## Significant Figures and Scientific Notation

*See also: numbers*

The calculator below formats a number to the required significant figure and the scientific notation.

Rounding will be done using Gaussian rounding (also known as banker's rounding):

- Round up, if
- the next digit is greater than 5, or
- the next digit is 5 and there is a non-zero digit after, or
- the next digit is 5 and the current digit to be rounded is odd

- Round down, if
- the next digit is less than 5, or
- the next digit is 5 followed by only zero digits or not followed by any other digits and the current digit to be rounded is even

Please see below for the rules of significant figures and scientific notation.

Please report any error to [email protected]. Thank you.

### Significant Figures

The **significant figures** (or **significant digits**) of a number are the digits that are known with some degree of confidence.

#### Rules for significant figures:

- All non zero digits are significant.

E.g. 325 has 3 significant figures, 52.34 has 4 significant figures. - Zeros between non zero digits are significant.

E.g. 1009 has 4 significant figures, 3.02 has 3 significant figures. - Leading zeros are insignificant.

E.g. 0.0005 has 1 significant figures, 0.030 has 2 significant figures. - Trailing zeros in a number containing a decimal point is significant.

E.g 25.00 has 4 significant figures, 3.50 has 3 significant figures. - Trailing zeros in a number not containing a decimal point can be either significant or insignificant. To indicate which zeros are significant, place a bar over the zeros which are significant or indicate the number of significant figures in the brackets.

E.g 2500 has 2 significant figures, 35000 has 3 significant figures, 12000 has 4 significant figures, 800 (2 sf) has 2 significant figures.

#### Rules for addition and subtraction:

*The result of addition or subtration should have the same number of decimal places as the least number of decimal places in any number involved.*

For example: 40.55 + 3.1 + 10.222 = 53.872

The number with the least decimal places involved is 3.1 (1 decimal place), therefore your result should be rounded to 53.9 (3 significant figures).

#### Rules for multiplication and division:

*The result of multiplication or division should have as many significant figures as the number involved which has the least number of significant figures.*

For example: ( 32.1 × 1.234 ) ÷ 1.2 = 33.0095

The number with the least number of significant figures is 1.2 (2 significant figures). Therefore your result should be rounded to 33 (2 significant figures).

### Scientific Notation

**Scientific Notation** or **standard form** is a way of expressing a number in terms of power of ten. In other words, it's expressed in the form

*a* × 10^{n}

where *a* is a real number that satisfies 1 ≤ |*a*| < 10 and *n* is an integer. *a* is called the *significand* and *n* is called the *exponent*.

Please note that the absolute value of *a* must be at least 1 and less than 10, hence 0.34 × 10^{2} and -11.23 × 10^{4} are not in standard form.

Examples of converting numbers to scientific notation

- 1234 becomes 1.234 × 10
^{3} - -0.000023 becomes -2.3 × 10
^{-5} - 50000000 becomes 5 × 10
^{7}

*See also: numbers*