Rational and Irrational Numbers

Hi, and welcome to this video on rational and irrational numbers!

Rational and irrational numbers comprise the real number system. This Venn diagram shows a visual representation of how real numbers are classified.

The natural numbers comprise the smallest subset, which is also known as the set of “counting” numbers. These are all positive, non-decimal values starting at one. Whole numbers are the natural numbers plus the value of zero. The integer set of numbers includes whole numbers and all negative, non-decimal values.

Rational numbers include the sets seen here in addition to the fractional values in between.

An easy way to remember this is that the word ratio is in the name of this classification. All numbers included in the rational number set can be written as a ratio of integers:

If \(a\) and \(b\) are integers: rational numbers can be written as \(\frac{a}{b}\), as long as \(b\neq 0\).

Clearly, the set of integers can be written as ratios because any integer divided by 1 results in the original integer. As illustrated here, integers can be expressed as fractions in infinite numbers of ways.

The integer 3 can be represented as the fractions \(\frac{3}{1}\), \(\frac{6}{2}\), \(\frac{-24}{-8}\)
The integer -5 can be represented as the following fractions \(\frac{-5}{1}\), \(\frac{5}{-1}\), \(\frac{-25}{5}\)
The integer 0 can be represented as the fractions \(\frac{0}{3}\), \(\frac{0}{-2}\), \(\frac{0}{123}\)

As a side note, these are not the only fractions that result in these integers, they are just a few of the many examples that exist.

Fractions can also be written as decimals. For example:

.1 is equivalent to \(\frac{1}{10}\) because the 1 is in the tenths decimal place

.13 is equivalent to \(\frac{13}{100}\) because the 3 is in the hundredths decimal place and the one is in the tenths decimal place

.237 is equivalent to \(\frac{237}{1,000}\) because the 7 is in the thousandths decimal place, and so on.

These decimals can be written as fractions, so they are considered rational.

Other decimals have repeating patterns. These are also considered rational because they can be expressed as a fraction based on the following proof:

The repeating decimal \(2.\overline{17}\) represents the digits \(2.1717171717\)…

Let’s try this as a practice problem.

Let \(x=2.\overline{17}\). The repeating decimal is two digits long, which represents the hundredths place.

So let’s multiply both sides of the equation by 100:

This results in \(100x=217.17171717\)…, which is equal to \(217.17\) repeating.

We move the decimal over two spots because we multiplied by 100. Now let’s subtract the original equation from this one:

\(100x\)\(=217.171717\)…
\(–\)\(x\)\(=002.171717\)…
\(99x\)\(=215\)

Notice that the repeating portion of the decimal is now eliminated.

Solving for \(x\) results in \(x=\frac{215}{99}\).

So, we have \(99x\) is equal to \(215\). Notice that the repeating portion of the decimal is now eliminated. Solving for \(x\) results in dividing both sides by \(99\): \(x = \frac{215}{99}\).

This is a fractional representation of \(x=2.\overline{17}\).

This proof shows that repeating decimals are also considered rational because they can be written as a fraction of integers. If you plug this into your calculator, you’ll get something close to, probably rounded, to 2.17 repeating.

It is important to note that not all decimals are repeating. Some decimals have an infinite number of non-repeating digits and, therefore, cannot be expressed as a fraction of integers. These types of real numbers are classified as irrational. While there are an infinite number of irrational numbers in the real number system, the most commonly used in mathematics are the square roots of non-perfect squares, like the square root of 2 for example, and the constants π and e. The notation for irrational numbers allows for efficiency in math applications.

For geometry, you may recall that π = 3.14159… for infinity. This is derived from the circumference of any circle and its diameter. Because the decimal value is non-repeating and infinite, we use an approximate value in math applications. Business applications regarding continuously compounded interest employ the irrational value of e, which has an approximate value of 2.718 again, for infinity.

Thanks for watching, and happy studying!

Frequently Asked Questions

Q

Are all integers rational numbers?

A

Yes, a rational number is any number that can be expressed as a fraction. All integers fit this definition.

Q

Are negative numbers rational?

A

Yes, most negative numbers are rational. A rational number is any number that can be written as a fraction. These include whole numbers, fractions, decimals that end, and decimals that repeat. Positive and negative do not affect rationality.

Q

Are all rational numbers whole numbers?

A

No, not all rational numbers are whole numbers. Rational numbers include all numbers that end or repeat. A whole number is any number without a fractional part that is greater than or equal to zero.
Ex. 2.7 is a rational number but not a whole number.

Q

What is the difference between rational and irrational numbers?

A

The difference between rational and irrational numbers is that a rational number can be represented as an exact fraction and an irrational number cannot. A rational number includes any whole number, fraction, or decimal that ends or repeats. An irrational number is any number that cannot be turned into a fraction, so any number that does not fit the definition of a rational number.

Practice Questions

Question #1:

 
Is π rational?

Yes

No

Sometimes

Cannot be determined

Answer:

The correct answer is no. Pi (π) is an irrational number because it is a never-ending decimal that cannot be simplified as an exact fraction.

Question #2:

 
Is \(1.\overline{3}\) a rational number?

Yes

No

Sometimes

Cannot be determined

Answer:

The correct answer is yes. \(1.\overline{3}\) can be represented as the fraction \(1\frac{1}{3}\), which means it is rational. Any number that can be represented as a fraction is considered rational.

Question #3:

 
Which of the following numbers is an example of a rational number?

π

\(\sqrt{2}\)

4.17

\(4-\sqrt{7}\)
Answer:

The correct answer is 4.17. This is the only number out of this list that can be turned into a fraction, \(4\frac{17}{100}\).

Question #4:

 
Which of the following is an irrational number?

\(\frac{17}{3}\)

13

\(2.\overline{97}\)
\(\sqrt{3}\)
Answer:

The correct answer is \(\sqrt{3}\). Square roots of non-perfect squares are not rational because they are equal to a never-ending decimal number, which means it is a number that cannot be turned into a fraction.

Question #5:

 
Is \(\frac{7}{9}\) rational?

Yes

No

Sometimes

Cannot be determined

Answer:

The correct answer is yes. A rational number is any number that can be turned into a fraction, and \(\frac{7}{9}\) is a fraction.

 

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by Mometrix Test Preparation | Last Updated: November 18, 2021