Arithmetic » Rounding off and significant digits

Contents

Decimals
Rounding off
    Special cases
    Negative numbers
Significant digits
Calculating with significant digits

Decimals

A digit behind the decimal sign is called a decimal.
For example: 23.5 has one decimal and 4.375 has three decimals.

Rounding off

Rounding off a number goes as follows:
You always have to look at the decimal directly behind the decimal you round off to.

Is that a 0, 1, 2, 3 or 4? Than the decimal you round off to remains unchanged. In this case you round down as you make the number smaller by removing the last digits.
Is that a 5, 6, 7, 8 or 9? Than the decimal you round off to is raised by one. In this case you round up as you make the number bigger.

Do you think it is difficult?
Put a coloured line directly behind the decimal you round off to.

'Is approximately equal to'-sign

When you round off a number, you use the ≈-sign.
You pronounce this as 'is approximately equal to'

Example 1

Round 15.4769 off to two decimals.
The third decimal is a 6, therefore we have to round up.
Answer: 15.47|69 ≈ 15.48

Example 2

Round 23.545 off to one decimal.
The second decimal is a 4, therefore we have to round down.
Answer: 23.5|45 ≈ 23.5

Example 3

Round 19.027 off to a whole number.
The first decimal is a 0, therefore we have to round down.
Answer: 19.|027 ≈ 19

Special cases

Example 4

Round 7.7498 off to three decimals.
The fourth decimal is a 8, therefore we have to round up.
When we raise the 9 with one, we will get a 10.
For this reason you add one to the decimal in front of the 9.
Answer: 7.749|8 ≈ 7.750
The 0 remains in your answer!
By doing this, you show that the number is precise to three decimals.

Example 5

Round 1.49951 off to three decimals.
The fourth decimal is a 5, therefore we have to round up.
When rounding the 9 up, we will get 10.
For this reason you add one to the decimal in front of the 9.
Because this is also a 9 we have to do this again.
Answer: 1.499|51 ≈ 1.500

Example 6

Round 12 453 789 off to thousands.
The thousands are the fourth digit from the right.
Therefore you look at the third digit from the right.
This is a 7, therefore we have to round up.
Answer: 12 453|789 ≈ 12 454 000

Negative numbers

For negative numbers the rules are exactly the same.

Example 7
Round –17.7488 off to two decimals.
The third decimal is a 8, therefore we have to round up.
The second decimal changes from 4 into 5.
Answer: –17.748|8 ≈ –17.75

Brain teaser
Although the value of the rounded up number has actually become smaller (because –17.75 < –17.7488) we still say that we 'rounded up', because we raised the digit 4 to 5.

Significant digits

Significant digits are also known as significant figures. Significant digits are the digits that carry meaning in the number. The digits tell you how precise the measurement was. Digits behind the decimal separator also count, even zeros.

Examples

numbernumber of significant digits
11
2153
73.43
173.405 (the zero at the end counts too)
410.8006 (the zeros at the end count too)
120 473.344    9
073.434 (the zero at the front does not count)
0.0442 (the zeros at the front do not count)

Let's measure something with a metric ruler (with a mark for every millimetre). Something measures exactly halfway between 15 and 16 mm. Therefore, we can estimate the length to be 15.5 mm. We could also write this as 1.55 cm. In both cases we have three significant digits. Even when we convert the length into metres (0.0155 m) we will still have three significant digits.

Now we measure something with a metric ruler with only marks every cm. We measure something at exactly 23 between 42 and 43 cm. We know that 23 ≈ 0.66667. We cannot write the length as 42.66667 cm as this would imply an accuracy that is not there. It is better to estimate the length to be 42.7 cm.

Problem with zeros at the end without a decimal separator

What if someone says the distance to his house is 12 km. We have two significant digits. Converted to metres this is 12 000 m. There are still two significant digits. This person will probably not live exactly at a distance of 12 km, but somewhere between 11.5 and 12.5 km.
If the same person says the distance to his house is 10 km, we again have two significant digits. Converted to metres this is 10 000 m. There are still two significant digits. The first zero is significant while the other zeroes are not. A zero at the end of a number without a decimal separator can therefore be significant or not. For example 72 400 can have 3, 4 or 5 significant digits.
Exception: The number 0 has one significant digit.

A solution is to convert the number to a different unit. With 1500 mg it is unclear how many zeroes are significant. If it had been written as 1,50 g we would have known the first zero is significant.

Another solution is to place an overline over the last digit that is significant.
So 1000 has two significant digits and 250 000 has four significant digits. 

Use the scientific notation

The easiest and most common solution is to use the scientific notation.
Then 1000 = 1.0 × 102 and 250 000 = 2.500 × 105.

Examples scientific notation

numbernumber of significant digits
 1.3 × 102    2 
 1.54 × 1043 
 7.2500 × 1015 (the zeros at the end count too) 

The measuring device also has a certain degree of accuracy

Every measuring device has its own accuracy. Hold two different rulers next to each other and sometimes you will notice they differ slightly. Digital instruments like a scale also have a certain accuracy. This can be found in the manual and is sometimes shown on the display.

Example
A digital scale shows 12.873 grams. The device has an accuracy of 0.01 grams. You write 12.87 ± 0.01 grams. This way you show that the number could also be 12.86 or 12.88.

Calculating with significant numbers

Adding and subtracting

When adding or subtracting with significant digits you round your final answer to the number of decimals of the given value with the smallest number of decimals.

Example 1
47.2 grams + 12.37 grams = 59.57 grams ≈ 59.6 grams
The smallest number of decimals in the given values is one: 47.2.

Example 2
15 grams + 2.15 grams = 52.15 grams ≈ 52 grams
The smallest number of decimals in the given values is zero: 15.

Other cases

For other calculations with significant digits you round your final answer to the least accurate given value, so the given value with the smallest number of significant digits.

Example 3
24.25 cm × 26.278 cm = 637.2415 cm2 ≈ 637.24 cm2
The smallest number of significant digits in the given values is two.

Example 4
A runner takes 62.27 seconds to run the 500 metres. We know the track has been laid accurately in centimetres. Calculate the speed of the runner in m/s?

Answer:
The track is accurately in centimetres so has a length of 500.00 metres. This is five significant digits. However, the measured time is in four significant digits. So, our final answer will have to be in four significant digits as well.
500.0062.27 = 8.029548... m/s ≈ 8.030 m/s

Example 5
Calculate the volume of a cube with edges measuring 25 cm.

Answer:
The original value has two significant digits. So our final answer should also be in two significant digits.
253 = 15 625 cm3 ≈ 1.6 × 104 cm3.

Watch out!

With mathematics normally significant numbers are not taken into account. I even think a lot of maths teachers will not be amused if you (only) give the answer 1.6 × 104 cm3 at example 5. With physics and chemistry most of the time you do have to look at the significant digits. Ask your teacher if you are unsure.


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