## Calculator Use

Add, subtract, multiply and divide with significant figures. Enter numbers, scientific notation or e notation and select the math operator. The calculator does the math and rounds the answer to the correct number of significant figures (sig figs).

You can use this calculator to double check your own calculations using significant figures.

Enter whole numbers, real numbers, scientific notation or e notation. Example inputs are 3500, 35.0056, 3.5 x 10^3 and 3.5e3.

Read more below for doing math with significant figures.

## What are Significant Figures?

Significant figures are the digits of a number that are meaningful in terms of accuracy or precision. These digits provide information about how precise a calculation or measurement might be.

## Significant Figures Rules

- Non-zero digits are always significant
- Zeros in between non-zero digits are always significant
- Leading zeros are never significant
- Trailing zeros are only significant if the number contains a decimal point

Examples of Significant Figures

How Many

Significant Figures?

Which Figures

are Significant?

81

2

8, 1

26.2

3

2, 6, 2

0.007

1

7

5200.38

6

5, 2, 0, 0, 3, 8

380.0

4

3, 8, 0, 0

78800

4

7, 8, 8, 0

78800.

5

7, 8, 8, 0, 0

## Rules for Adding and Subtracting with Significant Figures

- Find the place position of the last significant digit in the least certain number
- Add and/or subtract the numbers in your calculation as you normally would
- Round the answer to the place position of least significance that you found in step 1

### Example: Adding and Subtracting with Significant Figures

A step in your "Let's Make a Latte" chemistry lab assignment requires that you account for the volume of fluids in your latte.

You're starting with 7 oz. of milk, and your espresso machine uses 2.5 oz. of water to make a 2 oz. espresso shot -- the other 0.5 oz. remains in the espresso puck. Finally, your high tech milk steamer tells you how much water is used in the steaming process, out to 3 decimal places.

You make your espresso and see that you've pulled the perfect 2 oz. shot. You steam and froth your milk, and the steamer indicator says 0.063 oz. of water was used during the process. You need to add up 2 oz. espresso plus 7 oz. milk plus 0.063 oz. of steam. But because this is a chemistry lab assignment you have to do your math with significant figures.

Reviewing the rules for adding and subtracting with significant figures, **find the place position of the last significant digit of your least certain number**. Your milk and espresso are each one significant digit in volume, in the __ones__ place.

Adding the volumes of fluid in your latte you have:

7 oz. milk + 2 oz. espresso + 0.063 oz. water = 9.063 oz.

9.063 oz. rounded to the ones place = 9 oz.

Although you have a volume of fluids that seems accurate to the thousandths, you have to round to the ones place because that is the least significant place value. So following the rules of addition with significant figures you report that your latte is 9 oz. in volume.

## Rules for Multiplying and Dividing with Significant Figures

- For each number in your calculation find the number of significant figures
- Multiply and/or divide the numbers in your calculation as you normally would
- Round the answer to the fewest number of significant figures that you found in step 1

### Example: Multiplying and Dividing with Significant Figures

A word problem on a physics test goes like this: Marine scientists have identified a unique whale who calls at 52 hertz. We know that sound travels in air at about 343 meters per second. Given that the sound of speed travels 4.3148688 times faster in water than in air, what is the wavelength of the 52 Hz whale call?

The formula for wavelength is:

\( \lambda = \dfrac{v}{f} \)

Where

\( \lambda \) = wavelength, in meters

\( v \) = velocity, at meters per second

\( f \) = frequency, at hertz

So wavelength equals velocity divided by frequency. For this physics problem you have to multiply velocity of the speed of sound in air by 4.3148688 to get the velocity of the speed of sound in water. Then divide this number by 52 Hz to get the wavelength of the sound wave.

- \( \lambda = \dfrac{v}{f} \)
- \( \lambda = \dfrac{343 \times 4.3148688}{52} \)
- \( \lambda = \dfrac{1480}{52} \)
- \( \lambda = 28.4615384 \) meters

Following the rules for doing multiplication and division with significant figures you should round your final answer to the fewest number of significant figures given your original numbers. In this case 52 has the fewest number of significant digits, so you should round the final answer to 2 sig figs.

28.4615384 meters rounded to 2 sig figs = 28 meters. So in water, one wavelength of a 52 Hz whale call is 28 meters long.

### Note: Doing Math With Significant Figures

If you are entering a constant or exact value as you might find in a formula, be sure to include the proper number of significant figures.

For example, consider the formula for diameter of a circle, d = 2r, where diameter is twice the length of the radius. If you measure a radius of 2.35, multiply by 2 to find the diameter of the circle: 2 * 2.35 = 4.70

If you use this calculator for the calculation and you enter only "2" for the multiplier constant, the calculator will read the 2 as one significant figure. Your resulting calculation will be rounded from 4.70 to 5, which is clearly not the correct answer to the diameter calculation d=2r.

You can think of constants or exact values as having infinitely many significant figures, or at least as many significant figures as the the least precise number in your calculation. In this example you would want to enter 2.00 for the multiplier constant so that it has the same number of significant figures as the radius entry. The resulting answer would be 4.70 which has 3 significant figures.

### Related Calculators

To learn more about rounding significant figures see our Rounding Significant Figures Calculator.

For more about rounding numbers in general see our Rounding Numbers Calculator.

To practice identifying significant figures in numbers see our Significant Figures Counter.

### References

- Khan, Salman "Significant Figures,"
*The Khan Academy*. - Weisstein, Eric W. "Significant Digits,"
*MathWorld*--A Wolfram Web Resource.