Significant Figures and Standard Forms

What is Significant Figure?

In measurement, significant figures indicates the certainty of the measurement. As the number of significant figures increases, the certainty of the measurement increases. In other words, we are more certain about what we have measured.

Identifying Significant Digits

The rules for identifying significant digits when writing or interpreting numbers are as follows:

• All non-zero digits are considered significant. Example: 1, 20, and 300 all have one significant figure. They are 1, 2, and 3 respectively. 123.45 has five significant figures: 1, 2, 3, 4 and 5.
• Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.
• Leading zeros are not significant. For example, 0.00012 has two significant figures: 1 and 2.
• Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures. This convention clarifies the precision 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.
• A number with all zero digits (e.g. 0.000) has no significant digits, because the uncertainty is larger than the actual measurement.

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Number of Significant Digits After an Operation

Standard Form

Standard form is a way of writing down very large or very small numbers easily. 10³ = 1000, so 4 × 10³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.
Small numbers can also be written in standard form. However, instead of the index being positive (in the above example, the index was 3), it will be negative.

The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

Example

Write 81 900 000 000 000 in standard form:

81 900 000 000 000 = 8.19 × 10¹³

It’s 10¹³ because the decimal point has been moved 13 places to the left to get the number to be 8.19

Example

Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10^-6

It’s 10^-6 because the decimal point has been moved 6 places to the right to get the number to be 1.2

On a calculator, you usually enter a number in standard form as follows:
Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.

Manipulation in Standard Form

This is best explained with an example:

Example

The number p written in standard form is 8 × 10⁵
The number q written in standard form is 5 × 10^-2
Calculate p × q. Give your answer in standard form.

Multiply the two first bits of the numbers together and the two second bits together:
8 × 5 × 10⁵ × 10^-2

= 40 × 10³ (Remember 10⁵ × 10^-2 = 10³)

The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.

= 4 × 10⁴

Calculate p ÷ q. Give your answer in standard form.

This time, divide the two first bits of the standard forms. Divide the two second bits.

(8 ÷ 5) × (10⁵ ÷ 10^-2)

= 1.6 × 10⁷