Question

What is fractional part in -0.7,0.7, -1.5 & -2.0

Answer 1

Hint: First, we will write the formula for relation between greatest integer function and fractional part of it. Then, we will put the given values in the relation and solve for each number and find the value of the fractional part of the number.

Complete step-by-step answer:
We know that,
A number can be written as the sum of the greatest integer of that number and the fractional part of it.
 \[ \Rightarrow x = \left[ x \right] + \left\{ x \right\}\]
First, we will find the fractional part of 0.7.
Which can be written as
 \[ \Rightarrow x = \left[ x \right] + \left\{ x \right\}\]
Putting the value of x as 0.7
 \[ \Rightarrow - 0.7 = [ - 0.7] + \{ - 0.7\} \]
The greatest integer of -0.7 is -1
 \[ \Rightarrow - 0.7 = - 1 + \{ - 0.7\} \]
We have to find the value of fractional part of -0.7
 \[ \Rightarrow \{ - 0.7\} = 1 - 0.7\]
On simplification we get,
 \[ \Rightarrow \{ - 0.7\} = 0.3\]
Now, we will find the value of fractional part of 0.7
 \[ \Rightarrow x = \left[ x \right] + \left\{ x \right\}\]
Here, we will put the value of\[x\] as 0.7
 \[ \Rightarrow 0.7 = [0.7] + \{ 0.7\} \]
The value of greatest integer of 0.7 is 0
 \[ \Rightarrow 0.7 = 0 + \{ 0.7\} \]
On simplification we get,
 \[ \Rightarrow 0.7 = \{ 0.7\} \]
The value of fractional part of 0.7 is 0.7
Now, we will find the value of fractional part of -1.5
Here we use the formula
 \[ \Rightarrow x = \left[ x \right] + \left\{ x \right\}\]
Put the value of x as -1.5
 \[ \Rightarrow - 1.5 = [ - 1.5] + \{ - 1.5\} \]
The value of $[ - 1.5]$ is -2
 \[ \Rightarrow - 1.5 = - 2 + \{ - 1.5\} \]
On rearranging we get,
 \[ \Rightarrow \{ - 1.5\} = 2 - 1.5\]
On simplification we get,
 \[ \Rightarrow \{ - 1.5\} = 0.5\]
Therefore, the fractional part of -1.5 is 0.5
Now, we will find the value of fractional part of -2.0
 \[ \Rightarrow x = \left[ x \right] + \left\{ x \right\}\]
Here, we will put the value of x as -2.0
 \[ \Rightarrow - 2.0 = [ - 2.0] + \{ - 2.0\} \]
As \[[ - 2.0] = - 2\], so we have,
 \[ \Rightarrow - 2 = - 2 + \{ - 2.0\} \]
On rearranging we get,
 \[ \Rightarrow \{ - 2.0\} = - 2 + 2\]
On simplification we get,
 \[ \Rightarrow \{ - 2.0\} = 0\]
Hence, the fractional part of -2.0 is 0.

Note: Greatest integer of a number is the integer less than or equal to the number. For example, the greatest integer of 1.8 would be the integer less than the given number 1.8 and the integer less than 1.8 is 1. So, the greatest integer of 1.8 is 1. And it is denoted by [x], where x is the number whose greatest integer is to be determined.