0.1 Significant digits

Multiplication and division

In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.

The product, 4.56 × 3.456, involves the multiplication of a number with three significant digits and another with four significant digits. The product should be expressed as a number with three significant digits. Using a calculator the number, 15.75936, appears on the display of the device. It should then be rounded to three significant digits. So the product expressed using three significant digits is 15.8. Note that in this example, upward rounding was employed.

Addition and subtraction

The results that one obtains through the operations of addition and subtraction follow the following rule. The result of addition and subtraction should have as many decimal places as the number with the smallest number of decimal places. For example the sum 13.678 + 2.59 should be expressed as 16.27.

Something to remember

When performing calculations, you are encouraged to keep as many digits as is practical. This practice will help you obtain more precise results by eliminating some of the error introduced by the rounding operation. Once you have a final answer, then one should go about applying the rules presented in this section to produce a result that is consistent from the standpoint of significant digits.

Example: finding current using ohm’s law

Let us put to work the ideas put forth earlier in the context of an example that involves calculations centering on Ohm’s Law. Let us suppose that we are presented with the circuit shown in Fig. 1.

Suppose that the voltage ( V ) is 15 V and that the resistance is ( R ) 3.3 kΩ. What is the value of the current ( I )?

Ohm’s Law tells us that the current is the ratio of the voltage and the resistance.

Substituting the values for V and R into the equation yields the result

Now, let us consider this result in the context of significant digits. Both the voltage and the resistance were expressed using two significant digits. We realize that the resulting value for the current ( I ) should contain two significant digits. We can accomplish this by expressing the result after rounding to two significant digit precision.

Thus the answer to this example should be expressed as 0.0045 A.

We may choose to express this number using engineering notation and the result becomes

This is the preferred manner for expressing the result.

Example: finding acceleration by means of newton’s law of motion

Sir Issac Newton was the first to formulate the relationship between force, mass and acceleration of an object. He found that a force ( F ) exerted on an object of mass ( m ) will produce an acceleration ( a ) according to the following equation

Let us assume that we have a mass that weighs 120.6 kg. How much force is required to accelerate the mass at a rate of 26.1 m/s ² ?

We recognize that the value for mass is expressed using four significant digits, while the value for the acceleration is expressed using only three significant digits. We substitute the values for mass and acceleration into Newton’s Law of Motion

The number (3147.66) represents what is displayed on the calculator upon calculation of the product. This number has six significant digits. We wish our result to be expressed by the smaller of the number of significant digits contained in the two numbers that were multiplied. In this case the number should be expressed using just three significant digits. So the result expressed using three significant digits is

In order to remove any uncertainty relating to the trailing 0, the result can be written in engineering notation as

This is the preferred form for the solution.

Exercises

1. Express the quantity 5.342 015 m using 3 significant digits.
2. Express the quantity 5.347 015 m using 3 significant digits.
3. How many significant digits are present in the number 0.1256? Repeat for 0.01256? Repeat for 0.012560?
4. A 5 V source is connected to a 22 Ω resistor. Express the current using 3 significant digits.
5. A 9 V source is connected to a resistor whose value is unknown. The current of the circuit is measured and found to be 125 mA. Express the value of the resistance using 3 significant digits.
6. A current measured to be 35.7 mA passes through a 2.36 MΩ resistor. Express the value of the voltage across the resistor using 3 significant digits.
7. The power delivered by an electrical source is equal to the product of the voltage of the source and the current that flows out of the source. Suppose that the voltage is 120 V and that the current is 856.7 mA. Express the power delivered by the source using 3 significant digits.
8. A mass of 5.00 kg undergoes an acceleration of 2.37 m/s ² . Express the force that is needed to produce this acceleration using 3 significant digits.
9. A force equal to 2.69 N is applied to a mass of 8.23 kg. Express the resulting acceleration using 3 significant digits.
10. A 78.9 kg object is accelerated at a rate of 2.10 m/s ² . How much force is required to produce this acceleration?