Numbers are all we have in mathematics. What is a number? It is an arithmetic value that indicates the quantity of any figure, word or symbol. This has implications such as measurements, counting, etc.
Numbers can be of various kinds such as real, whole, natural, integers and complex numbers. Real numbers are divided into two types of numbers which are rational and irrational.
Now, you know what types are rational and irrational numbers. If you want to learn more about Rational and Irrational Numbers, here is the article to guide you related to rational and Irrational numbers.
We will help you understand the difference between these numbers and show the difference between them with examples. You will also understand how to identify these numbers in the examples.
Firstly, let us understand the definition of Rational and Irrational Numbers:
Rational Numbers
Have you ever wondered what type of number you will get if you started with all integers and then included all the fractions? The result formed would be called rational numbers. It can be written as a ratio of two integers.
A rational number is defined as a ratio of two integers. An irrational number is defined as any real number that is not rational. Because the irrationals are defined in this way, these two sets of numbers can’t share even a single element. Another example of such a relationship between two types of numbers is even and odd numbers.
Suppose we define the “even” numbers as all the integers that are “evenly” divided by 2 (i.e. no remainders after the division) and define the “odd” numbers as all the integers that are not even. In that case, there will be no integers that are both even and odd.
Not only that, but there are also no integers that are neither even nor odd. Together these two sets include all the integers but do not share any integers in common. Every integer must be either one or the other.
Rational numbers are the numbers that are integers and fractions. A number is rational if it can be expressed as a fraction, where the numerator & the denominator are both integers. For example, .3333333… is rational because.
It is because it can be easily expressed by the fraction (1/3). Note that any integer can simply be expressed as itself/1. Therefore, all integers are a rational number. On the other hand, irrational is any number that is not rational.
Also, Read: How to Multiply Mixed Fractions
Irrational Numbers
These numbers are one of those real numbers that cannot be represented in a ratio. These are the set of real numbers that cannot be expressed in the form of fractions. Example p/q, where p and q are integers. In this q is not equal to 0.
It is a contradiction of rational numbers. Irrational numbers are numbers that have a never-ending decimal representation, but no pattern repeats itself forever.
It is the main difference. An irrational number has infinitely many random digits without any noticeable pattern after the decimal point. Irrational numbers can never be exactly measured.
For example, you can never exactly evaluate the square root of 21, as a result, is irrational. Even a trillionth place after the decimal point is just an approximation to an irrational number.
Some of the examples of irrational numbers are stated below:
ㄫ(pi) is an irrational number. π=3⋅14159265… The decimal value never stops at any point. Since the value of ㄫ is closer to the fraction 22/7, we take the value of pi as 22/7 or 3.14
√2 is an irrational number. Consider a right-angled isosceles triangle, with the two equal sides AB and BC of length 1 unit. By the Pythagoras theorem, the hypotenuse AC will be √2. √2=1⋅414213⋅⋅⋅⋅
Euler’s number e is an irrational number. e=2⋅718281⋅⋅⋅⋅
| Irrational number | value |
| π | 3.14159265…. |
| e | 2.7182818….. |
| √2 | 1.414213562… |
| √3 | 1.73205080… |
| √5 | 2.23606797…. |
| √7 | 2.64575131…. |
| √11 | 3.31662479… |
| √13 | 3.605551275… |
| -√3/2 | -0.866025…. |
| ∛47 | 3.60882608 |
Difference between Rational and Irrational Numbers
- Rational numbers are the numbers that can be expressed in the ratio of two integers. In comparison, irrational numbers are numbers that cannot be written as the ratio of two integers.
- Rational numbers are expressed in fractions where the denominator is not equal (≠ ) to 0 while irrational numbers are the numbers that cannot be expressed in a fractions.
- Rational numbers include perfect squares. On the other hand, irrational numbers include Surds.
- If you talk about decimal expansion in rational and irrational numbers, rational numbers have finite & recurring decimals while irrational numbers have non-finite & non-recurring decimals.
- An example of a rational number is 0.33333, 0.656565.., 1.75 and an example of irrational numbers is π, √13, e.
- In rational numbers, both numerator and denominator are whole numbers & the denominator is not equal to zero, while irrational numbers cannot be written in fractional form.
- The rational number includes only those decimals, which are finite and repeating. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern.
The examples of rational numbers are 1/2, 3/4, 11/2, 0.45, 10, etc.
The examples of irrational numbers are Pi (π) = 3.14159…., Euler’s Number (e) = (2.71828…), and √3, √2.
Identifying Rational and Irrational Numbers
Identifying Rational and Irrational numbers is quite straightforward. Rational numbers are defined as real numbers which are written in a ratio of two integers, while irrational numbers are the ones that cannot be expressed in a ratio of two or more integers.
Below is a perfect way to identify the difference between these two types of real numbers:
- If a number is a terminating number or repeating decimal, then it is rational, A simple example is, 1/2 = 0.5
- If a number is a non-terminating & non-repeating decimal, then it is referred to as an irrational number. A simple way to recognise irrational numbers is 0.31545673 & so on.
In other words, numbers that can be represented on a number line are rational and irrational numbers can be represented on a number line.
Conclusion
You now know what rational and irrational numbers are. You also understand the difference between the two. Rational numbers can be easily expressed as a fraction, a ratio of two integers. Irrational numbers are numbers that cannot be expressed as a fraction. Hope by now you understand the difference between the two number types.
