Difference Between Rational Numbers vs. Irrational Numbers

Main Difference

The main difference between Rational Numbers and Irrational numbers is that the rational numbers can be written in fraction form whereas irrational numbers cannot be written in a fractional form where denominator and numerator are not equal to zero.

Rational Numbers vs. Irrational Numbers

Arithmetical values or the mathematical numbers are divided into various groups and categories by their features and characteristics. The major categories involve whole numbers, 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 surd values of irrational numbers. Rational numbers can be written in the fractional form, but irrational numbers can never be expressed as a fraction. Upon decimal expansion, irrational numbers give infinite and non-recurring values whereas the rational numbers have recurring and finite values. Rational Numbers and Irrational numbers are two of the major categories of numbers on the whole. The numbers that can be written in the fractional form are referred to as rational numbers.

The numerator and the denominator in the fractional form of rational numbers are surely integers and whole. In other words, we can also say that the numbers which can be expressed as the ratio of two integers are termed as rational numbers. Unlike irrational numbers, rational numbers are perfect squares of numbers, and they possess a recurring or finite number of value after written in the decimal form. On the other hand, irrational numbers are opposite numbers in nature as compare to rational numbers. Irrational numbers can never be written in the fractional form, and they also cannot express as the ratio between two integers. Although the irrational numbers can be written in the decimal form, upon decimal expansion, they always give infinite and non-recurring values. As compare to rational numbers the irrational numbers give surd values despite the perfect squares of integers.

Comparison Chart

Rational NumbersIrrational Numbers
A rational number is a number that can be written as the ratio of two integers or a number that can be expressed in fractional form.An irrational number is a number that cannot be written as the ratio of two integers or a number that cannot be expressed in the fractional form.
Involves
Rational numbers includes perfect squares and finite decimal values.Irrational numbers involves surd values and infinite decimal values.
Decimal Extension
Rational numbers possess finite and recurring values upon decimal expansion.Irrational numbers possess infinite and non-recurring values upon decimal expansion.
Fraction
Rational numbers can be expressed in the fractional form.Irrational numbers cannot be written in the fractional form.
Integers
All the integers are rational numbers in nature.Not all integers are irrational numbers in 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 in fractional form. All the integers are rational numbers in nature. Rational numbers can be expressed in the fractional form, where the denominator is not equal to 0, and both numerator and denominator are integers. Rational numbers possess finite and recurring values upon 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 recurring decimals are rational.
  • √25 – As the square root can be simplified to 5, which is the quotient of fraction 5/1
  • 1/7 – Both numerator and denominator are integers.
  • 4 – Can be expressed as 4/1, whereas 4 is the quotient of integers 4 and 1.
  • 0.2 – Can be written as 2/10 where all the terminating decimals are rational.

What are Irrational Numbers?

An irrational number is a number that cannot be written as the ratio of two integers or a number that cannot be expressed in the fractional form. Not all integers are irrational numbers in nature. Irrational numbers cannot be written in the fractional form. Irrational numbers involve surd values and infinite decimal values. Irrational numbers possess infinite and non-recurring values upon decimal expansion. The infinite and non-recurring decimal values of irrational numbers are themselves irrational.

Examples

  • π – Being infinite and non-recurring upon expansion falls in the category of irrational numbers. The actual value of π is not exactly equal to any fraction. 22/7 in the fractional form is the approximated an estimated value of Pi.
  • 0.2673633379 – The decimal expansion values are not finite and are non-recurring, so it is the irrational value or number.
  • √3 – √3 cannot be simplified, and so, it is irrational.
  • √7/5 – The given number is a fraction, but it is not the only criteria to be called as the rational number. Both numerator and denominator need to integers, and √7 is not an integer. Hence, the given number is irrational.
  • 7/0 – Fraction with denominator zero, is irrational.

Key Differences

  • Rational numbers are the numbers that can be written in fractional form while irrational numbers are the number that cannot be written in the fractional form.
  • The numerator and denominator both are integers and not equal to zero in case of rational numbers whereas the denominator is always zero in case of 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 between two integers whereas irrational numbers can never be expressed as the ratio between the two integers.
  • The finite and recurring decimal values of the rational numbers themselves are rational, on the flip side, the infinite and non-recurring decimal values of irrational numbers are themselves irrational.

Comparison Video

Conclusion

Rational numbers are those numbers that are used to show the ratio between two integers, can be written in the fractional form, give perfect squares and possess finite and recurring values upon decimal expansion. Irrational numbers, on the other hand, are the numbers that cannot be expressed in the fractional form, does not depict ratio between two integers, give surd values and upon decimal expansion give non-recurring and infinite values.

Author:

Harlon Moss

Harlon currently works as a quality moderator and content writer for Turpy Media. He graduated from the University of California in 2010 with a degree in Computer Science. Follow him on Twitter @HarlonMoss or on Pinterest @HarlonMoss

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