Question

Explain why the sum of a rational number and an irrational number must be irrational.

Short answer

The sum of a rational and an irrational number is irrational because assuming otherwise leads to a contradiction.

Step-by-step solution

Define a Rational Number

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. For example, if we let the rational number be represented as \( r \), then \( r = \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b eq 0 \).

Define an Irrational Number

An irrational number is a number that cannot be expressed as the quotient of two integers. Irrational numbers have non-terminating, non-repeating decimal expansions. An example of an irrational number is \( \theta \) where \( \theta \) cannot be represented as \( \frac{p}{q} \) for any integers \( p \) and \( q \).

Assume the Sum is Rational

Assume, for the sake of contradiction, that the sum of a rational number and an irrational number is rational. Let \( r \) be a rational number and \( \theta \) be an irrational number. Assume their sum \( r + \theta \) is a rational number. This means \( r + \theta \) can be expressed as \( \frac{m}{n} \) where \( m \) and \( n \) are integers and \( n eq 0 \).

Isolate the Irrational Number

From the assumption in the previous step, we have \( r + \theta = \frac{m}{n} \). Subtract the rational number \( r \) from both sides of the equation to isolate the irrational number: \( \theta = \frac{m}{n} - r \). Since \( r = \frac{a}{b} \), substitute this in: \( \theta = \frac{m}{n} - \frac{a}{b} \).

Combine the Fractions

Combine the fractions on the right-hand side to get a single fraction: \( \theta = \frac{m}{n} - \frac{a}{b} = \frac{mb - an}{nb} \). This expression of \( \theta \) is now in the form of \( \frac{x}{y} \) where \( x \) and \( y \) are integers and \( y eq 0 \).

Contradiction

Since \( \theta \) is isolated as a fraction of two integers, we have expressed \( \theta \) as a rational number (\( \frac{x}{y} \)). This contradicts the definition of \( \theta \) being an irrational number. Thus, the assumption that the sum of a rational and an irrational number is rational must be false.

Conclusion

Therefore, the sum of a rational number and an irrational number must be irrational, as assuming otherwise leads to a contradiction.

Key concepts

rational numbers

Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers. Importantly, the denominator cannot be zero. For example, consider the number \(3\). It can be expressed as \(\frac{3}{1}\). Another example is \(\frac{1}{2}\). In general, if a number can be written as \(\frac{a}{b}\), where \(a\) and \(b\) are integers, and \(b \eq 0\), then the number is rational.

irrational numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. These numbers have non-terminating, non-repeating decimal expansions. A classic example is \(\pi\), which starts as 3.14159... and goes on forever without repeating. Another well-known example is the square root of 2 (\(\sqrt{2}\)), which cannot be exactly expressed as a fraction of two integers.

proof by contradiction

Proof by contradiction is a logical method where we assume the opposite of what we aim to prove, and then show that this assumption leads to a contradiction. For instance, to prove that the sum of a rational and an irrational number is always irrational, we assume that their sum is rational. From this assumption, we derive a result that contradicts a known fact, thereby showing our initial assumption must be false. Hence, the original statement (that the sum must be irrational) is proven true.

fraction subtraction

Fraction subtraction is the process of subtracting one fraction from another. To subtract fractions, they must have a common denominator. For example, to subtract \(\frac{a}{b}\) from \(\frac{m}{n}\), we find a common denominator \(bn\) and rewrite each fraction accordingly: \(\frac{m}{n} - \frac{a}{b} = \frac{mb - an}{bn}\). This common denominator helps express the subtraction as a single fraction, making it easier to handle in equations or proofs.