We know that quadratic equations can have rational solutions or irrational solutions. For example, the solutions to are -3 and 1, which are rational. The solutions to are , which are irrational.

Sometimes solutions to equations combine two numbers by addition or multiplication—for example, and . What kind of numbers are these expressions?

When we add or multiply two rational numbers, is the result rational or irrational?

• The sum of two rational numbers is rational. Here is one way to explain why it is true:

  • Any two rational numbers can be written and , where are integers, and and are not zero.
  • The sum of and is . The denominator is not zero because neither nor is zero.
  • Multiplying or adding two integers always gives an integer, so we know that and are all integers.
  • If the numerator and denominator of are integers, then the number is a fraction, which is rational.

• The product of two rational numbers is rational. We can show why in a similar way:

  • For any two rational numbers and , where are integers, and and are not zero, the product is .
  • Multiplying two integers always results in an integer, so both and are integers. Therefore,  is a rational number.

What about two irrational numbers?

• The sum of two irrational numbers could be either rational or irrational. We can show this through examples:

  • and are both irrational, but their sum is 0, which is rational.
  • and are both irrational, and their sum is irrational.

• The product of two irrational numbers could be either rational or irrational. We can show this through examples:

  • and are both irrational, but their product is , or 4, which is rational.
  • and are both irrational, and their product is , which is not a perfect square and is therefore irrational.

What about a rational number and an irrational number?

• The sum of a rational number and an irrational number is irrational. To explain why requires a slightly different argument:

  • Let be a rational number and an irrational number. We want to show that is irrational.
  • Suppose represents the sum of and (), and suppose is rational.
  • If is rational, then would also be rational, because the sum of two rational numbers is rational.
  • is not rational, however, because .
  • cannot be both rational and irrational, which means that our original assumption that is rational was incorrect. , which is the sum of a rational number and an irrational number, must be irrational.

• The product of a nonzero rational number and an irrational number is irrational. We can show why this is true in a similar way:

  • Let be rational and irrational. We want to show that is irrational.
  • Suppose is the product of and (), and suppose is rational.
  • If is rational, then would also be rational because the product of two rational numbers is rational.
  • is not rational, however, because .
  • cannot be both rational and irrational, which means our original assumption that  is rational was false. , which is the product of a rational number and an irrational number, must be irrational.