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How do you write a rational number as a decimal?
A rational number is the one that can be written as a ratio of two integers a and b, where b is not zero.
Answer: We write a rational number as a decimal; Terminating and Non-Terminating Repeating
While dividing a number a by b, if we get zero as the remainder, the decimal expansion of such a number is called terminating.
While dividing a number, if the decimal expansion continues and the remainder does not become zero, it is called non-terminating.
Below are the ways how we convert a rational number as decimals.
There are two types of decimal representations for rational numbers
- Non-Terminating Repeating
Conversion of Terminating rational number to decimal:
The rational number with a finite decimal part for which the long division terminates or ends after a definite number of steps is known as finite or terminating decimals.
Example: 2/5 when converted to decimal gives 0.4, 7/100, when converted to decimal, gives 0.007
Conversion of Non-Terminating rational number to decimal:
Non-terminating decimals are those which keep on continuing after the decimal point (i.e. they go on forever).
1) Assume the repeating decimal to be equal to some variable x
2) Write the number without using the bar and equal to x
3) Determine the number of digits having a bar on their heads or the number of digits before the bar for mixed recurring decimal.
4) If the repeating number is the same digit after decimal such as 0.2222... then multiply by 10, if repetition of the digits is in pairs of two numbers such as 0.7878... then multiply by 100 and so on.
5) Subtract the equation formed by step 2 and step 4.
6) then find the value of x in the simplest form.
Let's take an example of a repeating number 0.6666...
Let , x = 0.666... -------------- (1)
Multiplying 10 on both the sides we get,
10x = 6.666.. ----------------- (2) (This has to be chosen in such a way that on subtracting we get rid of the decimal)
Subtracting the two equations
10x - x = 6.666 - 0.666
9x = 6
x = 6/9 or 2/3
Let's take another example to understand this
Let x = 0.6565... --------------------- (1)
Multiplying 100 on both the sides
100x = 65.6565... -------------------- (2)
Subtracting the above equations
100x - x = 65.6565 - 0.6565
99x = 65
x = 65/99
Thus, we have learned to convert rational numbers in terms of the decimal.