Significant Figure
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Significant Figures are the digits of a number which are used for expressing the necessary
degree of accuracy that starts from the first nonzero 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
What are 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 roundoff 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.
Significant Figures Rules
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 nonzero digits are considered significant figures. For example, 345 has three significant
figures.
2. All zeros between nonzero 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 nonzero 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.
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.
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.
Adding Significant Figures
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
Subtracting Significant Figures
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.
W
hen 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
NOTE : For addition and subtraction the important consideration is the number of decimal places,
not the number of significant figures.
Multiplying 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
Dividing Significant Figures
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 3.7952.66
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.
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.
The answer in this case to the nearest third decimal place is 1.427
Rounding Significant Figures
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 roundoff numbers according to the following rules:
1. 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.
2. 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
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
Note : 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 roundoff.
Significant Figures Practice Problems
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?
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