Skip to main content

Examples of Equations with Both Rational and Irrational Numbers

By Kathleen Knowles, 01 Oct 2020

Both rational and irrational numbers can be referred to as real numbers, but when it comes to their properties, there are a few differences. You can represent a rational number in the form P/Q where P and Q are integers and Q ≠ 0.

For irrational numbers, you can't write them in simple fractions. 2/3 Is an example of a rational number, whereas √2 is an irrational number.


Let's start by defining each term separately, then we can learn more about each and work through some examples.

What Is a Rational Number?

Any number expressed as a fraction with positive numbers, negative numbers, and a zero is referred to as a rational number. Rational numbers are derived from the word 'ratio.' In other words, it's the ratio of two integers. For example, 3/2 is a rational number, which means 3 is divided by another integer 2.

What Is an Irrational Number?

Essentially, irrational numbers can be written as decimals but as a ratio of two integers. Irrational numbers tend to have endless non-repeating digits after the decimal point. Take this example: √8= 2.828.

Examples of Rational and Irrational Numbers

For Rational

  • 0.5 can be written as ½ or 5/10, and any terminating decimal is a rational number.
  • √81 as the square root can be simplified to 9, which is the quotient of the fraction 9/1
  • You can express 3 as 3/1, where 3 is the quotient of the integers 3 and 1.
  • 0.777777 is recurring decimals and is a rational number.
  • 1/5 is a rational number because both denominator and numerator are integers.

For Irrationals

  • √2 this number can't be simplified; hence it is an irrational number.
  • Π is an irrational number and has a value of 3.142…which is a never-ending and non-repeating number. Therefore the value of π is not precisely equal to any fraction. The fraction 22/7 is just an estimation.
  • 0.212112111…is an irrational number and is non-recurring and non-terminating, so it cannot be expressed as a quotient of a fraction.
  • Though number in √7/5 is given is a fraction, both the numerator and denominator must be integers.  But because √ 7 is not an integer, the number provided is irrational.
  • 5/0 is irrational. Any fraction with 0 as the denominator is irrational.

Properties of Rational and Irrational Numbers

These are the basic rules of arithmetic performed on the rational and irrational numbers

Rule 1: The result of the sum of two rational numbers is also rational

  • Example: ½ +1/3 = 5/6

Rule 2: The product of two rational numbers is rational

  • Example: ½ x 1/3 = 1/6

Rule 3: the result of the sum of two irrational numbers can either be rational or irrational

  • Take for example: √2 + √2 = 2√2 is irrational
  • while 2 + 2√5 + (-2√5) = 2 the result is rational

Rule 4: The result of the product of two irrational numbers can be either irrational or rational.

  • Take for example: √2 * √3 = √6 is irrational
  • while √2 * √2 = √4 = 2 rational

Let's now focus on the individual properties of rational and irrational numbers.

Distinctive Features of Rational Numbers

  • The sum of rational numbers is always a rational number. For example, if W and Z are two rational numbers, the sum of W and Z is rational.
  • The result of the division of a rational number by a non zero number is a rational number. For example, W÷Z= rational number.
  • The product of any two or three rational numbers results in another rational number. For instance, if you multiply W and Z, the answer you get should be rational.
  • The difference between the two rational numbers results in another number. For example, if you subtract Z from W, the answer you get is a rational number.

Since the result of the sum of any two rational numbers is a rational number, then rational numbers must always be closed. Consequently, rational numbers are equally closed for multiplication, subtraction, and division if the divisor is not equal to zero.

How to Represent Rational Numbers as Decimals

You can express any rational number as a terminating decimal or a non-terminating decimal. A terminating decimal is any decimal numeral where after a finite number of decimal points, the other succeeding place values are 0. For example 1/8= 0.125.

As you can clearly see from the given example, the division is exact. Such quotients are referred to as terminating decimals. Alternatively, rational numbers may also be expressed as non-terminating decimals. A non-terminating decimal is those decimals that continue endlessly after the decimal point.

Let's take a look at these examples:

  1. 3/7=0.42857142
  2. 18/23=0.78260869

In the two examples above, you realize that the division never ends, regardless of how long it may continue. Quotients of such divisions are what is referred to as terminating decimals.

In some cases, a non-terminating decimal may have a digit or a set of numbers repeating continuously. These non-terminating decimals are referred to as periodic, recurring, or circulating decimals. The set of repeating digits is referred to as the period of the recurring decimal.




Distinctive Features of Irrational Numbers

  • The product of irrational numbers can either be rational or irrational.
  • The result of a product of a non-zero rational number and an irrational number is always irrational.
  • The sum of irrational numbers can either be rational or irrational.
  • The sum of a rational and an irrational number is always irrational.
  • The difference between two irrational numbers may or may not be irrational.
  • The sum of a rational and an irrational number is always irrational.

Significant Differences Between Rational and Irrational Numbers

  • A rational number can be expressed as a ratio of two numbers in the (p/q form), while an irrational number cannot.
  • A rational number includes numbers that can end or repeat, while irrational numbers are non-terminating and non-repeating.
  • A rational number has perfect squares like 4, 9, 16, 25, and so on, while irrational numbers have surds like √2, √3, √5, √7.
  • For a rational number, the numerator and denominator are whole numbers where the denominator is not equal to zero: 3/2 = 1.5, 3.6767,
  • Irrational numbers cannot be written as a fraction: √5, √11.

Frequently Asked Questions (FAQs)

What are rational and irrational numbers?

You can express rational numbers in the form of a ratio (P/Q & Q ≠ 0), but for irrational numbers, you can't express them as a fraction. Even so, they're both real numbers you can include in a number line.

What is the significant difference between rational and irrational numbers?

Rational numbers are finite and repeating decimals, while irrational numbers are infinite and non-repeating.

Is pi a real number?

Pi (π) is an irrational number, so it's a real number. The value of (π) is 22/7 r 3.142…

Is 4 a rational number?

Yes, it is, because it satisfies all the conditions of a rational number. You can express it as a ratio, as long as the denominator is not equal to zero.

If you represent a decimal number by a bar, is it rational or irrational? A decimal number with a bar represents that the number after the decimal is repeating, so it's a rational number.

3.605551275… is rational or irrational?

The ellipsis (…) after 3.605551275 shows that the number is non terminating and has no repeating pattern. So it's irrational.


Rational numbers can be applied to calculate wear rate, variations, water current, or the speed of the wind. The above examples and explanations make it easy for anyone to tell the difference between a rational number and an irrational number.

Be the first to comment below.

Leave a comment

Comment Preview

HTML: You can use simple tags like <b>, <a href="...">, etc.

To enter math, you can can either:

  1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone):
    `a^2 = sqrt(b^2 + c^2)`
    (See more on ASCIIMath syntax); or
  2. Use simple LaTeX in the following format. Surround your math with \( and \).
    \( \int g dx = \sqrt{\frac{a}{b}} \)
    (This is standard simple LaTeX.)

NOTE: You can mix both types of math entry in your comment.


* indicates required

SquareCirclez is a "Top 100" Math Blog

SquareCirclez in Top 100 Math Blogs collection
From Math Blogs