# Algebra II : Irrational Numbers

## Example Questions

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### Example Question #1 : Irrational Numbers

Which of the following is an irrational number?

Explanation:

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

Â can be written as the fractionÂ . Â The termÂ  is a whole number. Â The square root ofÂ  is , also a rational number. , however, is not a perfect square, and its square root, therefore, is irrational.

### Example Question #2 : Irrational Numbers

Of the following, whichÂ is a rational number?

Explanation:

A rational number is any numberÂ that can be expressedÂ as a fraction/ratio, with both the numerator and denominator being integers. The one limitation to thisÂ definitionÂ is that the denominator cannot be equal to .

Â

Using the above definition, we see Â , Â and Â Â (which is ) cannot be expressed as fractions. These are non-terminating numbers that are not repeating, meaning the decimal has no pattern and constantly changes. When a decimal is non-terminating and constantly changes, it cannot be expressed as a fraction.

Â  is the correct answer because , which can be expressed as , fullfilling our above defintion of a rational number.Â

### Example Question #1 : Understand The Difference Between Rational And Irrational Numbers: Ccss.Math.Content.8.Ns.A.1

Of the following, which is an irrational number?

Explanation:

The definition of an irrational number is a number which cannot be expressed in a simple fraction, or a number that is not rational.

Â

Using the above definition, we see that Â is already expressed as a simple fraction.

Â

Â Â any numberÂ Â and

. All of these options can be expressed as simple fractions, making them all rational numbers, and the incorrect answers.

Â

Â  cannot be expressed as a simple fraction and is equal to a non-terminating, non-repeating (ever-changing) decimal, begining withÂ

This is an irrational number and our correct answer.

### Example Question #4 : Irrational Numbers

Which of the following numbers is an irrational number?

,Â

Explanation:

An irrational number Â is one that cannot be written as a fraction. All integers are rational numberes.

Repeating decimals are never irrational, Â can be eliminated because

.

Â Â andÂ Â are perfect squares making them both integers.

Therefore, the only remaining answer isÂ .

### Example Question #5 : Irrational Numbers

Which of the following is/are an irrational number(s)?Â

I.Â

II.Â

III.Â

IV.Â

All of them are rational numbers.Â

II.Â only

BothÂ IIÂ andÂ IV

IV.Â only

III.Â only

II.Â only

Explanation:

Irrational numbers are numbers that can't be expressed as a fracton. This elminates statementÂ III automatically as it's a fraction.

Statement I's fraction isÂ Â so this statement is false.

StatementÂ IV.Â may not be easy to spot but if you let that decimal beÂ Â and multiply that byÂ Â you will getÂ . This becomesÂ . Subtract it fromÂ Â and you get an equation ofÂ .Â

Â becomesÂ Â which is a fraction.

StatementÂ II can't be expressed as a fraction which makes this the correct answer.

### Example Question #1 : Irrational Numbers

IsÂ Â rational or irrational?

Rational, because there is a definite value.Â

Irrational, because it can't be expressed as a fraction.Â

Rational, because it can't be expressed as a fraction.Â

Irrational, because there are repeating decimals.Â

Irrational, because it can be expressed as a fraction.Â

Irrational, because it can't be expressed as a fraction.Â

Explanation:

Irrational numbers can't be expressed as a fraction with integer values in the numerator and denominator of the fraction.

Irrational numbers don't have repeating decimals.

Because of that, there is no definite value of irrational numbers.

Therefore,Â Â is irrational because it can't be expressed as a fraction.Â

### Example Question #5101 : Algebra Ii

What do you get when you multiply two irrational numbers?

Sometimes irrational, sometimes rational.

Imaginary numbers.

Always rational.

Integers.

Always irrational.

Sometimes irrational, sometimes rational.

Explanation:

Let's take two irrationals likeÂ Â and multiply them. The answer isÂ Â which is rational.

Â

But what if we took the product ofÂ Â andÂ .Â We would getÂ Â which doesn't have a definite value and can't be expressed as a fraction.

This makes it irrational and therefore, the answer is sometimes irrational, sometimes rational.Â

### Example Question #8 : Irrational Numbers

Which of the following is not irrational?

Explanation:

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

Â is surely irrational as we can't get an exact value.

The same goes forÂ Â andÂ .Â

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

AlthoughÂ Â is a square root, the sum inside however, makes it a perfect square so that meansÂ Â is rational.Â

### Example Question #9 : Irrational Numbers

Which concept of mathematics will always generate irrational answers?

The diagonal of a right triangle.

Finding an area of a square.

FindingÂ value ofÂ ;Â .

Finding an area of a triangle.

Finding volume of a cube.

FindingÂ value ofÂ ;Â .

Explanation:

Let's look at all the answer choices.

The area of a triangle is base times height divided by two. Since base and height can be any value, this statement is wrong. We can have irrational values or rational values, thus generating both irrational or rational answers.Â

The diagonal of a right triangle will generate sometimes rational answers or irrational values. If you have a perfect Pythagorean Triple Â or Â etc..., then the diagonal is a rational number. A Pythagorean Triple is having all the lengths of a right triangle being rational values. One way the right triangle creates an irrational value is when it's an isosceles right triangle. If both the legs of the triangle areÂ , the hypotenuse isÂ Â

,Â ,Â ,Â Â can't be negative since lengths of triangle aren't negative.Â

The same idea goes for volume of cube and area of square. It will generate both irrational and rational values.

The only answer is finding value ofÂ .Â Â is irrational and raised to any power except 0 is always irrational.Â

### Example Question #10 : Irrational Numbers

Which of the following numbers are irrational?