Significant Figures and Precision

The significant figures of a number are those digits that carry meaning contributing to its precision.

Thus the number of significant digits depends on the least count of the measuring instrument.

All the certain digits and the one uncertain digit are called the significant figures in the measured value.

RULES TO FIND SIGNIFICANT FIGURES

Specifically, the rules for identifying significant figures when writing or interpreting numbers are as follows:

All non-zero digits are considered significant.

91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).

Zeros appearing anywhere between two non-zero digits are significant.

101.1203 has seven significant figures
(1, 0, 1, 1, 2, 0 and 3.)

Leading zeros are not significant.

0.00052 has two significant figures: 5 and 2.

Trailing zeros in a number containing a decimal point are significant

12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0.

NOTE.The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant).

The significance of trailing zeros in a number not containing a decimal point can be ambiguous.

It may not always be clear if a number like 1300 is precise 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:

i) A bar may be placed over the last significant figure; any trailing zeros following this are insignificant.

For example: $13\bar { 0 } 0$ has three significant figures (and hence indicates that the number is precise to the nearest ten).

ii) The last significant figure of a number may be underlined.

For example: "$2000$" has two significant figures.

iii) A decimal point may be placed after the number.

For example "100." indicates specifically that three significant figures are meant.

Scientific notation: In most cases, 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.

0.00012 (two significant figures) becomes $1.2\times { 10 }^{ -4 }$, and 0.00122300 (six significant figures) becomes 1.22300×${ 10 }^{ -3 }$.

In particular, the potential ambiguity about the significance of trailing zeros is eliminated.

1300 to four significant figures is written as 1.300×${ 10 }^{ 3 }$, while 1300 to two significant figures is written as 1.3×${ 10 }^{ 3 }$.

The part of the representation that contains the significant figures (as opposed to the base or the exponent) is known as the significand or mantissa.

The exact numbers appearing in the mathematical formulae of various physical quantities have infinite number of significant figures.

Perimeter of a square is given by $4 \times side$. Here 4 is an exact number and has infinite number of significant figures. Therefore it can be written as 4.0 or 4.00 or 4.000 as per the requirement.