Proving That a Sum Is Irrational

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  • 0:04 Rational & Irrational Numbers
  • 1:21 Proving a Sum is Irrational
  • 3:25 Sum of Two Irrational Numbers
  • 4:25 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will briefly review the definitions of rational and irrational numbers. We will then look at how to prove that a sum is irrational based on its addends being rational or irrational.

Rational and Irrational Numbers

You are probably familiar with rational and irrational numbers, but did you know that according to legend, when irrational numbers were first discovered, someone died for it?

You see, rational numbers are numbers that can be written in the form a/b, where a and b are integers. Back in the time of Pythagorus, rational numbers were well known and the Pythagoreans were comfortable working with them.

Irrational numbers, on the other hand, are numbers that cannot be written as a fraction. They continue on forever past their decimal point with no repeating pattern, and before they were discovered, the Pythagoreans knew nothing about them.

One day, a Pythagorean named Hippasus was working on a math problem and came across the number √2. Today, we know that this value is an irrational number, but when Hippasus tried to calculate √2, he quickly realized that it just kept going.

When he reported his findings to his fellow Pythagoreans, they were entirely freaked out by this type of number, because they didn't know how to work with it. Because of this, they took Hippasus out to sea and drowned him!

Poor Hippasus! Thankfully, today if we discover something about rational or irrational numbers, it is celebrated, and we don't have to worry about losing our lives over it! Speaking of discovery, let's talk a bit about sums of rational and irrational numbers.

Proving a Sum is Irrational

In some cases, we can determine whether or not a sum is going to be irrational by simply looking at its addends and seeing whether they are rational or irrational. For instance, to add two rational numbers together, we just use the rule for adding fractions:

a/b + c/d = (ad + bc) / bd (a rational number)

We see that the sum of two rational numbers will always be a rational number, and we say that the rational numbers are closed under addition.

Now, consider the sum of a rational number and an irrational number. When we are adding a rational and an irrational number, we will always get an irrational number.


The sum of a rational number and an irrational number is an irrational number
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To prove this, we can use an indirect proof, also called a 'proof by contradiction'. In an indirect proof, we prove that something is true by assuming that it is not true and finding a contradiction.

Thus, to prove that the sum of a rational and an irrational number is always irrational, we assume the opposite (that a sum of an irrational and a rational number is rational) and this will lead to a contradiction.

Proof:

Assume that a is rational, b is irrational, and a + b is rational. Since a and a + b are rational, we can write them as fractions.

Let a = c/d and a + b = m/n

Plugging a = c/d into a + b = m/n gives the following:

c/d + b = m/n

Now, let's subtract c/d from both sides of the equation.

b = m/n - c/d, or
b = m/n + (-c/d)

Since the rational numbers are closed under addition, b = m/n + (-c/d) is a rational number. However, the assumptions said that b is irrational and b cannot be both rational and irrational. This is our contradiction, so it must be the case that the sum of a rational and an irrational number is irrational.

And that's our proof!

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