Significant Digits/Figures
Within all areas of science, measurements are made and values are reported. These reported
values all have a degree of accuracy that is limited by the equipment that the experiment can use and
the procedure of the experiment conducted. For example, a scale used to measure your weight is
usually given up to 2 decimal places. Someone that measures 150.23 pounds will not know the exact
value of the third decimal place, therefore they can only report what they know for sure, which is 150.23
pounds.
To determine which numbers to report in science, scientists developed a system called “Significant
Digits”. This system allows us to determine the maximum accuracy that the final reported values can be.
Identifying Significant Digits
1)
2)
3)
4)
All non-zero digits are significant.
Zeros found between two non-zero digits are significant.
In a number WITH a decimal, all zeros AFTER the LAST non-zero number are SIGNIFICANT.
In a number WITHOUT a decimal, all zeros AFTER the LAST non-zero number are NOT
SIGNIFICANT.
5) All zeros BEFORE the FIRST non-zero digit are NOT SIGNIFICANT.
Examples:
1) 123.45
2) 120.403
3) 1 250.000
4) 1 034 000
5) 0.0000152
This number has 5 sig digs.
This number has 6 sig digs.
This number has 7 sig digs. (There is a decimal)
This number has 4 sig digs. (There is no decimal)
This number has 3 sig digs.
Scientific Notation
Very often in science, math, and computers, you will encounter some very large and some very
small numbers. For example, the distance from the Earth to the Sun is 152 000 000 000 m while the
mass of an electron is quoted as 0.0000000000000000000000000000000911 kg (that’s 30 zeros after
the decimal!).
These numbers can be very tedious and time consuming to write this way. Thus, scientists developed a
kind of shorthand for writing out these numbers called “Scientific Notation”. This notation allows us to
omit the zeros while still reporting the magnitude of the number. In essence, scientific notation is a way
to report a number using the least amount of information necessary and takes the form:
a x 10N
Where ‘a’ is a number between 1-10
Where N is an integer number
Conversion from Standard Notation to Scientific Notation
1) Start after the first non-zero digit from the left.
2) Count the number of decimal places you move to reach the ORIGINAL NUMBER.
3) Write down all the significant digits with your decimal after the first digit and add “ x 10N ”
where N is the number from step 2.
4) If you moved the decimal to the left then N is negative. If you moved the decimal to the right
then N is positive.
Example: 0.000000000000000000000000000000911
Step 1: 9.11
Step 2: From 9.11 to the original number, you count 31 decimal places to the left.
Step 3 and 4: 9.11 x 10-31
Example: 152 000 000 000
Step 1: 1.52
Step 2: From 1.52 to the original number, you count 11 decimal places to the right.
Step 3 and 4: 1.52 x 1011
Rounding
Due to limitations on the accuracy of measurements within scientific experiments, scientists
must follow the rules of significant digits as mentioned above when reporting their final calculations.
However, the mathematics themselves can give more accuracy than what is certain. Thus the last
significant digit that we are allowed to report is rounded to the number that it is more likely to be. For
example, a calculation that gives the number 150.23111 but the accuracy only allows 4 significant digits
will be rounded down to 150.2 while a calculation that gives 0.00523666 but only allows 3 significant
digits will be rounded up to 0.00524.
Rounding, when in conjunction with significant digits, aims to increase the certainty of our
experimental values in order to come closer to the true values.
Rules of Rounding
1)
2)
3)
4)
5)
Find the digit you want to round.
Look at the digit after it.
If the digit from step 2 is less than 5 then keep the digit from step 1 the same.
If the digit from step 2 is greater than 5 then round the digit from step 1 to the next number up.
If the digit from step 2 is exactly 5 then round the digit from step 1 to the closest even number.
Examples: Round each of the following to the nearest hundredth decimal place.
1) 1.2321
2) 2.486
3) 4.215023
4) 3.235
Rounds to 1.23 (The digit after the 3 is 2, therefore 3 does not change)
Rounds to 2.49 (The digit after the 8 is 6, therefore 8 changes to 9)
Rounds to 4.22 (The digit after the 1 is 5, but it is not EXACTLY 5, therefore 1
changes to 2)
Rounds to 3.24 (The digit after the 3 is EXACTLY 5, therefore 3 changes to the
closest even number which Is 4)
Arithmetic Using Significant Digits
Since measured numbers are usually used in calculations within science, the final values must
follow the rules of significant digits as well, giving an accuracy that is certain with as little error from the
true value as possible. The rule of thumb is to complete the calculations fully until the very end before
any rounding is done.
Significant Digits in Addition/Subtraction
When doing addition or subtraction, the final answer is reported with the least number of
decimals.
Example:
1) 15.23 + 2.3685 = 17.5985
Final Answer: 17.60 (The most accurate this answer can be is 2 decimal places, therefore
9 is rounded up.)
2) 1.2354 – 0.253 = 0.9824
Final Answer: 0.982 (The most accurate this answer can be is 3 decimal places, therefore
2 does not change.)
Significant Digits in Multiplication/Division
When doing multiplication or division, the final answer is reported with the least number of
significant digits.
Example:
1) (1.25 x 104)(2.5326 x 10-5) = 3.16575 x 10-1
Final Answer: 3.17 x 10-1 (The most accurate this answer can be is 3 sig digs, therefore 6
is rounded up.)
2) 14.53 x 2.65785 = 38.6185605
Final Answer: 38.62 (The most accurate this answer can be is 4 sig digs, therefore 1 is
rounded up.)