Algebra II - Types of Numbers

Which of the following is a complex number?

$\frac{4}{3}$

$\sqrt{7}$

Explanation

By definition, a complex number is a number with an imaginary term denoted as i.

A complex number is in the form,

where represents the real part of the number and represents the imaginary portion of the complex number.

Therefore, the complex number which is the solution is .

Which of the following describes the number $\pi$ ?

real, irrational

real, rational

imaginary, irrational

real, natural

Explanation

$\pi$ is a real number, because you can represent it on the Cartesian coordinate plane, but it is irrational because it cannot be represented by a fraction of two integers. Natural numbers are integers greater than 0.

Simplify .

Explanation

Multiplying out using FOIL (First, Inner, Outer, Last) results in,

Remember that

True or false:

The following set comprises only imaginary numbers:

$\left { i^{10}, i^{14}, i^{18}, i^{22}, i^{26}\right }$

False

True

Explanation

To raise to the power of any positive integer, divide the integer by 4 and note the remainder. The correct power is given according to the table below.

Powers of i

Every element in the set is equal to raised to an even-numbered power, so when each exponent is divided by 4, the remainder will be either 0 or 2. Therefore, each element is equal to either 1 or . Consequently, the set includes no imaginary numbers.

Which of the following are natural numbers?

$\textup{One-million.}$

$\textup{None of the answers are natural numbers.}$

$\textup{Negative square root of sixteen}$

$\textup{Fifty percent}$

$\textup{Two-thirds}$

Explanation

The definition of natural numbers states that the number may not be negative, and must be countable where:

$\textup{Number(s)} =[0,1,2,3,4,...]$

Decimal places and fractions are also not allowed.

The value of fifty percent equates to $\frac{1}{2}$ .

The only possible answer is: $\textup{One-million.}$

Which of the below is an irrational number?

$\sqrt{2}$

$\frac{4}{5}$

Explanation

Irrational numbers are defined by the fact that they cannot be written as a fraction which means that the decimals continue forever.

Looking at our possible answer choices we see,

$1.7392=\frac{1087}{625}$

$\frac{4}{5}$ is already in fraction form

$3-2i=3-2\sqrt{-1}$ which is an imaginary number but still rational.

Therefore since,

$\sqrt{2} = 1.414213562......$

we can conclude it is irrational.

Which of the following sets of numbers contain only natural numbers.

$\sqrt{2}$

Explanation

Natural numbers are simply whole, non-negative numbers.

Using this definition, we see only one set of numbers within our answer choices containing only whole, non-negative numbers. Any set containing decimals or negative numbers, will violate our defintion of natural numbers and thus be an incorrect answer.

What sets do the numbers $2,-7,\frac{11}{2}$ have in common?

$\mathbb{Q}$

$\mathbb{R},\mathbb{I}$

$\mathbb{N},\mathbb{Z}$

$\mathbb{N},\mathbb{Q},\mathbb{R}$

Explanation

Step 1: Define the different sets:

Rational: Any number that can be expressed as a fraction (improper/proper form) (example: )
Irrational: Any number whose decimal expansion cannot be written as a fraction. (example: $\sqrt{2}$ )
Real numbers: The combination of all numbers that belong in the Rational and the Irrational set. (Example: $\frac {-11}{2}$ )
Integers: All whole numbers from $(-\infty, \infty)$ . (Example: )
Natural Numbers (AKA Counting numbers): All numbers greater than or equal to 1, $[1,+\infty)$

Step 2: Let's categorize the numbers given in the question to these sets above:

belongs to the set of rational numbers, natural numbers, and integers.
belongs to the set of rational numbers and integers.
$\frac{11}{2}$ belongs to the set of rational numbers.

Step 3: Analyze each number closely and pull out any sets where all three numbers belong..

All three numbers belong to the set of rational numbers.

In math, we symbolize rational numbers as $\mathbb{Q}$ .
So, all three numbers belong to the sets $\mathbb{Q}$ .

Try without a calculator.

True or false:

The set $\left { 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{10}, \frac{1}{12} \right }$ includes only rational numbers.

True

False

Explanation

A rational number, by definition, is one that can be expressed as a quotient of integers. Each of the fractions in the set - $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{10}, \frac{1}{12}$ - is such a number. The sole integer, 1, is also rational, since any integer can be expressed as the quotient of the integer itself and 1.