Significant Figures Examples
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The Significant Figures (also called significant digits) of a number are those digits that carry
meaning contributing to its precision. This includes all digits except:
leading and trailing zeros where they serve merely as placeholders to indicate the scale of the
number.
spurious digits introduced, for example, by calculations carried out to greater accuracy than that of
the original data, or measurements reported to a greater precision than the equipment supports.
The concept of significant digits is often used in connection with rounding. Rounding to n
significant digits is a more generalpurpose technique than rounding to n decimal places, since it
handles numbers of different scales in a uniform way.
For example : the population of a city might only be known to the nearest thousand and be
stated as 52,000, while the population of a country might only be known to the nearest million and
be stated as 52,000,000.
The former might be in error by hundreds, and the latter might be in error by hundreds of
thousands, but both have two significant digits (5 and 2).
This reflects the fact that the significance of the error (its likely size relative to the size of the
quantity being measured) is the same in both cases.
Computer representations of floating point numbers typically use a form of rounding to significant
digits, but with binary numbers.
The term "significant digits" can also refer to a crude form of error representation based around
significantdigit rounding; for this use, see significance arithmetic.
Identifying significant digits
The rules for identifying significant digits when writing or interpreting numbers are as follows:
All nonzero digits are considered significant. For example, 91 has two significant digits (9 and 1),
while 123.45 has five significant digits (1, 2, 3, 4 and 5).
Zeros appearing anywhere between two nonzero digits are significant. Example: 101.12 has five
significant digits: 1, 0, 1, 1 and 2.
Leading zeros are not significant. For example, 0.00052 has two significant digits: 5 and 2.
Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has
six significant digits: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant
digits (the zeros before the 1 are not significant). In addition, 120.00 has five significant digits.
This convention clarifies the accuracy of such numbers; for example, if a result accurate to four
decimal places is given as 12.23 then it might be understood that only two decimal places of
accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four
decimal places.
The significance of trailing zeros in a number not containing a decimal point can be ambiguous.
For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and
just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the
nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:
Scientific notation
Generally, the same rules apply to numbers expressed in scientific notation. However, in the
normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits
are significant.
For example : 0.00012 (two significant digits) becomes 1.2x104, and 0.00122300 (six signific
ant digits) becomes 1.22300x103. In particular, the potential ambiguity about the significance of
trailing zeros is eliminated. For example, 1300 to four significant digits is written as 1.300x103,
while 1300 to two significant digits is written as 1.3x103.
Rounding
To round to n significant digits:
If the first nonsignificant digit is a 5 followed by other nonzero digits, round up the last significant
digit (away from zero). For example, 1.2459 as the result of a calculation or measurement that
only allows for 3 significant digits should be written 1.25.
If the first nonsignificant digit is a 5 not followed by any other digits or followed only by zeros,
rounding requires a tiebreaking rule. For example, to round 1.25 to 2 significant digits, Round
half up rounds up to 1.3, while Round half to even rounds to the nearest even number 1.2.
Replace any nonsignificant digits by zeros.
Arithmetic
For multiplication and division, the result should have as many significant digits as the measured
number with the smallest number of significant digits.
For addition and subtraction, the result should have as many decimal places as the measured
number with the smallest number of decimal places.
In a logarithm, the numbers to the right of the decimal point is called the mantissa and the number
of significant figures must be the same as the number of digits in the mantissa. When taking
antilogarithms, the resulting number should have as many significant figures as the mantissa in
the logarithm.
When performing a calculation, do not follow these guidelines for intermediate results; keep as
many digits as is practical to avoid rounding errors.
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
ad
ded.
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
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
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.
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