Real Number System

Consider the numbers $4\ and\ \sqrt{5}$ . Which operation(s) will result in an irrational number?

• the sum only

• the product only

• neither the sum nor the product

• both the sum and the product

Answer explanation

4 is a rational number and  $\sqrt{5}$  is an irrational number. 
The sum of a rational and irrational is ALWAYS irrational
The product of a rational and irrational is ALWAYS irrational.
When in doubt use your calculator to test the operation in each answer.

Which sum(s) represent(s) a rational number?

• $5+\frac{1}{2}$

• $\frac{\pi}{2}+\frac{2}{\pi}$

• $\sqrt{2}+\frac{1}{\sqrt{2}}$

• $\frac{\sqrt{5}}{3}+\frac{1}{3}$

• $\sqrt{16}+\frac{1}{2}$

Answer explanation

Rational + Rational = ALWAYS Rational

Rational + Irrational is NEVER Rational
Irrational + Irrational is SOMETIMES Rational (Use your calculator to determine)
$5+\frac{1}{2}=5.5\ -\ Rational\$ (decimal terminates)
$\frac{\pi}{2}+\frac{2}{\pi}=2.207416\ -\ Irrational\ \left(mumble\ jumble\right)$
$\sqrt{2}+\frac{1}{\sqrt{2}}=2.12132\ -\ Irrational\ \left(mumble\ jumble\right)$
$\frac{\sqrt{5}}{3}+\frac{1}{3}=1.078689\ -\ Irrational\ \left(mumble\ jumble\right)$

$\sqrt{16}+\frac{1}{2}=4.5\ -\ Rational\ \left(decimal\ ends\right)$

How is the product of a rational, nonzero number and an irrational number different from the product of 2 irrational numbers?

• The product of a rational, nonzero number and an irrational number is always irrational, but the product of 2 irrationals is sometimes rational.

• The product of a rational, nonzero number and an irrational number is always irrational, but the product of 2 irrationals is never rational.

• The product of a rational, nonzero number and an irrational number is never irrational, but the product of 2 irrational numbers is sometimes rational.

• The product of a rational, nonzero number and an irrational number is never irrational, but the product of 2 irrationals is always rational.

Answer explanation

The product of 2 Rational numbers is ALWAYS rational

The product of a Rational and Irrational number is ALWAYS Irrational

The product of 2 Irrational numbers is SOMETIMES Rational, SOMETIMES Irrational (Use your calculator to find out)

Which product(s) represent(s) a rational number?

• $5\times\frac{1}{2}$

• $\sqrt{2}\times\frac{1}{\sqrt{2}}$

• $\frac{\sqrt{5}}{3}\times\frac{1}{3}$

• $\sqrt{16}\times\frac{1}{2}$

• $\frac{\pi}{2}\times\frac{2}{\pi}$

Answer explanation

Use your calculator to verify $5\times\frac{1}{2}=5.5\ -\ Rational$ $\sqrt{2}\times\frac{1}{\sqrt{2}}=1\ -\ Rational$
$\frac{\sqrt{5}}{3}\times\frac{1}{3\ }=0.248452\ -\ Irrational\ \left(mumble\ jumble\right)$

$\sqrt{16}\times\frac{1}{2\ }=2\ -\ Rational$

$\frac{\pi}{2}\times\frac{2}{\pi}=1\ -\ Rational$

David is sorting numbers into columns for rational and irrational numbers.

Which number(s) did David sort incorrectly?

• $\frac{3}{8},\ 2\pi$

• $5+\sqrt{2},\ \sqrt{5},\sqrt{7}$

• $1.\overline{67},\ \frac{3}{8}$

• $2\pi,\ 1.\overline{67}$

Answer explanation

$2\pi$ - Irrational
$1.\overline{67}$ - Rational (the decimal repeats)

Which BEST describes  $\frac{3}{2}\times\pi$  simplified, and why?

• The simplified value is irrational because the product of a non-zero rational number and an irrational number is always irrational.

• The simplified value is irrational because the sum of a rational number and an irrational number is always irrational.

• The simplified value is rational because the product of a rational number and a non-zero irrational number is always rational.

• The simplified value is rational because the sum of a non-zero rational number and an irrational number is always rational.

Answer explanation

This is multiplication.  You are finding the product.
The product of 2 Rational numbers is ALWAYS Rational
The product of a Rational and Irrational number is ALWAYS Irrational
The product of a 2 Irrational numbers is SOMETIMES Rational, SOMETIMES Irrational (use your calculator to verify)

Is $\frac{\pi}{3}$  irrational or rational, and why?

• The value of  $\frac{\pi}{3}$  is irrational because the difference between an irrational and a non-zero rstional number is irrational.

• The value of  $\frac{\pi}{3}$ is rational because the difference between an irrational number and a non-zero rational number is raional.

• The value of  $\frac{\pi}{3}$ is irrational because the quotient of an irrational number and a non-zero rational number is irrational.

• The value of  $\frac{\pi}{3}$  is rational because the quotient of an irrational number and a non-zero rational number is rational.
  

Answer explanation

You are finding the Quotient. Quotient is the answer to a division problem. 

Difference is the answer to a subtraction problem. 

The quotient of an Irrational number and non-zero Rational number is ALWAYS Irrational 

The quotient of 2 Rational numbers is ALWAYS Rational

The quotient of 2 Irrational numbers is SOMETIMES Irrational, SOMETIMES Rational (use your calculator to verify.)