IS IT RATIONAL OR IRRATIONAL?

Fill-in-the-blank:

Rational or Irrational?•1) The sum of two rational numbers is ________.•2) The product of two rational numbers is ______.•3) The sum of a rational and an irrational is ____.•4) The product of a non-zero rational and an irrational is _____.•5) The product of two irrational numbers is _____.

1) The sum of two rational numbers is rational.

• Example: 3 7 10 5

4 4 4 2

Proof:a

bc

dLet and be two rational numbers.

Then, , which is a rational number!a c ad bc

b d bd

2) The product of two rational numbers is rational.

• Example: 3 1 3

4 2 8

Proof:a

bc

dLet and be two rational numbers.

Then, , which is a rational number!a c ac

b d bd

3) The sum of a rational and an irrational is irrational.

• Example: 3

24

Proof: Let x be irrational and r be rational.

Assume x + r = t is rational. (So, t is rational.)

Then, x = t – r.

Since t and r are rational, their difference is rational.

However, this contradicts the assumption that x is irrational.

So, proof by contradiction, x + r is irrational.

4) The product of a non-zero rational and an irrational is irrational.

• Example: 1 3

32 2

Proof: Let x be irrational and be rational.

Assume x is rational. Then, x

So, x = , which by definition is a rational number.

This contradicts the assumption that x is irrational.

Proof by contradiction … x is irrational.

a

ba

b

a p

b q

p b

q a

a

b

5) The product of two irrational numbers is sometimes rational and sometimes irrational.

• Examples: 2 2 4 2

2 3 6