When a whole number cannot be evenly divided by another, the answer can be expressed as a decimal, remainder, or fraction.
Each option has its advantages and disadvantages depending on the context. However, decimal answers are recognised as the standard.
While most people are familiar with fractions, there is often a point of confusion between decimals and remainders.
We are going to explore the differences between the two as we jump into: are remainders the same as decimals?
Remainders and decimals can both be used to express a non-integer answer to a division question. They are not the same, rather they are different ways of representing the same thing.
Whether you express an answer as a remainder or a decimal can be a personal choice, demanded by an examination, or be required to perform further steps in a longer question.
So let's now find out what exactly a decimal is.
What is a decimal?
A decimal is a number that consists of both a whole and a fractional part.
The demarcation between the whole number and the fractional part is represented by a decimal point (.), which is the same as a full stop or period. You may also occasionally see a comma used in place of the point, though this is not standard practice.
As you move from left to right through the columns of a digit, you go from the largest value to the smallest. So with the number 123.45, the 1 is in the 100s column, the 2 is in the 10s columns, and the three is in the units column. The decimal point then marks the shift from whole to fractional numbers. So the 4 is in the 10ths column, and the 5 is in the 100ths column.
Decimals can be used to denote both integers and non-integers.
For example, in a running race, you may see the runners' times written as 10.00 seconds. This is still a decimal even though it is an integer, as it shows there is no value in the fractional part beyond the whole.
The runner's time may also be shown as 10.12 seconds. This is also a decimal and denotes a non-integer as the number shown is not a whole number as the .12 represents that fractional part.
When it comes to division, decimals are often used to show a precise answer that is not an integer.
- 25 / 2 = 12.5
- This is true because 12 x 2 = 24 and 0.5 x 2 = 1.
- 1 + 24 = 25.
The decimal system is the standard system and is used and understood around the world. It is sometimes also known as decimal notation.
What is a remainder?
In maths, the remainder is the amount that is left over from an integer (whole number) after performing a computation, usually division. The remainder is also an integer.
When you divide a number by another number, you may either be left with a whole number or a decimal number. You can either write the decimal answer or represent the same answer as an integer with a remainder.
The remainder is represented by the letter "r."
- 15 / 4 = 3 r 3
In this example, 15 can be split into 4 equal parts 3 times. Once the equal split has been done, you will be left with an unequal part of 3, the remainder.
You can also think of it the other way around. So 4 x 3 = 12. Then to get from 12 to 15, you need to add a group of 3 because 15 - 12 = 3.
How do remainder and decimal answers differ?
The answer to any division question that involves two integers can be expressed in either remainder or decimal form.
Let's take our previous example, 15 / 4.
- We have already seen that the solution to 15 / 2 can be written as 3 r 3.
- But it can also be expressed as 15 / 4 = 3.75
So, 4 goes into 15 a total of 3.75 times, or 4 goes into 15 3 times but leaves a remainder of 3 (as 15 - 12 = 3).
Should you use a decimal or a remainder answer?
Decimal answers to division questions are the standard that is recognised across the world. So, if in doubt, use a decimal.
However, if you have an exam question that specifically asks you to represent the answer using remainder, then, of course, you should use the remainder.
So why do we have remainders?
Remainders are useful in everyday life outside the maths classroom for dividing things that cannot be split.
For example, if you have 10 sweets and 3 children and don't want one to be favoured over the others, then you should give each child 3 sweets with a remainder of 1.
Why are some numbers not evenly divisible?
Odd whole numbers can never be divided evenly by even numbers. So any division calculation that involves a whole odd integer being divided by a whole even integer will always result in a final answer that is either a decimal, remainder, or fraction.
The same is not true the other way round. You can divide even numbers by whole numbers evenly.
Dividing prime numbers by any number other than 1 or itself will also always result in a decimal value, remainder, or fraction.
Remainders are not the same as decimals. They are two different ways of expressing a non-integer answer to a division question. While decimals are the standard in mathematics, remainders can also be very helpful for real-life situations.