Error Arithmetic Operations of Significant Figures

What Are Significant Figures?

In Mathematics, we encounter questions where the solution comes out as 7.73. 

We have a practice of rounding off the figures; here, we may get confused that can 7.73 be written as such or 7.7 or 7.730 or 7.70?

Let’s take another example:

Suppose your friend’s salary got incremented to ₹55,000. Now, he tells his salary as ₹55,000, however close to the exact amount. 

Which means he told the rounded-off amount to you.

The above examples were to explain the significance of the figures or digits.

So, what is the definition of significant figures?

Significant figures in the measured value of a physical quantity which tells the number of digits in which we have a surety.

What Do You Understand by Significant Figures?

Let’s say you have a rod, and its length is measured 25.3 cm.

Here, the digit after the decimal point is called the ‘Doubtful digit’.

It’s because the measurement can be 25.2, or 25.28 or 25.345 cm as we are not confident about the value to be preferred, while the digits in the number ‘25’ are accurate because we have confidence upon these digits.

Significant Digits = Adding all the accurate digits + First doubtful digit.

There are common rules for counting significant figures that are discussed further.

What Are the Rules for Determining Significant Figures?

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Following are the common rules for counting significant figures in a given expression:

1. Rule 1: All non-zero digits are significant

For example, we have a number 7.744

It has four significant digits.

2. Rule 2: All zeroes occurring between two non-zero digits are significant

For example, 2007 has four significant digits, and 1.08079 has six significant digits.

3. Rule 3: Ending zeroes: The ending zero will be significant, if and only if, it is after the decimal.

For example, in a number, 3.400, the ending zero is after the decimal is significant. Now, according to rule 2, if the zeroes occur between the two significant digits, then they are also significant.

So, the number, 3.400, has four significant figures, i.e., 3, 4, 0, 0.

4. Rule 4 or Rule of Initial Zeroes: In a number less than one, all zeros to the right of the decimal point and the left of non-zero digits are NOT significant.

In simple words, the zeroes at the initial zeroes are not significant.

Let’s take a number, 0.00078

Here, we can see 0 at the initial (before the decimal point), which is not significant and the three non-ending zeroes. 

So, 0.00078 has two significant digits.

Let’s say, a number, 0.9080

Here initial zero is insignificant, and 9080 has four significant digits.

5. Rule 5: All zeroes on the right of the last non-zero digits in the decimal part are significant.

For example, a number, 0.00700, has three significant figures 7, 0, 0.

6. Rule 6: All zeroes on the right of non-zero digits are NOT significant.

Here, we have numbers like 420, 4300; the ending zero is not after the decimal.

So,  zero is not significant, and they have two significant digits only.

7. Rule 7: Power of 10 is insignificant

For example, 3.4 x 108 is the number, having 108 ending zeroes, but the power of 10 is always insignificant and having only two significant digits, i.e., 3 and 4.

Rules for Arithmetic Operation of Significant Figures

1. Addition 

Let’s say the sum of three measurements of length; 3.2 m, 1.88 m, and 1.056 m  is 6.136 m, rounded off to 6.1 m.

2. Subtraction

If we take x = 15.87 m, and y = 15.8 m, then,

x  - y = 15.87 - 15.8 = 0.07 m, which is rounded off to 0.1 m.

In the subtraction of quantities, the magnitude and accuracy are almost destroyed.

Here, 15.87 has four significant figures and 15.8 has three. So, on subtracting these two numbers, we get 0.01 m which has one significant digit.

3.  Multiplication

For example, x = 3.5 and y = 2.125, then xy = 7.4375

Here, out of 3.5 and 2.125, 3.5 has the least significant digits, i.e., 2. 

So, rounding off 7.4375 to two significant digits, which is 7.4.

4. Division

If x = 9600, y = 11.25, then

x/y = 9600/11.25 = 853.33..

Here, 9600 has the least significant digits, i.e., 2. So, the answer will also have two significant digits.

Therefore, 853.33 is rounded off to 830.

Error Arithmetic Operations of Significant Figures

Relative error or fractional error and percentage error

Relative error or fractional error is calculated by the formula:

 =  mean absolute error/ mean value = Δμ’/μ

Percentage error is calculated by the formula:

= Δμ’/μ x 100 %

The measurements of different rods have got these values 4.2, 3.5, 4.7, 4.4, and 5.4, respectively. Find the mean value, relative, and the percentage error.

Here, mean (μ) = (4.2 + 3.5 + 4.7+ 4.4 + 5.4)/5 = 4.44 ≈ 4.4

 Absolute errors in measurement are:

Δμ1 = 3.5 - 4.2 = - 0.7

           Δμ2 = 4.7 - 3.5 = 1.2

           Δμ3 = 4.4 - 4.7 = - 0.3

           Δμ4 = 5.4 - 4.4 = 1.0

So, mean absolute error, Δμ’ = (- 0.7 + 1.2 - 0.3 + 1.0)/4 = 0.3

Relative error: Δμ’/μ = 0.3 / 4.4 = +/- ≈ 0.068 ≈ 0.1

Percentage error: 0.1 x 100% = +/- 10 %