Rational and irrational number properties

7 State whether the following statements are true or false; if false, provide a counter-example.

1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational.

3. a. "If \(p\) and \(q\) are irrational numbers, where \(p \neq q , q \neq 0\), then \(\frac { p } { q }\) is also irrational." Disprove this statement by means of a counter example.
b. (i) Sketch the graph of \(y = | x | - 2\).
(ii) Explain why \(| x - 2 | \geq | x | - 2\) for all real values of \(x\).

1. (a) "If \(m\) and \(n\) are irrational numbers, where \(m \neq n\), then \(m n\) is also irrational."

Disprove this statement by means of a counter example.
(b) (i) Sketch the graph of \(y = | x | + 3\) (ii) Explain why \(| x | + 3 \geqslant | x + 3 |\) for all real values of \(x\).

3 Give a counter example to disprove the following statement.
If \(x\) and \(y\) are both irrational then \(x + y\) is irrational.

4 Millie is attempting to use proof by contradiction to show that the result of multiplying an irrational number by a non-zero rational number is always an irrational number. Select the assumption she should make to start her proof.
Tick ( \(\checkmark\) ) one box. Every irrational multiplied by a non-zero rational is irrational. □ Every irrational multiplied by a non-zero rational is rational. □ There exists a non-zero rational and an irrational whose product is irrational. □ There exists a non-zero rational and an irrational whose product is rational. □

State whether the following statements are true or false; if false, provide a counter-example.

1. If \(a\) is rational and \(b\) is rational, then \(a + b\) is rational.
2. If \(a\) is rational and \(b\) is irrational, then \(a + b\) is irrational.
3. If \(a\) is irrational and \(b\) is irrational, then \(a + b\) is irrational. [3]

Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]

A student argues that when a rational number is multiplied by an irrational number the result will always be an irrational number.

1. Identify the rational number for which the student's argument is not true. [1 mark]
2. Prove that the student is right for all rational numbers other than the one you have identified in part (a). [4 marks]

Suppose \(x\) is an irrational number, and \(y\) is a rational number, so that \(y = \frac{m}{n}\), where \(m\) and \(n\) are integers and \(n \neq 0\). Prove by contradiction that \(x + y\) is not rational. [4]