Algebra II - Irrational Numbers

Which of the following is an irrational number?

$\sqrt{79}$

$\sqrt{10000}$

$\sqrt{\frac{175}{{7}}}$

Explanation

An irrational number is any number that cannot be written as a fraction of whole numbers. The number pi and square roots of non-perfect squares are examples of irrational numbers.

can be written as the fraction $\frac{13}{4}$ . The term $\sqrt{\frac{175}{7}} = \sqrt{25} = 5$ is a whole number. The square root of is , also a rational number. , however, is not a perfect square, and its square root, therefore, is irrational.

Which of the following numbers are irrational?

$\sqrt{169}$

Explanation

The definition of irrational numbers is that they are real numbers that cannot be expressed in a common ratio or fraction.

The term is imaginary which equals to $\sqrt{-1}$ .

The other answers can either be simplified or can be written in fractions.

The only possible answer shown is .

Which of the following is considered an irrational number?

$\sqrt{30}$

$\sqrt{144}$

$0.\bar{6}$

Explanation

The irrational numbers do not have a representation of a ratio between two numbers. They cannot be expressed by a fraction.

Repeating decimal numbers are not irrational because they can be rewritten as a fraction.

For instance: $0.\bar{3} = \frac{1}{3}$

The number may represent the short version of $\pi$ , but is not irrational, because is a fixed number and be rewritten as a ratio between two numbers.

The answer is: $\sqrt{30}$

Which of the following is equal to

$\frac{\sqrt{-16}}{5+\sqrt{-9}}$

$\frac{6}{17}+\frac{10i}{17}$

$\frac{-6}{17}+\frac{10i}{17}$

$\frac{12}{17}+\frac{10i}{17}$

$\frac{-12}{17}+\frac{10i}{17}$

$\frac{12}{17}+\frac{20i}{17}$

Explanation

$\frac{\sqrt{-16}}{5+\sqrt{-9}}$

Simplify the radicals

$\frac{4i}{5+3i}$

We notice that we have a complex number in the denominator. To get rid of this we multiply the numerator and denominator by the complex conjugate of the denominator.

$\frac{4i}{5+3i} \cdot\frac{5-3i}{5-3i}$

Distribute across the numerator and multiply the binomials in the denominator. You may use the FOIL method.

$\frac{20i-12i^{2}}{25-15i+15i-9i^{2}}$

We know that $i^{2}=-1$ so we replace $i^{2}$ with -1

$\frac{20i+12}{25-15i+15i+9}$

Combine like terms

$\frac{20i+12}{34}$

Reduce and put in standard form

$\frac{6+10i}{17}$ or $\frac{6}{17}+\frac{10i}{17}$

Which of the following is not irrational?

$\sqrt{3+6}$

$45\pi$

$\sqrt[3]{45}$

$1+\sqrt{5}$

Explanation

Some answers can be solved. Let's look at some obvious irrational numbers.

$45\pi$ is surely irrational as we can't get an exact value.

The same goes for and $1+\sqrt{5}$ .

is not a perfect cube so that answer choice is wrong.

Although $\sqrt{3+6}$ is a square root, the sum inside however, makes it a perfect square so that means $\sqrt{3+6}=\sqrt{9}=3$ is rational.

Try without a calculator.

True or false: the set

$\left { 7 \pi , 8 \pi, 9 \pi, 10 \pi, 11 \pi \right }$

comprises only irrational numbers.

True

False

Explanation

$\pi$ is an irrational number, as is any integer multiple of $\pi$ . All of the elements are integer multiples of $\pi$ , so all of them are irrational.

Solve the following equation for x. Express your answer with complex numbers.

$3+2x^{2}=-17$

$i\sqrt{10}$ or $-i\sqrt{10}$

$i\sqrt{10}$

$-i\sqrt{10}$

$2i\sqrt{5}$ or $-2i\sqrt{5}$

Explanation

$3+2x^{2}=-17$

Our first goal is to isolate the X. So we subtract the 3 on both sides.

$2x^{2}=-20$

Now we divide by 2 on both sides

$x^{2}=-10$

We square root both sides

$x=\pm \sqrt{-10}$

Simplify the radical

$x=\pm i \sqrt{10}$

Which of the following represents an irrational number?

$-\sqrt{144}$

$-\frac{1}{1000}$

$0.\overline{3}$

$\sqrt{26}$

None of the numbers is irrational.

Explanation

First, recall the definition of irrational numbers. Irrational numbers cannot be expressed as a ratio of integers.

The answer $0.\overline{3}$ is incorrect because it is a repeating decimal that can be rewritten as the following fraction:

$\frac{1}{3}$

In a similar vein, the following choice is already written as a simple fraction (negative numbers are not irrational):

$-\frac{1}{1000}$

It is important to note that not all square roots are irrational. For instance, the following square root can be simplified quite easily:

$-\sqrt{144} = -12$

The only answer that cannot be expressed as a simple fraction is:

$\sqrt{26}=5.09901951359278449161$

The answer is:

$\sqrt{26}$

Which set does NOT contain an irrational number?

$\left{-2.5, \frac{3}{7}, 1.23456789, \sqrt{4}, 200\right}$

$\left{-10, 4/5, 26, \sqrt{200}\right}$

$\left{-25, 1.23, 3/2, \sqrt{14}\right}$

$\left{-22.68, -14/3, 2.2, \pi , 37\right}$

Explanation

Irrational numbers are nonrepeating decimals-- they cannot be written as fractions.

$\left{-2.5, \frac{3}{7}, 1.23456789, \sqrt{4}, 200\right}$ has only real numbers because the square root of 4 is 2, a rational number.

Simplify

$\frac{5-2i}{5+2i}$

$\frac{21-20i}{29}$

$\frac{21}{29}$

$\frac{29-20i}{21}$

$\frac{21-20i}{21}$

Explanation

To simplify we need to remove the complex number from the denominator. To do this our first step is to multiply the expression by the complex conjugate of the denominator.

$\frac{5-2i}{5+2i} \cdot \frac{5-2i}{5-2i}$