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## Example Questions

### Example Question #3 : Irrational Numbers

Which of the following is an irrational number?

**Possible Answers:**

**Correct answer:**

A rational number can be expressed as a fraction of whole numbers, while an irrational number cannot.

can be written as .

is simply , which is a rational number.

The number can be rewritten as a fraction of whole numbers, , which makes it a rational number.

is also a rational number because it is a ratio of whole numbers.

The number, , on the other hand, is irrational, since it has an irregular sequence of numbers (...) that cannot be written as a fraction.

### Example Question #1 : Irrational Numbers

Which of the following is an irrational number?

**Possible Answers:**

**Correct answer:**

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 . The term 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.

### Example Question #2 : Irrational Numbers

Of the following, which is a rational number?

**Possible Answers:**

**Correct answer:**

A rational number is any number that can be expressed as a fraction/ratio, with both the numerator and denominator being integers. The one limitation to this definition is that the denominator **cannot** be equal to .

Using the above definition, we see , and (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.

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

### Example Question #3 : Irrational Numbers

Of the following, which is an irrational number?

**Possible Answers:**

**Correct answer:**

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

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

any number and

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

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

This is an irrational number and our correct answer.

### Example Question #4 : Irrational Numbers

Which of the following numbers is an irrational number?

,

**Possible Answers:**

**Correct answer:**

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

Repeating decimals are never irrational, can be eliminated because

.

and are perfect squares making them both integers.

Therefore, the only remaining answer is .

### Example Question #5 : Irrational Numbers

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

**I. **

**II. **

**III. **

**IV. **

**Possible Answers:**

All of them are rational numbers.

Both **II **and **IV**

**III. **only

**II. **only

**IV. **only

**Correct answer:**

**II. **only

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

Statement **I**'s fraction is so this statement is false.

Statement **IV. **may not be easy to spot but if you let that decimal be and multiply that by you will get . This becomes . Subtract it from and you get an equation of .

becomes which is a fraction.

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

### Example Question #6 : Irrational Numbers

Is rational or irrational?

**Possible Answers:**

Irrational, because there are repeating decimals.

Irrational, because it can be expressed as a fraction.

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

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

Rational, because there is a definite value.

**Correct answer:**

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

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

### Example Question #7 : Irrational Numbers

What do you get when you multiply two irrational numbers?

**Possible Answers:**

Integers.

Sometimes irrational, sometimes rational.

Imaginary numbers.

Always rational.

Always irrational.

**Correct answer:**

Sometimes irrational, sometimes rational.

Let's take two irrationals like and multiply them. The answer is which is rational.

But what if we took the product of and . We would get 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.

### Example Question #8 : Irrational Numbers

Which of the following is not irrational?

**Possible Answers:**

**Correct answer:**

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

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

The same goes for and .

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

Although is a square root, the sum inside however, makes it a perfect square so that means is rational.

### Example Question #9 : Irrational Numbers

Which concept of mathematics will **always** generate irrational answers?

**Possible Answers:**

Finding an area of a triangle.

Finding volume of a cube.

Finding an area of a square.

Finding value of ; .

The diagonal of a right triangle.

**Correct answer:**

Finding value of ; .

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.

The diagonal of a right triangle will generate sometimes rational answers or irrational values. If you have a perfect Pythagorean Triple or 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 are , the hypotenuse is

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

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 of . is irrational and raised to any power except 0 is always irrational.

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