Mathematicians don't like working with very long decimals because they are particularly unwieldy. This is why they use a technique called “rounding” or “estimation” to eliminate digits after the decimal point. To round, you have to know first how many digits after the decimal point you want to keep, then you have to look at the first digit to the right of them. If it is greater than or equal to 5, the last digit to be kept is increased by one, otherwise we leave this last digit as it is.
Part 1 of 2: rounding decimals
Step 1. Understand the meaning of a decimal point number based on its position
In any number, each digit refers to a different value. For example, the number “1” represents the number of thousands in 1,872, the number “8” the number of hundreds, and the number “2” the number of ones. When a number contains a comma, the digits to the right of it represent the fractions of the unit.
- Each position to the right of the decimal point is named according to the unit fraction it represents. The first digit to the right of the decimal point represents the tenths, the second, the hundredths, the third, the thousandths, etc.
- For example, in the number 2, 37589, the number "2" corresponds to the number of units, the number "3" (in the first position after the decimal point) represents the number of tenths, the number "7" (second position after the comma), the number of hundredths, the number "5" (3rd position), the thousandths, the number "8" (4th position), the ten-thousandths, and the number "9" (5th position), the hundred -thousandths.
Step 2. Determine to which position you want to round
In a school assignment, this position is indicated in the statement of the exercise. For example, the statement may specify that it is necessary to round to two digits after the decimal point.
- If you are asked to round the number 12 to the nearest thousandth (to three decimal places), 9789, you must first determine which digit after the decimal point is thousandths. Progressing to the right from the decimal point, the digits represent tenths (1st position), hundredths (2nd position), thousandths (3rd position), ten-thousandths (4th position) and hundred-thousandths (5th position). This means that the number "8" which corresponds to thousandths is the last number that we will keep.
- When the statement indicates, for example, that it is necessary to round to 2 digits after the decimal point (to the nearest hundredth), things are simpler, because there is no ambiguity.
Step 3. Look at the number just to the right of the last number you are going to keep
It is this figure that will determine the possible modification that must be made to round off.
In our example with the number 12, 9789, we must keep the thousandths (3rd digit after the decimal point), that is why we must be interested in the number "9" which is in the fourth position
Step 4. If this number is greater than or equal to 5, round the number by increasing the last digit to be kept by one
- In our example with the number 12, 9789, the digit in the fourth position being equal to 9, and therefore greater than 5, we must increase the thousandths digit by one. After rounding to the nearest thousandth, the number 12, 9789 takes its new value which is 12, 979. All digits after thousandths have been deleted.
Step 5. If this number is less than 5, leave the last number you want to keep as it is
This means that if the digit used for rounding is equal to 0, 1, 2, 3 or 4, we limit ourselves to eliminating all the digits which are located after the last digit to be kept. There is no further modification to the number to be rounded.
- For example, if we were to round the number 12, 9784 to the nearest thousandth (instead of 12, 9789), we would get the rounded number 12, 978.
- If this method sounds familiar to you, you have probably used it quite often in school or on other occasions where you have had to deal with numbers.
Step 6. Use this same technique to round a decimal number to a whole number
When we apply the rounding method based on the first digit after the decimal point, we speak of "rounding or unit truncation ". We use the technique described above, but this time we remove all the digits after the decimal point.
- We are interested in the digit which is located just after the decimal point, and if this is greater than or equal to 5, we increase the ones digit by one. If the digit after the decimal point is less than 5, the ones digit is kept as it is.
- For example, if you have to round the number 12, 9789 to the unit, you must be interested in the number which is located just after the decimal, which is equal to 9. Since 9 is greater than 5, you must increase by one. the number "2", and you get the number "
Step 13. As the rounding result. Since this is an integer, you should not keep the comma.
Step 7. Take all the instructions into account
In general, you are not asked for more than what is required in the examples given above. However, it happens that a additional instruction is given which modifies the way we must have to round the number.
- For example, if you are asked to round the number 4, 59 to the tenth inferior, you get the number 4, 5 after rounding, even if the hundredths digit ("9") is greater than 5.
- Likewise, if you are asked to round the number 180, 1 to unity superior, the fact that the tenths digit ("1") is less than 5 does not affect the result which is 181.
Part 2 of 2: Sample Problems
Step 1. Round 45.783 to the nearest hundredth
See the solution below.
- Start by determining the number that corresponds to hundredths. This is the second digit after the decimal point (45, 7
- Consider the number just after the hundredths (45, 78
- Since this ("3") is less than 5, you get the number 45, 78 as a result of rounding.
Step 2. Round 6.2979 to three decimal places
The statement is more explicit. It is equivalent to the following statement: Round 6.2979 to the nearest thousandth.
- Start by locating the third digit after the decimal point (6, 29
- Consider the number immediately after the number "7" (6, 297
- Since this ("9") is greater than 5, you get the number 6, 298 as a result of rounding.
Step 3. Round 11.90 to the nearest tenth
Don't be fooled by the number "0", and just take it as a number less than 5.
- Start by determining the number that corresponds to tenths. This is the first digit after the decimal point (11,
- Consider the number immediately after the tenths (11, 90).
- Since this ("0") is less than 5, you get the number 11, 9 as a result of rounding.
Step 4. Round off -8.7 to the unit
Don't be confused by the negative sign, because the rounding method works with negative numbers in exactly the same way as with positive numbers.
- Start by determining the number that corresponds to the ones (-
Step 8., 7).
- Consider the number that is just after the units, that is, just after the decimal point (-8,
- As this ("7") is greater than 5, you get -9 as a result of rounding. Obviously, you have to keep the negative sign.
- If you're having trouble interpreting the digits after the decimal point, check out this guide to fractions in decimals.
- You also have this practical tool for rounding decimals which will make it easier for you to handle particularly long numbers.