Questions tagged [significant-figures]

The term significant figures refers to the number of important single digits ($0$ through $9$ inclusive) in the coefficient of an expression in scientific notation .

The number of significant figures in an expression indicates the confidence or precision with which an engineer or scientist states a quantity.

• The first significant figure of a number is the first digit which is not zero. Hence the first significant figure of $20,499$ is $2$ and the first significant figure of $0.0020499$ is $2$.

• The second significant figure of a number is the digit after the first significant figure. This is true even if the digit is zero. Hence the second significant figure of $20,499$ is $0$, as is the second significant figure of $0.0020499$.

• The third significant figure of a number is the digit after the second significant figure. This is true even if the digit is zero, and so on. Hence the third significant figure of $20,499$ is $4$ and the fourth is $9$, as are the third and fourth significant figures of $0.0020499$.

We round a number to three significant figures in the same way that we would round to three decimal places. We count from the first non-zero digit for three digits. We then round the last digit. We fill in any remaining places to the right of the decimal point with zeros. This is because we need them to hold the correct place value for the significant digits.

For example, $20,499$ to three significant figures is $20,500$. We round up because the first figure we cut off is $9$. $\quad 0.0020499$ to three significant figures is $0.00205$. We do not put any extra zeros in to the right after the decimal point. This is because we do not need them to hold the correct place value for the significant digits.

If the last significant digit of a number is $0$, we include this. For example, $0.0020499$ to two significant figures is $0.0020$. The first significant digit is $2$, the second significant digit is $0$. The next digit is $4$, so we round down.

In any calculation, the number of significant figures in the solution must be equal to, or less than, the number of significant figures in the least precise expression or element.

Consider the following product:

$$2.56 \times 10^{67} \times (-8.33) \times 10^{-54}$$

To obtain the product of these two numbers, the coefficients are multiplied, and the powers of $10$ are added. This produces the following result:

$$2.56 \times (-8.33) \times 10^{{67}+(-54)}$$ $$= 2.56 \times (-8.33) \times 10^{67-54}$$ $$= -21.3248 \times 10^{13}$$

The proper form of common scientific notation requires that the absolute value of the coefficient be larger than $1$ and less than $10$. Thus, the coefficient in the above expression should be divided by $10$ and the power of $10$ increased by one, giving:

$$-2.13248 \times 10^{14}$$

Because both multiplicands in the original product are specified to only three significant figures, a scientist or engineer will round off the final expression to three significant figures as well, yielding:

$$-2.13 \times 10^{14}$$

as the product.

Reference:

https://en.wikipedia.org/wiki/Significant_figures