Subsection 2.1.1 Prep Text: Significant Figures
Significant figures (or sig figs) are the trustworthy digits in a measurement. They help ensure precision and honesty in your scientific spells and calculations. 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.
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Rounding
To round properly, follow these enchanted guidelines:
If digit to the right of your last desired digit is 6, 7, 8 or 9 then round the last desired digit up by one. Example: 2.46 → 2.5 (if keeping two sig figs)
If digit to the right of your last desired digit is 4, 3, 2, or 0 then keep the desired digit unchanged. Example: 2.14 → 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, then round the last desired digit up by one. Example: 2.2501 → 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, then round to the nearest even digit. Example: 2.250 → 2.2 (if keeping two sig figs). Example: 2.35 → 2.4 (if keeping two sig figs).
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Math with Care
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, \(2.5 × 3.42 = 8.55\text{,}\) 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 miscasting a potion halfway through.
Checkpoint 2.1.1.
How many significant figures are in the number: 42.02?
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, 2, and 2. 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 2.1.2.
How many significant figures are in the number: 0.007?
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. 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.
Why Use It?
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, 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.
Bonus Trick: Using Sig Figs in 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 their powers:
\(3.0 \times 10^2\) → 2 sig figs
\(3.00 \times 10^2\) → 3 sig figs
\(3 \times 10^2\) → 1 sig fig
Each extra zero you intentionally include after the decimal adds more confidence to your measurement.
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.
