Difference Between Rational Numbers and Irrational Numbers
Core difference
The main difference between rational numbers and irrational numbers is that rational numbers can be written in fractional form whereas irrational numbers cannot be written in fractional form where the denominator and numerator are not equal to zero.
Rational numbers versus irrational numbers
Arithmetic values or mathematical numbers are divided into various groups and categories based on their features and characteristics. The main categories include integers, real numbers, natural numbers, rational numbers, irrational numbers, etc. The basic difference between rational numbers and irrational numbers includes perfect squares of rational numbers in contrast to exorbitant values of irrational numbers. Rational numbers can be written in fraction form, but irrational numbers can never be expressed as a fraction. After decimal expansion, irrational numbers give infinite and non-recurring values, while rational numbers have finite and recurring values.
The numerator and denominator in the fractional form of rational numbers are surely integers and integers. In other words, we can also say that numbers that can be expressed as the ratio of two integers are called rational numbers. Unlike irrational numbers, rational numbers are perfect squares of numbers and have a recurring or finite number value after they are written in decimal form. On the other hand, irrational numbers are opposite numbers in nature compared to rational numbers. Irrational numbers can never be written in fractional form, and they can never be expressed as the ratio of two whole numbers. Although irrational numbers can be written in decimal form, in decimal expansion, they always give infinite and non-recurring values.
Rational numbers | Irrational numbers |
A rational number is a number that can be written as the ratio of two integers or a number that can be expressed as a fraction. | An irrational number is a number that cannot be written as the ratio of two integers or a number that cannot be expressed as a fraction. |
involves | |
Rational numbers include perfect squares and finite decimal values. | Irrational numbers involve odd values and infinite decimal values. |
decimal extension | |
Rational numbers have finite and recurring values in decimal expansion. | Irrational numbers have infinite and non-recurring values in decimal expansion. |
Fraction | |
Rational numbers can be expressed in fractional form. | Irrational numbers cannot be written in fractional form. |
integers | |
All integers are rational numbers by nature. | Not all integers are irrational numbers by nature. |
What are rational numbers?
A rational number is a number that can be written as the ratio of two integers or a number that can be expressed as a fraction. All integers are rational numbers by nature. Rational numbers can be expressed in fractional form, where the denominator is not equal to 0 and both the numerator and denominator are whole numbers. Rational numbers have finite and recurring values in decimal expansion. Rational numbers include perfect squares and finite decimal values. The finite and recurring decimal values of the rational numbers themselves are rational.
Examples
- 0.9999999– All repeating decimals are rational.
- √25: Since the square root can be simplified to 5, which is the quotient of the fraction 5/1
- 1/7: Both the numerator and denominator are whole numbers.
- 4 – Can be expressed as 4/1, while 4 is the ratio of the integers 4 and 1.
- 0.2 – Can be written as 2/10 where all final decimals are rational.
An irrational number is a number that cannot be written as the ratio of two integers or a number that cannot be expressed as a fraction. Not all integers are irrational numbers by nature. Irrational numbers cannot be written in fractional form. Irrational numbers involve odd values and infinite decimal values. Irrational numbers have infinite and non-recurring values in decimal expansion. The non-recurring infinite decimal values of irrational numbers are themselves irrational.
Examples
- π – Being infinite and non-recurring when expanded falls into the category of irrational numbers. The real value of π is not exactly equal to any fraction. 22/7 in fractional form is the approximate estimated value of Pi.
- 0.2673633379 – Decimal expansion values are non-finite and non-recurring, so it is the irrational value or number.
- √3 – √3 cannot be simplified, so it is irrational.
- √7 / 5: The given number is a fraction, but it is not the only criterion for being called a rational number. Both the numerator and denominator must be whole numbers, and √7 is not a whole number. Therefore, the given number is irrational.
- 7/0 – Fraction with denominator zero, it is irrational.
Key differences
- Rational numbers are numbers that can be written in fractional form, while irrational numbers are numbers that cannot be written in fractional form.
- Both the numerator and denominator are integers and are not equal to zero for rational numbers, while the denominator is always zero for irrational numbers.
- When written in decimal form, rational numbers give finite and recurring values, on the other hand, irrational numbers give infinite and non-recurring values when written in decimal form.
- Rational numbers can be written as the ratio of two integers, while irrational numbers can never be expressed as the ratio of the two integers.
- The finite and recurring decimal values of rational numbers are themselves rational, on the other hand, the infinite and non-recurring decimal values of irrational numbers are themselves irrational.
Conclusion
Rational numbers are those numbers that are used to show the ratio between two integers, can be written in fractional form, give perfect squares, and have finite and recurring values in decimal expansion. Irrational numbers, on the other hand, are the numbers that cannot be expressed in fractional form, do not represent the relationship between two integers, give strange values, and in decimal expansion they give infinite and non-recurring values.