Why The Sum Of A Rational And Irrational Number Always Results In An Irrational Number – Math Explanation.

sum of rational number and irrational number is

irrational

The sum of a rational number and an irrational number is always an irrational number.

To understand why, let’s first understand the definitions of rational and irrational numbers.

A rational number is a number that can be represented as a ratio of two integers, i.e., p/q, where p and q are integers and q is not equal to zero. For example, 2/3, -7/8, and 0.5 are all rational numbers.

An irrational number is a number that cannot be represented as a ratio of two integers. Instead, it has a non-repeating and non-terminating decimal expansion. For example, √2, π, and e are all irrational numbers.

Now, let’s consider the sum of a rational number and an irrational number, say x and y, respectively. We can represent the rational number x as a fraction p/q, and the irrational number y has a non-repeating and non-terminating decimal expansion.

So, x + y = p/q + y. Since y is irrational, we cannot simplify this expression to a ratio of two integers. Therefore, x + y is a non-repeating and non-terminating decimal (since y is irrational), and it cannot be expressed as a ratio of two integers (since x is rational). Hence, x + y is an irrational number.

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