Exact Numbers

Exact numbers are numbers that result from counting. For example, a ten is defined as 10 objects, and a kilogram is defined as 1000 grams. An exact number cannot be further simplified. Because they are not measured, these numbers have a certainty. This means that exact numbers have infinite significant figures.

Some examples of exact numbers are:

– Conversions between the different prefixes of the metric system: there are exactly 100 cm inside a metre

– Percentages: 1% means exactly 1 in 100

– Counted objects: the number of tables in a classroom

Measured Numbers

Measured numbers have a certain uncertainty, which depends on the measurement itself. These numbers always have a limited number of significant figures. For example, the weight of a bag of rice may be measured as 0.125 g, but it may be 0.128 g or 0.123 g because there is an inherent uncertainty.

Significant figures

Significant figures are used to represent the number in digits. The digits significantly represent the numbers to which they refer. The number of significant figures is the number of values after the first non-zero digit counting from the left. For example, 1.25 has three significant figures.

Define Significant Figures

Significant figures are the digits that make a measurement more accurate. For example, 1.672 has four significant figures.

Application of Significant Figures

1. Zeros are only significant if they are between non-zero digits. For example, 908,657.2 contains seven significant figures.
2. Zeros that are to the right or left of a different significant digit are not significant. For example, 0.012 has two significant digits. 
3. If a number has zeros to the right of a decimal point, these zeros are significant if they are not followed by one or more non-zero digits. For example, 14.00 contains four significant digits.
4. Non-zero zeros to the right of the decimal point are significant. For example, 0.00109800 contains six significant figures.
5. In terms of measurements, zeros to the right of the last non-zero digit are significant. For example, 230 m contains three significant figures.
6. In the case of a number that ends in zeroes without any decimal digits, the zeroes are not necessarily significant: 230 pounds can be defined as a number which contains either 2 or 3 significant figures. This confusion is very easy to clarify. Remember you can always use a standard exponential to represent numbers. Some clear examples are 190,000 km which can be taken as a 6 significant figures number. This number can also be transformed to a 2-significant figure number using an exponential representation. In this case, 190,000 km could be also expressed as 19×104 km.

Significant Figures Examples

Analyse next examples of significant figures.

230 – 3 significant figures

1.03 – 3 significant figures

4,703,000 – 4 significant figures

235.00 – 5 significant figures

0.0077 – 2 significant figures

Rounding of Significant Figures

Numbers can be rounded to obtain a smaller number of significant figures. If the first digit on the left has a value smaller than 5, the last digit must remain constant. Conversely, if it is greater than 5, the last digit can be rounded up to the largest integer value. If the leftmost digit is 5, rounding can be done either up or down. 

The Rules of Rounding

1. If the number after the digit to which you want to round up to is less than 5, you can exclude the numbers to the right side of this number.

For example: if we want to round 6.2536 to three significant digits, the value would be 6.25.

2. If the digit to the right of the rounding digit is a number greater than 5, then add 1 to the rounding digit, excluding the other numbers. 

For example: if we want to round 12.268 to four significant figures, its value would be 12.27.

More Examples

Some other examples are shown in this section:

Let’s check first the number of significant figures in different numbers. Remember five rules:

Now it is time to test the rounding skills. Check the rounding of the next numbers in the desired number of significant figures.

Operations

Depending on the type of operations, the manipulations between significant figures can vary. For instance, when it is subtraction or addition, the result will have as many significant figures after the decimal point as the bigger number. For example, if you have 5.21 and 6.1 the result will have at least 2 significant figures after the decimal point.

5.21+6.1=11.31