Show that 5−3 is irrational.
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Step-by-Step Solution
Step 1: Assume for contradiction it is rational
To prove that 5−3 is irrational, we will use the method of contradiction. We start by assuming that 5−3 is a rational number. If it is rational, it can be expressed in the form ba, where a and b are integers, b=0, and a, b are coprime (have no common factors other than 1).
Step 2: Isolate the irrational term
Now we rearrange the equation to isolate the irrational term, 3, on one side. We subtract ba from both sides and add 3 to both sides to achieve this.
Step 3: Simplify the rational expression
We combine the terms on the right-hand side of the equation into a single fraction. Since a and b are integers and b=0, 5b - a will be an integer and b will be a non-zero integer. Therefore, the expression b5b−a represents a rational number.
Step 4: State the contradiction
From reference-1, we know that 3 is an irrational number. However, the right-hand side of our equation, b5b−a, is a rational number because it is expressed as a ratio of two integers. This leads to a contradiction: an irrational number cannot be equal to a rational number.
Step 5: Conclude the proof
Since our initial assumption that 5−3 is rational leads to a contradiction, this assumption must be false. Therefore, 5−3 must be an irrational number.