Question

The sum of $\bar 2.75$ and $\bar 3.78$ is?

Answer 1

Hint: The bar on the number before the decimal point means that the number is repeated so many times on the left hand side of the decimal. That is the number $\bar 2.75$ essentially means (2222222…….).75.
We follow the rule of addition of p – adic numbers to solve this question.

Complete step-by-step answer:
Given data, the numbers are
$\bar 2.75$
$\bar 3.78$

That means these numbers can be expressed as:
$\bar 2.75$= (222222222……. and so on).75
$\bar 3.78$= (333333333…….. and so on).78
These types of numbers are called the p – adic numbers.

In mathematics, P – adic numbers are a form of rational numbers which extend in a different way from the rational number system to the real and complex number system.
The P in the p – adic number generally refers to a prime number.

Therefore, the addition of these numbers is as follows:
$\bar 2.75$+$\bar 3.78$
The above operation is expressed as:
= (-2 + 0.75) + (-3 + 0.78)
= ((-2) + (-3) + (0.75 + 0.78))
= (-5 + 1.53)
= (-5 + 1 + 0.53)
= (-4 + 0.53)
= (-4.53)
This can again be expressed in its actual number form that is $\bar 4.53$.
Therefore the sum of $\bar 2.75$ and $\bar 3.78$ is $\bar 4.53$.

Note: In order to solve this type of questions the key is to know the concept of p – adic numbers and how to perform operations on them.
It is possible to prove that every p – adic number has a positive negation, which is why we perform their addition by subtracting the repeating term before the decimal. Because it is unknown how many repetitions the number under the bar undergoes.
Similarly, p – adic numbers are eligible to perform operations like subtraction, multiplication and division.