Significant Digits
How Many Digits Should We Show in our Measurements?
Significant digits or significant figures are numbers that are important in a measurement or calculation.
Every time you perform a measurement, you need to estimate the number of significant digits to show, and that depends on the instrument that is used for that purpose.
A very useful method is to calculate the amount measured between two consecutive lines of the instrument and then divide that amount by two.
The answer will give you an idea on how many decimals you must show in your measurements and how much would be the error involved.
Example #1: When using a 50 cm3 Burette
A) 18.4 cm3 – 18.3 cm3 = 0.1 cm3
⇒ ÷2 = 0.05
B) 0.2 cm3 – 0.1 cm3 = 0.1 cm3
⇒ ÷2 = 0.05
C) 23.4 cm3 – 23.3 cm3 = 0.1 cm3
⇒ ÷2 = 0.05
When using a 50 cm3 burette you should ALWAYS show two decimals.
The last decimal is always estimated.
A = 18.75 cm3
B= 0.45 cm3
C= 23.00 cm3
Note that:
- The scale in the burette is upside down.
- All measurements have 2 decimals and the last digit will be 0 or 5. (This is for Cambridge only)
- Most measurements will have 2, 3 or 4 significant digits.
- The last digit will always be 0 or 5 depending on the position of the lower meniscus.
- When using a graduated cylinder, the accuracy depends on the graduated scale that it contains.
Example #1: When using a graduated cylinder
A) 23.4 cm3 – 23.2 cm3 = 0.2 cm3
⇒ ÷2 = 0.1
B) 9.7 cm3 – 9.6 cm3 = 0.1 cm3
⇒ ÷2 = 0.05
C) 79 cm3 – 78 cm3 = 1 cm3
⇒ ÷2 = 0.5
When using a graduated cylinder, the amount of decimals will depend on the graduated cylinder accuracy.
The last decimal is always estimated.
A = 24.0 cm3
B= 9.25 cm3
C= 75.5 cm3
Note that:
- All measurements given have 3 significant digits.
- Not all measurements have the same number of decimals.
How Many Digits Should We Show in our calculations?
Many times, your calculator will display an answer containing more digits than the number of significant figures you obtained in the measurements.
Example:
You calculate through a titration that 20.35 mL of HCl solution neutralize 25.0 mL of a 0.110 mol.dm-3 solution of NaOH, you should perform the following calculations:
- Find the number of moles of sodium hydroxide used.
- Find the concentration of the acid in the solution
Data:
- Burette reading for HCl volume used = 20.35 cm-3 (4 sig dig)
- Volume of NaOH = 25.0 cm-3 (3 sig dig)
- Concentration of NaOH = 0.110 mol/dm-3 (3 sig dig)
Calculations:
1- Find the number of moles of sodium hydroxide used.
# moles of NaOH = CC (mol. dm-3) x VOL (dm3)
# moles NaOH = 0.110 mol. dm-3 x 25.0 / 1000 dm3
# moles NaOH = 0.00275 mol NaOH (3 sig fig)
2- Using the mole ratio, calculate the # of moles of HCl present in the volume used.
The mole ratio is 1:1 since the formula for the neutralization reaction isHCL + NaOH ==>; H2O + NaCl
==> number of moles of HCl are also 0.00275
3- Find the concentration of the acid in the solution.
CC HCl = # moles HCl / Vol (dm3)
the concentration would be then 0.135M
How Many Digits Should We Show in our calculations?
Significant digits based on calculations
Addition and Subtraction
If the calculation you are performing is an addition or subtraction, the answer will have the amount of decimals of the number that has fewest.
Example, you measure masses of three different compounds using different balances. how many decimals should you use?
1.467 g + 3.2 g + 3.507 g =
Calculator shows = 8.1787
(4) (1) (3) => # of decimals
Correct answer= 8.2 g
Multiplication and division
If instead you are performing a multiplication or division, your answer will show the same number of significant figures as the number that hast the fewest.
Example, you measure masses of three different compounds using different balances. how many decimals should you use?
1.4767 cm x 3.20 cm x 3.507 cm =
Calculator shows = 16.5485
(5) (3) (4) => # of sig. fig.
Correct answer= 16.5 cm3 – 3 s.d.
How to Round Off
When performing calculations, you need to know which digits are important (significant) and which are not. The significant figures are kept and the insignificant figures are dropped. You need to know what to do with the last significant digit in our number, and that depends on the FIRST NON SIGNIFICANT DIGIT.
If the number AFTER the last significant digit is 5 or more, you will round up the last significant digit to the next number.
If the number AFTER the last significant digit is lower than 5, you just drop the numbers
Examples:
25.0 x 0.110 / 20.00 = 0.1375 → 0.138
25.0 x 0.200 / 11.50 = 0.43478 → 0.435
25.0 x 0.110 / 21.75 = 0.12644 → 0.126
25.0 x 0.010 / 12.25 = 0.02041 → 0.0204 (3 Sig dig) → 0.020 (2 sig fig)
Final Clarification: In your Cambridge Examination, you will be asked for the correct amount of sig fig in your answer. If nothing is stated, you should use 3 significant figures.