Correct meaning, definition , and differences between rational and irrational which I can understand clearly
Correct meaning, definition , and differences between rational and irrational which I can understand clearly
Definition of Rational Numbers The term ratio is derived from the word ratio, which means the comparison of two quantities and expressed in simple fraction. A number is said to be rational if it can be written in the form of a fraction such as p/q where both p (numerator) and q (denominator) are integers and denominator is a natural number (a non-zero number). Integers, fractions including mixed fraction, recurring decimals, finite decimals, etc., are all rational numbers.
Examples of Rational Number
- 1/9 – Both numerator and denominator are integers.
- 7 – Can be expressed as 7/1, wherein 7 is the quotient of integers 7 and 1.
- √16 – As the square root can be simplified to 4, which is the quotient of fraction 4/1
- 0.5 – Can be written as 5/10 or 1/2 and all terminating decimals are rational.
- 0.3333333333 – All recurring decimals are rational.
Definition of Irrational Numbers A number is said to be irrational when it cannot be simplified to any fraction of an integer (x) and a natural number (y). It can also be understood as a number which is irrational. The decimal expansion of the irrational number is neither finite nor recurring. It includes surds and special numbers like π (‘pi’ is the most common irrational number) and e. A surd is a non-perfect square or cube which cannot be further reduced to remove square root or cube root.
Examples of Irrational Number
- √2 – √2 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.
- 3/0 – Fraction with denominator zero, is irrational.
- π – As the decimal value of π is never-ending, never-repeating and never shows any pattern. Therefore, the value of pi is not exactly equal to any fraction. The number 22/7 is just and approximation.
- 0.3131131113 – The decimals are neither terminating nor recurring. So it cannot be expressed as a quotient of a fraction.
The difference between rational and irrational numbers :-
- Rational Number is defined as the number which can be written in a ratio of two integers. An irrational number is a number which cannot be expressed in a ratio of two integers.
- In rational numbers, both numerator and denominator are whole numbers, where the denominator is not equal to zero. While an irrational number cannot be written in a fraction.
- The rational number includes numbers that are perfect squares like 9, 16, 25 and so on. On the other hand, an irrational number includes surds like 2, 3, 5, etc.
- 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 term ratio is derived from the word ratio, which means the comparison of two quantities and expressed in simple fraction. A number is said to be rational if it can be written in the form of a fraction such as p/q where both p (numerator) and q (denominator) are integers and denominator is a natural number (a non-zero number). Integers, fractions including mixed fraction, recurring decimals, finite decimals, etc., are all rational numbers.
Examples of Rational Number
- 1/9 – Both numerator and denominator are integers.
- 7 – Can be expressed as 7/1, wherein 7 is the quotient of integers 7 and 1.
- √16 – As the square root can be simplified to 4, which is the quotient of fraction 4/1
- 0.5 – Can be written as 5/10 or 1/2 and all terminating decimals are rational.
- 0.3333333333 – All recurring decimals are rational.
A number is said to be irrational when it cannot be simplified to any fraction of an integer (x) and a natural number (y). It can also be understood as a number which is irrational. The decimal expansion of the irrational number is neither finite nor recurring. It includes surds and special numbers like π (‘pi’ is the most common irrational number) and e. A surd is a non-perfect square or cube which cannot be further reduced to remove square root or cube root.
Examples of Irrational Number
- √2 – √2 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.
- 3/0 – Fraction with denominator zero, is irrational.
- π – As the decimal value of π is never-ending, never-repeating and never shows any pattern. Therefore, the value of pi is not exactly equal to any fraction. The number 22/7 is just and approximation.
- 0.3131131113 – The decimals are neither terminating nor recurring. So it cannot be expressed as a quotient of a fraction.
The difference between rational and irrational numbers :-
- Rational Number is defined as the number which can be written in a ratio of two integers. An irrational number is a number which cannot be expressed in a ratio of two integers.
- In rational numbers, both numerator and denominator are whole numbers, where the denominator is not equal to zero. While an irrational number cannot be written in a fraction.
- The rational number includes numbers that are perfect squares like 9, 16, 25 and so on. On the other hand, an irrational number includes surds like 2, 3, 5, etc.
- 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.
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