Why is the product of a rational and an irrational irrational?

If you want to play a bit with logical arguments, I offer you an alternative proof based on the contrapositive of the statement you are trying to show. Instead of proving $P\to Q$, I'll prove $\lnot Q\to \lnot P$.

We want to show that

$$(\forall x)(\forall y)[x\in \mathbb{Q}^* \land y\not \in\mathbb{Q} \to xy\not\in\mathbb{Q}],\quad(*)$$

since any number can either be rational or irrational; $\mathbb{Q}^*=\mathbb{Q}-\{0\}$. But

\begin{align*} x\in \mathbb{Q}^* \land y\not \in\mathbb{Q} \to xy\not\in\mathbb{Q} \equiv \lnot\left(xy\not\in\mathbb{Q}\right)\to \lnot\left(x\in \mathbb{Q}^* \land y\not\in\mathbb{Q}\right)\\ \equiv xy\in\mathbb{Q}\to \left(x\not\in \mathbb{Q}^* \lor y\in\mathbb{Q}\right),\quad\quad(**) \end{align*}

The original problems translate to proving (*). For, let $z = xy$ and assume $x\in \mathbb{Q}^+$. Then

$$ z\frac{1}{x}= (xy)\frac{1}{x} = y\in \mathbb{Q}. $$

We used the fact that $x\neq 0$ has an inverse and that the product of two rationals is rational. This proves that (**) is true, so its equivalent version (*) must also be true. Q.E.D.

Experiment with sums and products of two numbers from the following list to answer the questions that follow: $$ 5,\tfrac{1}{2},0,\sqrt{2},-\sqrt{2},\tfrac{1}{\sqrt{2}},\pi. $$

Based on the above information, conjecture which of the statements is ALWAYS true, which is SOMETIMES true, and which is NEVER true?

1. The sum of a rational number and a rational number is rational.
2. The sum of a rational number and an irrational number is irrational.
3. The sum of an irrational number and an irrational number is irrational.
4. The product of a rational number and a rational number is rational.
5. The product of a rational number and an irrational number is irrational.
6. The product of an irrational number and an irrational number is irrational.

Following operations between rational and irrational numbers result in an irrational number. Whatever the order of operations, the outcome is always an irrational number.

1. Rational + Irrational: [ 3 + √2 ], [ 4 + √7 ], …
2. Rational − Irrational: [ 5 – √2 ], [ √3 – 6 ], …
3. Rational × Irrational: [ 4 × π = 4π ], [ 6 × √3 = 6√3 ], …
4. Rational ÷ Irrational: [ 2 ÷ √2 ], [ π ÷ 2 ], …

Product of two irrational numbers

We know that product of two rational numbers is rational. The only remaining case is the product of two irrational numbers. In this case, the resulting number may be rational or irrational, depending on the multiplicand and multiplier.

Let us see some examples.

Example 5

√2 × √6

=12​

= 2√3

This product is an irrational number due to the presence of the square root of √3, which is an irrational number.

Example 6

√3 × √3

= √9

= 3

3 is a rational number. The product of two irrational numbers can be rational.

Example 7

(2 + √3) × (2 − √3)

The resulting number in each bracket above is a sum of a rational and an irrational number. So it is an irrational number. Let us see what happens to the product.

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Are repeating decimals rational? The answer is yes. But before we talk about why, let's review rational numbers. A rational number is a fraction in its lowest term. It's written in form a/b, where both a and b are integers, and b is a non-zero denominator.

Now, let’s talk about why repeating decimals are considered rational numbers.

Are Repeating Decimals Rational?

Repeating or recurring decimals are decimal representations of numbers with infinitely repeating digits. Numbers with a repeating pattern of decimals are rational because when you put them into fractional form, both the numerator a and denominator b become non-fractional whole numbers.

For example, when you use long division to divide 1 by 3, the resultant quotient is 0.33333.... However, when put it into fractional form, it's made of positive integers that don’t have decimal points:

In infinite decimal expansion, decimal digits repeat on forever with no end. Numbers with repeating decimals can have an overline above the last number:

The overline is an easier, shorter way to indicate infinite decimal expansion without having to write a bunch of repeating numbers. Though repeating digits like this don't seem like rational numbers, they can take the form of a rational expressions when converted to their fraction form:

This is because the repeating part of this decimal no longer appears as a decimal in rational number form. Instead, it’s represented by non-repeating, natural numbers 4 and 9. Remember — irrational numbers cannot be written as fractions.

What Is a Terminating Decimal?

When converted to decimal form, some rational numbers have a terminating decimal. This means that there's a finite number of digits after the decimal point:

The terminating decimal of this rational number is 5 since no decimal numbers follow it.

What Is a Non-Terminating Decimal?

You can never count the decimal places of a number with repeating decimals. Because of this, repeating decimals are called non-terminating decimals.

Here's an example of a non-terminating decimal:

In this sequence of digits, the same number combination of 09 will be infinitely repeated.

Repeating Decimals Are Rational

Because rational numbers are used at all levels of math, it's important to know what makes a number rational. It might not seem like numbers with repeating decimals are rational numbers. But when you use your algebraic skills to convert repeating decimals into the fractional form a/b, you prove that repeating decimals are indeed rational.

How to prove the product of a rational and irrational number is irrational?

Is it true that the product of two irrational numbers is also irrational prove your answer?

What is the product of a rational and irrational number a both rational and irrational?

Why √ 2 and π are called irrational numbers?