text{?}\)

Four
Correct! \(4202\) has four significant figures.
Three
Your targeting system glitched, try again. All non-zero digits count, and zeros between non-zeros digits count.
Two
Your enchanted click missed the mark, try again. All non-zero digits count, and zeros between non-zeros digits count.
One
Your digital blade missed the mark, try again. All non-zero digits count, and zeros between non-zeros digits count.

Checkpoint 2.1.5.

How many significant figures are in the number: \(0.00560\text{?}\)

Three
Correct! \(0.00560\) has three significant figures.
Two
Your targeting system glitched, try again. Leading zeros are not significant, but trailing zeros after a decimal and digit are.
One
Your enchanted click missed the mark, try again. Leading zeros are not significant, but trailing zeros after a decimal and digit are.
Four
Your digital blade missed the mark, try again. Leading zeros are not significant, but trailing zeros after a decimal and digit are.

Checkpoint 2.1.6.

How many significant figures are in the number: \(10\underline{0}\text{?}\)

Three
Correct! \(10\underline{0}\) has three significant figures.
Two
Your targeting system glitched, try again. All non-zero digits count. Zeros to the right of a non-zero digit and before a decimal are undefined unless an underline is included.
One
Your enchanted click missed the mark, try again. All non-zero digits count. Zeros to the right of a non-zero digit and before a decimal are undefined unless an underline is included.
Four
Your digital blade missed the mark, try again. All non-zero digits count. Zeros to the right of a non-zero digit and before a decimal are undefined unless an underline is included.

Checkpoint 2.1.7.

How many significant figures are in the number: \(40.070\text{?}\)

Five
Correct! \(40.070\) has five significant figures.
Three
Your targeting system glitched, try again. All non-zero digits count. Zeros between non-zero digits count. Trailing zeros count.
Two
Your enchanted click missed the mark, try again. All non-zero digits count. Zeros between non-zero digits count. Trailing zeros count.
Four
Your digital blade missed the mark, try again. All non-zero digits count. Zeros between non-zero digits count. Trailing zeros count.

Checkpoint 2.1.8.

How many significant figures are in the number: \(0.03040\text{?}\)

Four
Correct! \(0.03040\) has four significant figures.
Six
Your targeting system glitched, try again. All non-zero digits count. Zeros between non-zero digits count. Trailing zeros count. Leading zeros do not count.
Five
Your enchanted click missed the mark, try again. All non-zero digits count. Zeros between non-zero digits count. Trailing zeros count. Leading zeros do not count.
Two
Your digital blade missed the mark, try again. All non-zero digits count. Zeros between non-zero digits count. Trailing zeros count. Leading zeros do not count.

Round the number to the desired significant figures.

Checkpoint 2.1.9.

Round \(1.25\) to two significant figures.

\(1.2\)
Correct!
\(1.3\)
Your targeting system glitched, try again. When digit to right after the last desired digit is \(5\text{,}\) you round to the nearest even digit.
\(1\)
Your enchanted click missed the mark, try again. Round to two significant figures, not one.
\(1.4\)
Your digital blade missed the mark, try again. When digit to right after the last desired digit is \(5\text{,}\) you round to the nearest even digit.

Checkpoint 2.1.10.

Round \(6.750\) to three significant figures.

\(6.75\)
Correct!
\(6.76\)
Your targeting system glitched, try again. The digit after the last desired digit is \(0\text{,}\) so you keep the last digit unchanged.
\(6.74\)
Your enchanted click missed the mark, try again. The digit after the last desired digit is \(0\text{,}\) so you keep the last digit unchanged.
\(6.8\)
Your digital blade missed the mark, try again. Round to three significant figures, not two.

Checkpoint 2.1.11.

Round \(0.004509\) to three significant figures.

\(0.00451\)
Correct!
\(0.00450\)
Your targeting system glitched, try again. There is a \(9\) next to the \(0\text{,}\) round the thrid sig fig up.
\(0.0045\)
Your enchanted click missed the mark, try again. Round to three significant figures, not two.
\(0.0046\)
Your digital blade missed the mark, try again. You rounded too early—only the third sig fig (after \(4\) and \(5\text{,}\) i.e. the \(0\)) should be affected.

Checkpoint 2.1.12.

Round \(35\) to one significant figure.

\(4\)
Correct!
\(3\)
Your targeting system glitched, try again. The digit after the last desired digit is exactly \(5\text{,}\) so you round to the nearest even digit.
\(20\)
Your enchanted click missed the mark, try again. The digit after the last desired digit is exactly \(5\text{,}\) so you round to the nearest even digit. There is a closer even number to \(35\) than \(20\text{.}\)
\(35\)
Your digital blade missed the mark, try again. Round to one significant figure, not two.

Solve the following expression, rounding your final answer the appropiate amount of significant figures.

Checkpoint 2.1.13.

Solve the expression: \(2.00 \times 366.25\text{,}\) and round your answer to the appropriate number of significant figures.

\(732\)
Correct!
\(733\)
Your targeting system glitched, try again. \(2.00 \times 366.25 = 732.5\) The digit after the last desired digit is exactly \(5\text{,}\) so you round to the nearest even digit.
\(732.5\)
Your enchanted click missed the mark, try again. \(2.00 \times 366.25 = 732.5\) When multiplying or dividing, your answer should have the same number of significant figures as the measurement with the fewest. \(2.00\) has three sig figs, and \(366.25\) has five sig figs.
\(700\)
Your digital blade missed the mark, try again. \(2.00 \times 366.25 = 732.5\) When multiplying or dividing, your answer should have the same number of significant figures as the measurement with the fewest. \(2.00\) has three sig figs, and \(366.25\) has five sig figs.

Checkpoint 2.1.14.

Solve the expression: \(15.00 \div 4.5\text{,}\) and round your answer to the appropriate number of significant figures.

\(3.3\)
Correct!
\(3.333\)
Your targeting system glitched, try again. \(15.00 \div 4.5 = 3.33\overline{3}\) When multiplying or dividing, your answer should have the same number of significant figures as the measurement with the fewest. \(15.00\) has four sig figs, and \(4.5\) has two sig figs.
\(3.33\)
Your enchanted click missed the mark, try again. \(15.00 \div 4.5 = 3.33\overline{3}\) When multiplying or dividing, your answer should have the same number of significant figures as the measurement with the fewest. \(15.00\) has four sig figs, and \(4.5\) has two sig figs.
\(3\)
Your digital blade missed the mark, try again. \(15.00 \div 4.5 = 3.33\overline{3}\) When multiplying or dividing, your answer should have the same number of significant figures as the measurement with the fewest. \(15.00\) has four sig figs, and \(4.5\) has two sig figs.

Checkpoint 2.1.15.

Solve the expression: \(81.201 - 3.02\text{,}\) and round your answer to the appropriate number of decimal places.

\(78.18\)
Correct!
\(78.2\)
\(81.201 - 3.02 = 78.181\) When adding or subtracting, your answer should be rounded to the same number of decimal places as the measurement with the fewest decimal places. \(81.201\) has three decimal places, and \(3.02\) has two decimal places.
\(78.181\)
Your enchanted click missed the mark, try again. \(81.201 - 3.02 = 78.181\) When adding or subtracting, your answer should be rounded to the same number of decimal places as the measurement with the fewest decimal places. \(81.201\) has three decimal places, and \(3.02\) has two decimal places.
\(78\)
Your digital blade missed the mark, try again. \(81.201 - 3.02 = 78.181\) When adding or subtracting, your answer should be rounded to the same number of decimal places as the measurement with the fewest decimal places. \(81.201\) has three decimal places, and \(3.02\) has two decimal places.

Checkpoint 2.1.16.

Solve the expression: \(12.1 + 0.35\text{,}\) and round your answer to the appropriate number of decimal places.

\(12.4\)
Correct!
\(12.5\)
Your targeting system glitched, try again. \(12.1 + 0.35 = 12.45\) The digit after the last desired digit is exactly \(5\text{,}\) so you round to the nearest even digit.
\(12.45\)
Your enchanted click missed the mark, try again. \(12.1 + 0.35 = 12.45\) When adding or subtracting, your answer should be rounded to the same number of decimal places as the measurement with the fewest decimal places. \(12.1\) has one decimal places, and \(0.35\) has two decimal places.
\(12\)
Your digital blade missed the mark, try again. \(12.1 + 0.35 = 12.45\) When adding or subtracting, your answer should be rounded to the same number of decimal places as the measurement with the fewest decimal places. \(12.1\) has one decimal places, and \(0.35\) has two decimal places.

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