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Algebra II : Types of Numbers

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Example Questions

Example Question #1 : Types Of Numbers

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

Possible Answers:

real, natural

real, rational

real, irrational

imaginary, irrational

Correct answer:

real, irrational

Explanation:

$\pi$ $\displaystyle \pi$ 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.

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Example Question #2 : Types Of Numbers

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

Possible Answers:

$\displaystyle 1.3, 2, 5, 7.9$

$\displaystyle 3, 5, 2, 1$

$\displaystyle -1, 3, 4, 100$

$\sqrt{2}$ $\displaystyle \sqrt{2}$$\displaystyle , 1, 5, 6$

Correct answer:

$\displaystyle 3, 5, 2, 1$

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.

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Example Question #3 : Types Of Numbers

What is the value of i^6 $\displaystyle i^6$ ?

Possible Answers:

$\displaystyle -1$

$\displaystyle i$

$\displaystyle -i$

$\displaystyle 1$

Correct answer:

$\displaystyle -1$

Explanation:

There is a repeating pattern of four exponent values of $\displaystyle i: i^1=i, i^2=-1, i^3=-1, i^4=1$.

i^6 $\displaystyle i^6$ is the same as i^2 $\displaystyle i^2$.

$i^6=i^2\cdot i^2 \cdot i^2=(-1)(-1)(-1)=-1$ $\displaystyle i^6=i^2\cdot i^2 \cdot i^2=(-1)(-1)(-1)=-1$

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Example Question #4 : Types Of Numbers

Simplify $\displaystyle (i-4)(2i+2)$.

Possible Answers:

-6i-10 $\displaystyle -6i-10$

6i+4 $\displaystyle 6i+4$

$\displaystyle 15$

-6i-6 $\displaystyle -6i-6$

Correct answer:

-6i-10 $\displaystyle -6i-10$

Explanation:

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

$\displaystyle 2i^2-8i+2i-8$.

Remember that i^2=-1 $\displaystyle i^2=-1$

$\displaystyle 2(-1)-8i+2i-8=-6i-10$

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Example Question #5 : Types Of Numbers

Which of these numbers is prime?

$\displaystyle 10, 13, 15, 18$

Possible Answers:

$\displaystyle 15$

$\displaystyle 10$

$\displaystyle 13$

$\displaystyle 18$

Correct answer:

$\displaystyle 13$

Explanation:

For a number to be prime it must only have factors of one and itself.

10 has factors 1, 2, 5, 10.

15 has factors 1, 3, 5, 15.

18 has factors 1, 2, 3, 6, 9, 18.

The only factors of 13 are 1 and 13. As such it is prime.

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Example Question #6 : Types Of Numbers

Which of the below is an irrational number?

Possible Answers:

1.7392 $\displaystyle 1.7392$

3-2i $\displaystyle 3-2i$

$\frac{4}{5}$ $\displaystyle \frac{4}{5}$

$\sqrt{2}$ $\displaystyle \sqrt{2}$

Correct answer:

$\sqrt{2}$ $\displaystyle \sqrt{2}$

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,

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

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

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

Therefore since,

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

we can conclude it is irrational.

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Example Question #7 : Types Of Numbers

$x=i^2\sqrt{3}$ $\displaystyle x=i^2\sqrt{3}$

Which of the following describes the type of $\displaystyle x$?

Possible Answers:

$\displaystyle x$ is real, rational

$\displaystyle x$ is real, irrational

$\displaystyle x$ is imaginary, rational

$\displaystyle x$ is imaginary, irrational

None of these options describe $\displaystyle x$.

Correct answer:

$\displaystyle x$ is real, irrational

Explanation:

An irrational number is a number that cannot be written in fraction form. In other words a nonrepeating decimal is an irrational number.

The $\sqrt{3}$ $\displaystyle \sqrt{3}$ is an irrational number.

i^2 $\displaystyle i^2$ is a real number with a value of $\displaystyle -1$.

Therefore, $x=-\sqrt{3}$ $\displaystyle x=-\sqrt{3}$. This is a real but irrational number.

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Example Question #1 : Types Of Numbers

What is the most specific classification for the x-intercepts to the equation graphed:

Exam transform 2

Possible Answers:

rational

integer

irrational

real

imaginary

Correct answer:

imaginary

Explanation:

The graph shown never intersects with the x-axis. This means that the x-intercepts must be imaginary.

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Example Question #3 : Types Of Numbers

What is the most specific classification for $\sqrt{5}$ $\displaystyle \sqrt{5}$

Possible Answers:

imaginary

rational

natural number

irrational

counting number

Correct answer:

irrational

Explanation:

The square root of 5 is irrational since it is a non-terminating, non-repeating decimal that cannot be expressed as a fraction.

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Example Question #6 : Types Of Numbers

Which of the following is a rational number?

Possible Answers:

$\sqrt{8}$ $\displaystyle \sqrt{8}$

$\sqrt{13}$ $\displaystyle \sqrt{13}$

$\sqrt{65}$ $\displaystyle \sqrt{65}$

$\sqrt{17}$ $\displaystyle \sqrt{17}$

$\sqrt{9}$ $\displaystyle \sqrt{9}$

Correct answer:

$\sqrt{9}$ $\displaystyle \sqrt{9}$

Explanation:

$\sqrt{9}=3$ $\displaystyle \sqrt{9}=3$ A rational number is a number that can be expressed in the form p/q. in this case p=3 and q=1. The other answers are irrational because they cannot be expressed as whole numbers or fractions.

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Algebra II : Types of Numbers

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Types Of Numbers

Example Question #2 : Types Of Numbers

Example Question #3 : Types Of Numbers

Example Question #4 : Types Of Numbers

Example Question #5 : Types Of Numbers

Example Question #6 : Types Of Numbers

Example Question #7 : Types Of Numbers

Example Question #1 : Types Of Numbers

Example Question #3 : Types Of Numbers

Example Question #6 : Types Of Numbers

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