Question

Consider the following statements. Select all that are always true.

Answer 1

Answer:

- The sum of a rational number and a rational number is rational.

- The sum of a rational number and an irrational number is irrational.

- The product of a rational number and a rational number is rational.

- The product of a rational number and an irrational number is irrational.

I THINK

Answer 2

Answer:

A, B, D and E

Step-by-step explanation:

Given

Options A to F

Required

Determine which is true

Option A:

This is always true;

Take for instance the following rational numbers

$a = \frac{1}{2}$    $b = \frac{1}{3}$

$a + b = \frac{1}{2} + \frac{1}{3}$

$a + b = \frac{3 + 2}{6}$

$a + b = \frac{5}{6}$

This will always result in a rational number

Option B:

This is always true;

Take for instance the following rational number

$a = 0.5$      

And the following irrational number

$b = 3.142857$

$a + b =0.5+ 3.142857$

$a + b =3.642857$

This will always result in an irrational number

Option C:

This is not always true;

1. Take for instance the following irrational numbers

$a = 0.33333$        $b = 3.142857$

$a + b =0.33333 + 3.142857$

$a + b =3.476187$

2. Take for instance the following irrational numbers

$a = 3 + \sqrt5$        $b = -\sqrt5$

$a + b = 3 + \sqrt5 - \sqrt5$

$a + b = 3$

From the above examples, this implies that the statement is not always true

D.

This is always true;

Take for instance the following rational numbers

$a = \frac{1}{2}$    $b = \frac{1}{3}$

$a * b = \frac{1}{2} * \frac{1}{3}$

$a * b = \frac{1}{6}$

This will always result in a rational number

E.

This is always true;

Take for instance the following rational number

$a = 0.5$      

And the following irrational number

$b = 3.142857$

$a * b =0.5 * 3.142857$

$a * b =1.5714285$

This will always result in an irrational number

F.

This is not always true;

1. Take for instance the following irrational numbers

$a = 0.33333$        $b = 3.142857$

$a * b =0.33333 * 3.142857$

$a * b =1.04760852381$

2. Take for instance the following irrational numbers

$a = 3 + \sqrt5$        $b = 0$

$a * b = (3 + \sqrt5) * 0$

$a * b = 0$

From the above examples, this implies that the statement is not always true