SIGNIFICANT DIGITS

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This chapter presents the concept of significant digits and the application of significant digits in a calculation.

EO 1.8DETERMINE the number of significant digits in a given number.

EO1.9  Given a formula, CALCULATE the answer with the appropriate number of significant digits.

Calculator Usage, Special Keys

Most calculators can be set up to display a fixed number of decimal places. In doing so, the calculator continues to perform all of its internal calculations using its maximum number of places, but rounds the displayed number to the specified number of places.

INV key

To fix the decimal place press the INV key and the number of the decimal places desired. For example, to display 2 decimal places, enter INV 2.

Significant Digits

When numbers are used to represent a measured physical quantity, there is uncertainty associated with them. In performing arithmetic operations with these numbers, this uncertainty must be taken into account. For example, an automobile odometer measures distance to the nearest 1/10 of a mile. How can a distance measured on an odometer be added to a distance measured by a survey which is known to be exact to the nearest 1/1000 of a mile? In order to take this uncertainty into account, we have to realize that we can be only as precise as the least precise number. Therefore, the number of significant digits must be determined.

Suppose the example above is used, and one adds 3.872 miles determined by survey to 2.2 miles obtained from an automobile odometer. This would sum to 3.872 + 2.2 = 6.072 miles, but the last two digits are not reliable. Thus the answer is rounded to 6.1 miles. Since all we know about the 2.2 miles is that it is more than 2.1 and less than 2.3, we certainly don't know the sum to any better accuracy. A single digit to the right is written to denote this accuracy.

Both the precision of numbers and the number of significant digits they contain must be considered in performing arithmetic operations using numbers which represent measurement. To determine the number of significant digits, the following rules must be applied:

Rule 1:  The left-most non-zero digit is called the most significant digit.

Rule 2:  The right-most non-zero digit is called the least significant digit except when there is a decimal point in the number, in which case the right-most digit, even if it is zero, is called the least significant digit.

Rule 3:  The number of significant digits is then determined by counting the digits from the least significant to the most significant.

Example:

In the number 3270, 3 is the most significant digit, and 7 is the least significant digit.

Example:

In the number 27.620, 2 is the most significant digit, and 0 is the least significant digit.

When adding or subtracting numbers which represent measurements, the right-most significant digit in the sum is in the same position as the left-most least significant digit in the numbers added or subtracted.

Example:

15.62 psig + 12.3 psig = 27.9 psig

Example:

401.1 + 50 = 450

Example:

401.1+50.0=451.1

When multiplying or dividing numbers that represent measurements, the product or quotient has the same number of significant digits as the multiplied or divided number with the least number of significant digits.

Example:

3.25 inches x 2.5 inches = 8.1 inches squared

Summary

The important information from this chapter is summarized below.

Significant Digits Summary

Significant digits are determined by counting the number of digits from the most significant digit to the least significant digit.

When adding or subtracting numbers which represent measurements, the right­most significant digit in the sum is in the same position as the left-most significant digit in the numbers added or subtracted.

When multiplying or dividing numbers that represent measurements, the product or quotient has the same number of significant digits as the multiplied or divided number with the least number of significant digits.