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How many significant figures?

As we have studied in the measurement instruments section, when measuring any magnitude, mistakes are always made. But we not only commit them when reading the data from the measuring instruments but also in the results of arithmetic operations in which decimal numbers are involved. In this section, we will focus on how to treat the data obtained so that our calculations are as accurate as possible. In this article, we read about How many significant figures? Specifically, we will focus on:

The concept of a significant figure

What is rounding and how is it used?

Are you ready? Let’s go there.

Significant numbers

When carrying out measurement with a measuring instrument, it returns a value made up of a series of figures. This series of figures are called significant figures.

The set of digits that are known with certainty in a measure are called significant figures (cs).

Of all the significant figures there is always one, the last, that will be affected by an error. For this reason, the rest of the figures are called exact figures.

Digital thermometer

Digital thermometers used in practical medicine use 3 significant figures. The first two are exact figures and the last one is a significant figure affected by error since the real temperature will probably be made up of infinite decimals that are impossible to represent and which are also not necessary to determine whether the patient has a fever or not.

Rules for determining significant figures

Any number other than zero is considered significant.

Examples

25.36 m has 4 cs or 154 has 3 cs

Zeros located between two digits other than zero and those located after the decimal point are considered significant figures.

Examples

2005.20 has 6 cs or 34.00 has 4 cs

However, the zeros located at the beginning of a number are not considered significant figures, including those located to the right of the decimal point until reaching a digit other than zero.

Example

0.000560 has 3 cs (560)

The zeros at the end of a number without a decimal point are also not considered significant, except if they are indicated with a period.

Examples

450 has 2 cs (45), however, 450. has 3 cs

Example

In an electronic circuit, a multimeter is used that returns the following current measurements at different points in the circuit: 0.050 A, 10.050 A, and 0.101 A. How many significant figures do these measurements have?

Rounding

When we carry out some type of mathematical operation, it may be interesting to reduce the number of decimal places that we obtain to avoid working with excessively large values. Rounding can help this task by making the results as accurate as possible.

Rounding is the process of eliminating the figures to the right of the last significant figure.

Rules for rounding

When the first of the discarded digits is five or greater than five, the previous figure is increased by one.

Example

45.367892 rounded to 4 cs is 45.37. Since we have to stay with 4 figures, we must discard from the 5th onwards, that is, from 7. 7 is greater than 5 so we increase the previous one by one unit. Therefore: 45.37

When the first of the discarded digits is less than five, the previous figure remains the same.

Example

123.643421 rounded to 5 cs is 123.64. Since we have to stay with 5 figures, we must discard from the 6th onwards, that is, from 3. 3 is less than 5 so we leave the previous figure the same. Therefore: 123.64.

When we perform mathematical operations with decimal values, the result must be rounded to a specified number of significant figures.

When we add or subtract, the result must have the same number of decimal places as the value that has the least:

Example

12.07 + 3.2 = 15.27

When we multiply or divide, the result must have the same number of significant figures as the value that has the least:

Example : 12.07 · 3.2 = 39 (Not 38.624 since 3.2 has 2 cs)

Example

Given the following operations, round them correctly:

1. a) 150.00 / 8.65
2. b) 34.045 + 2.05

Also read: how fast does the earth spin

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