Question

Without calculating the result, how do you find the number of significant figures in the following products and quotients?
a.${\text{0}}{{.005032 \times 4}}{\text{.0009}}$
b.${\text{0}}{{.0080750/10}}{\text{.037}}$

Answer 1

Hint: In the above question, we are asked to find the significant digits of a given number. Significant digits can be defined as the digits which are used for measurement purposes. To answer this question, we have to look at the rules of significant digits. The answer will have significant numbers equal to the lower significant number of the two operands.

Complete step-by-step answer:We know that the significant figures are sometimes referred to as the significant digits or precision of a number. They are digits which carry meaningful contributions to its measurement resolution.
Let us now look at the rules of significant figures:
1.All non-zero digits are considered significant. For example, 77 has two significant figures.
2.Zeros appearing anywhere between two significant figures are significant. For example, 101 has 3 significant figures. Zero is counted as a significant figure as it lies between 2 significant digits (1).
3.Zeros which are present to the left side of the significant figures are not significant. In other words, leading zeros are insignificant. For example, 0.00037 has two significant figures, that is, 3 and 7.
4.Zeros which are present to the right of the non-zero digits, that is, the trailing zeros are significant if they are to the right of the decimal point as these are necessary to indicate precision.
5.In multiplication and division, the significant digits is equal to number having least significant number.
In the above question, the number given to us is:
a.${\text{0}}{{.005032 \times 4}}{\text{.0009}}$: ${\text{0}}{\text{.005032}}$ has leading zeros which are insignificant. So, it has 4 significant numbers. ${\text{4}}{\text{.0009}}$ has zeros between 2 significant numbers, 4 and 9. So, all are significant. So, it has 5 significant numbers. Hence, their product will have 4 significant figures.
b.\[{\text{0}}{\text{.0080750/10}}{\text{.037}}\]: ${\text{0}}{\text{.0080750}}$ has leading zeros which are insignificant. So, it has 5 significant numbers. \[{\text{10}}{\text{.037}}\] has zeros between 2 significant numbers, 1 and 3. So, all are significant. So, it has 5 significant numbers. Hence, their product will have 5 significant figures.

Note:The basic concept of significant figures is generally used in connection with rounding. Rounding to significant figures is far better than rounding to n decimal places, since it handles numbers of different scales in a uniform way.