Significant Figures are the digits of a number which are used for expressing the necessary degree of accuracy that starts from the first non-zero digit.

The number of significant figures in a result indicates the number of digits that can be used with confidence. The idea of a significant figure is simply a matter of applying common sense when dealing with numbers. An important characteristic of any numerical value is the number of digits, or significant figures it contains. A significant figure is defined as any digit in the number ignoring leading zeros and the decimal point.

Examples of Significant Figures

  • 821 has 3 significant figures
  • 0.0310 has 3 significant figures

The digits that are used to express a number are called significant digits or significant figures. The use of significant figure is very important.

“Significant figures is the number of digits in a number that actually convey any meaning. This does not include any information on the order or power of ten in the number representation, as this can be conveyed by other means.”
The term "significant figure" refers to the number of digits in a written number that can be trusted by implication. Factors that can reduce trust include the possibility of round-off errors and any explicit expression of uncertainty. Unless specified otherwise, all digits in a written number are considered significant. Also, whole numbers generally have an infinite number of significant figures unless uncertainty is expressed explicitly.
Often you will be given a set of numbers, with units, that you will have to use algebraically to determine some final result. The number of significant figures in the answer is determined by the number of significant figures in the initial numbers.

There are few simple rules for significant figures
  1. All non-zero digits are considered significant figures. For example, 345 has three significant figures.
  2. All zeros between non-zero digits are significant. Ignore the decimal point, if necessary. For example, the number 305 also has three significant figures. So does 3.05, because the zero is between non-zero digits.
  3. Zeros at the beginning of a decimal number or at the end of a large number are not significant. For example, 546,000 has only three significant figures. The three zeros simply serve to put the 5 in the 100,000s place, and the 6 in the 1000s place.
    Significant Figures
    Similarly, the decimal number 0.000928 has only three significant figures, since the four zeros serve only to place the 9,2 and the 8 in the correct column.
    Significant Figures Example
    The number 0.0006098 has four significant figures, because we do include the zero between the 6 and the 9. However, we do not include the first four zeros as significant.
  4. Zeros written at the end of a number after the decimal point are significant. Otherwise, they would not even be written. For example, 0.67 has two significant figures, but 0.670 has three significant figures. If that final zero were not significant, it should not have been written down.
In adding significant figures only similar units that are written to the same number of decimal places may be added. Also, the number with the fewest number of decimal places,and not necessarily the fewest number of significant figures, places a limit on the number that the sum can justifiably carry.

When adding significant figures (including negative numbers), the rule is that the least accurate number will determine the number reported as the sum. In other words, the number of significant figures reported in the sum cannot be greater than the least significant figure in the group being added.

Examples on Adding Significant Figures


Given below are some examples on adding significant figures.

Example 1:

Add the following numbers
446mm + 185.22cm + 18.9m.

Solution:

First convert the quantities to similar units, which in this case is the mater (second row below). Next, choose the least accurate number, which s 18.9. It has only one number to the right of the decimal so the other two values will have to be rounded off (third row below).
Given
Converted values
Addition
446mm
0.446m
0.4m
185.22cm 1.8522m
1.8m
18.9m 18.9m
18.9m

Answer
= 0.4 + 1.8 + 18.9 = 21.1m

Example 2:

Add 5.80 and 3

Solution:

5.80
3 +
8.80

We can round off the answer as 9. Therefore, 5.80 + 3 = 9
In subtraction of significant figures the final answer should contain digits only as far to the rightmost decimal column found in the least precise number used in the calculation.
When the number is subtracted, the number of decimal places in the result should equal the smallest number of decimal places of any term.

The least significant digit of the difference occupies the same relative position as the least significant digit of the quantities being subtracted. In this case the number of significant figures is not important; it is the position that matters.

Examples on Subtracting Significant Figures


Given below are some examples on subtracting significant figures.

Example 1:

Subtract: 31.2 - 5.56

Solution:

31.2
- 5.56
25.64

So, 31.2 - 5.56 = 25.64

Example 2:

Subtract: 72.2 - 3.56

Solution:

72.2
- 3.56
68.64

So, 72.2 - 3.56 = 68.64

For addition and subtraction the important consideration is the number of decimal places, not the number of significant figures.
In multiplication the following rules should be used to determine the number of significant figures.

  • The product or quotient should contain the number of significant digits that are contained in the number with the fewest significant digits.
  • The number of significant digits in a product or quotient is the same as that in the least significant of the values used to calculate the product or quotient.

Examples on Multiplying Significant Figures


Given below are some examples that explains multiplication of significant figures.

Example 1:

Solve 16 x 35.6

Solution:

The product of 16 x 35.6 = 569.6 has only two significant digits since 16 is the least significant of the values used in forming the product.
The product should be written as 16 x 35.6 = 57.
However, if it were predetermined that 16 was an integer good for an infinite number of decimal places, the product would have three significant figures based on the value 35.6.
The product would be correctly written as 16 x 35.6 = 570. In significant figure multiplication, the result must have as many significant digits as those of the number with the least significant digits entering the operation.

Example 2:

Solve 11.63 x 5.74

Solution:

The product of 11.63 x 5.74 is 66.7562. But, according to the rules of multiplication in significant figures, the product must have only 3 significant figures. Therefore, the answer is 66.8.

11.63 x 5.74 = 66.8
In the division of measurements, the number of significant figures in the result is not greater than the number of significant figures in the measurement with the fewest significant figures.

Examples on Dividing Significant Figures


Given below are some examples on dividing significant figures.

Example 1:

Divide $\frac{3.795}{2.66}$

Solution:

Step 1: First convert the divisor into a whole number by moving the decimal point two places to the right, so that 2.66 becomes 266. Equalize this by moving the decimal point in the dividend the same number of places to the right, so that 3.795 becomes 379.5.

Step 2: Now proceed as in ordinary division but place the decimal point in the answer immediately after bringing down the first figure in the decimal.
Significant Figures Division
Step 3: The division is incomplete, and may be carried as far as necessary by bringing down a succession of 0's from the decimal part of the dividend.

Step 4: If the answer is required to the third decimal place, and the division is still not complete, then it should be carried to the fourth place.

Step 5: If the fourth figure is less than 5 it is ignored; if it is 5 or over, then 1 is added to the third place to obtain the nearest correct answer to the third decimal place.
Significant Figures Division Example
The answer in this case to the nearest third decimal place is 1.427
In numerical computations, we come across numbers which have large number of digits and it will be necessary to cut them to a usable number of figures. This process is called rounding off. It is simply the dropping of figures starting on the right until the appropriate number of significant figures remain.

Rules for Rounding Significant Figures


It is usual to round-off numbers according to the following rules:

  • When a figure less than five is dropped, the next figure to the left remains unchanged. Thus the number 11.24 becomes 11.2 when it is required that the four be dropped.
  • When the figure is greater than five that number is dropped and the number to the left is increased by one. Thus 11.26 will become 11.3
  • When the figure that needs to be dropped is a five, round off to the nearest even number. This prevents rounding bias. Thus 14.25 becomes 14.2 and 57.75 becomes 57.7
At each step of the computation retain at least one more significant figure than the one given in the data, perform the last operation and than round-off.

Given below are some practice problems on significant figures.

Practice Problem 1:

Round to the correct decimal position
  • 23.4 x 42.87 =
  • 8.56 x 39.2 =
  • 146 x 15.58 =
  • 1.4756 x 2.4 =
  • 3.4 x 4.21 x 1.8 =
  • 7.34 / 4.9 =
  • 24 / 4.34 =
  • 5.3214 / 3.87 =
Practice Problem 2:

If a scale says a person weighs 182 pounds, how many kilograms(counting significant digits) is that person?