Approximation and significant figures

All measurements are approximations. There is no measuring device that can provide great measurements without experimental ambiguity. Conventionally, a mass measured to 13.2 g is said to possess an absolute ambiguity of plus or minus 0.1 g and is said to have been measured to the nearest 0.1 g. In other words, we are rather unsure about that last digit—it could be a "2"; then again, it could be a "1" or a "3". A mass of 13.20 g shows an absolute ambiguity of plus or minus 0.01 g.

A "significant figure" is merely the number of figures that are known with some degree of precision and accuracy. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures.

Rules for computing the number of significant figures in a measured quantity:

(1) All nonzero figures are significant:

1.234 g is made up of 4 significant figures,

1.2 is made up of 2 significant figures.

(2) Zeroes between nonzero digits are significant:

1002 kg has 4 significant figures,

3.07 mL has 3 significant figures. 

(3) Preceding zeros to the left of the first nonzero digits are non significant; zeroes like that just show the position of the decimal point: 

0.001 oC is made up of just 1 significant figure,

0.012 g is made up of just 2 significant figures.

(4) Trailing zeroes that are as well to the right of a decimal point in a number are significant: 

0.0230 mL has 3 significant figures,

0.20 g has 2 significant figures. 

(5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not essentially significant: 

190 miles may be 2 or 3 significant figures,

50,600 calories may be 3, 4, or 5 significant figures. 

The possible ambiguity in the last rule can be avoided by making use of standard exponential, or "scientific," notation. For instance, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as:

5.06 × 104 calories (3 significant figures)

5.060 × 104 calories (4 significant figures), or

5.0600 × 104 calories (5 significant figures).

By writing a number in scientific notation, the number of significant figures is totally shown with the number of numerical figures in the 'digit' term as illustrated by the following examples. This approach is a reasonable convention to follow.

An "exact number"

A few numbers are exact due to the fact that they are known with absolute certainty.

The majority of numbers are integers: precisely 12 inches are in a foot, there may be precisely 23 students in a class. Exact numbers are frequently obtained as conversion factors or as counts of objects.

Exact numbers can be taken to possess infinite number of significant figures. Therefore, the number of obvious significant figures in any exact number can be ignored as a limiting factor in deciding the number of significant figures in the result of a calculation.

Rules for mathematical operations

While doing calculations, the general rule is that the preciseness of a calculated result is limited by the least correct measurement involved in the calculation. 

(1) In addition and subtraction, the result is given to the last common digit that exists farthest to the right in every constituent. Another way to mention this rule is as follows: in addition and subtraction, the result is rounded so that it has the same number of digits as the measurement that has the fewest decimal places (counting from left to right). For instance, 

100 (assume 3 significant figures) + 23.643 (5 significant figures) = 123.643,

Which ought to be rounded to 124 (3 significant figures). Observe, nevertheless, that it is possible two numbers possess no common digits (significant figures in the same digit column).

(2) In multiplication and division, the result ought to be summarized in order to have equivalent number of significant figures as in the constituent with the least number of significant figures. For instance, 

3.0 (2 significant figures ) × 12.60 (4 significant figures) = 37.8000

which ought to be rounded to 38 (2 significant figures).

Rules for approximating numbers

(1) If the digit to be rounded is greater than 5, the last digit that is retained ought to be raised by one. For instance, 13.6 is rounded off as 14.

(2) If the digit to be rounded is below 5, the last digit to be retained is left untouched

For instance, 

13.4 is rounded off as 13.

(3) If the digit to be rounded off is 5, and if the digit after it is not zero, the last remaining digit is increased by one. For instance, 

12.51 is rounded to 13.

(4) If the digit to be rounded off is 5 and is followed alone by zeroes, the last remaining digit is increased by one if it is odd, but left as it is if even. For instance, 

11.5 is approximated to 12, 

12.5 is approximated to 12.

This rule implies that if the digit to be rounded off is 5 that is followed alone by zeroes, the result is constantly rounded to the even digit. The rationale for this rule is to do away from encountering bias in approximating: half of the time we approximate up and half of the time we approximate down.

General guidelines for using calculators

When making use of a calculator, if you work the whole calculation without writing down any intermediate results, you may be unable to tell if an error is made. Further, even if you realize that there is an error, you may not be capable of telling where the error is.

In a long calculation that involves mixed operations, bear as many digits as possible all through the whole set of calculations and then approximate the final result correctly. For instance,

(5.00 / 1.235) + 3.000 + (6.35 / 4.0)=4.04858... + 3.000 + 1.5875=8.630829...

The first division ought to result in 3 significant figures. The last division ought to result in 2 significant figures. The three numbers added together ought to result in a number that is rounded off to the last regular significant digit that occurs to the farthest right; in this case, the final result ought to be rounded with 1 digit after the decimal. Therefore, the correct rounded end result ought to be 8.6. This final result has been limited by the preciseness in the last division.

Most modern calculators give you room to perform all the results of intermediate calculations in the display when doing a complicated series of calculations. By so doing, you can retain the results of each individual calculation step, and avoid entering back into the intermediate results. This practice may encourage rounding off too early. In this manner, you can totally refrain from truncation errors introduced by rounding intermediary results.

Note: Moving all digits through to the end result before approximating is essential for a lot of mathematical operations in statistics. Approximating intermediate results when calculating sums of squares can seriously affect the accuracy of the result.

QUESTIONS on significant figures

1. 37.76 + 3.907 + 226.4 = ?

2. 319.15 - 32.614 = ?

3. 104.630 + 27.08362 + 0.61 = ?

4. 125 - 0.23 + 4.109 = ?

5. 2.02 × 2.5 = ?

6. 600.0 / 5.2302 = ?

7. 0.0032 × 273 = ?

8. (5.5)3 = ?

9. 0.556 × (40 - 32.5) = ?

10. 45 × 3.00 = ?

11. What is the average of 0.1707, 0.1713, 0.1720, 0.1704, and 0.1715?

12. What is the standard deviation of the numbers in question 11?

13. 3.00 x 105 - 1.5 x 102 = ? (Provide the exact numerical result, and then express that result to the exact number of significant figures).

Answers to questions on significant figures

1. 37.76 + 3.907 + 226.4 = 268.1

2. 319.15 - 32.614 = 286.54

3. 104.630 + 27.08362 + 0.61 = 132.32

4. 125 - 0.23 + 4.109 = 129 (supposing that 125 is made up of 3 significant figures).

5. 2.02 × 2.5 = 5.0

6. 600.0 / 5.2302 = 114.7

7. 0.0032 × 273 = 0.87

8. (5.5)3 = 1.7 x 102

9. 0.556 × (40 - 32.5) = 4

10. 45 × 3.00 = 1.4 x 102

This answer supposes that 45 is made up of two significant figures; nevertheless, that is not unmistakable, due to the fact that it is not made up of any decimal point, and is also not expressed in scientific notation. If 45 is an exact number (e.g., a count), then the result ought to be 1.35 x 102.

11. For the question that asks what is the average of 0.1707, 0.1713, 0.1720, 0.1704, and 0.1715?

The average of these numbers is computed as 0.17118, which approximates to 0.1712 .

12. What is the standard deviation of the numbers in question 11?

The result that you obtain in calculating the standard deviation of these numbers depends on the number of digits retained in the intermediate digits of the calculation. For instance, if you make use of 0.1712 rather than the more precise 0.17118 as the mean in the standard deviation calculation, that will provide you with a wrong answer. Do not approximate your results in between as that will bring in proliferation of error into your calculations. A calculator gives 6.37965516309e-04

These figures ought to be rounded to 0.0006380, which would give 6.380 x 10-4 when expressed in scientific notation.

13. 3.00 x 105 - 1.5 x 102 =? (Provide the exact numerical result, and then express that result to the correct number of significant figures).

Identifying Rationals from Decimal Approximations

Assuming we had a decimal approximation of an unknown rational number, and we wanted to get back the original rational number that gave rise to the decimal approximation. One method is to make use of the fact that the digits of the decimal approximation of a rational number are finally periodic. If we have enough decimal places, we can find out the repeating part and decipher the original rational number from there.

For instance, if we had the repeating decimal

0.121212...,

we could let x = 0.121212..., multiply by 100, and then subtract x:

100x – x = 12.121212... – 0.121212... = 12.

Therefore we see that x = 12/99 = 4/33.

The above procedure works fine if we have enough of the decimal expansion to spot the repeating part. But suppose we don't have enough decimal places available to discern any repetition. For instance, 0.46017699115044247787 is the first 20 decimal places of a rational number with a moderately small denominator. Can you guess what the number is?

A good method is to look at the convergents from the continued fraction expansion of our number, due to the fact that they will be rational numbers which provide good approximations to the value. psalmfresh.blogspot.com