What Are The Properties Of Irrational Numbers

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Written By Berry Mathew


Arithmetic, on the whole, can be seen as a result of the basic activity of counting. And interestingly, counting is nothing but repeated creation of new numbers from the previous ones resulting in an infinite series of integers, which are grouped into several types like real numbers, whole numbers, rational and irrational numbers, prime numbers, etc.

The sequence of these numbers is an extremely valuable tool for the human mind; it contains an inexhaustible wealth of extraordinary laws derived from the application of the four basic arithmetic operations. We call a number that is not rational as irrational. All points on the number line are defined by the rational and irrational numbers together. As a consequence, the real numbers are made up of two kinds of numbers, rational and irrational.

Rational numbers are those numbers that can be represented as x/y, where x and y are integers, and y is not equal to zero. These can be positive or negative, and the decimal form of the number may either terminate or repeat itself. The counting numbers, integers, non-integers, and whole numbers are all rational numbers.

Counting numbers being the natural numbers like 1, 2, 3, 4, … . Whole numbers comprise counting numbers along with zero. The integers comprise of the counting numbers and their opposites as well as zero like ….-2, -1, 0, 1, 2…..

Non-integers are the fractions that are written as terminated or repeated decimal.

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A terminating decimal is one that reaches a full stop, while a repeating decimal has the same digits or a block of digits repeated over forever like 5.25, 1/3, etc.

While irrational numbers cannot be expressed as a fraction, the first irrational number to be discovered was the square root of two, which has a value of 1.41421….. Pi is another example of an irrational number, with a value of 3.147……. The decimal part is not replicated in this irrational number.

There are two types of irrational numbers: algebraic and transcendental numbers. Algebraic ones are those which have roots of the algebraic equation as the square root of 2. The other category of numbers is transcendental numbers, which is represented by pi and e, where pi is a trigonometric function, and e is an exponential function. Irrational numbers come in a multitude of ways.

Irrational numbers are non-repeating and non-terminating as the decimal part never ends and never repeats itself. The most common irrational number is the value of pi.

Let us see some theorems about rational and irrational numbers:

-The value of the square root of any prime number is an irrational number.

-If there is at least one prime that occurs an odd number of times in the prime factorization of the natural number X, then the square root of X is irrational.

-If each prime appears an even number of times in the prime factorization of the natural number Y, then Y is the square of a natural number, and hence the square root of Y is rational.

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-If a number’s decimal representation repeats, then the number is rational.

Math worksheets based on numbers are a great resource for strengthening upon these concepts, and many such worksheets are available online. One such online platform is Cuemath, from where the students can get access to a variety of interactive and engaging worksheets for their everyday practice. Also these are grouped age wise and also as per the understanding level of the students thus giving a wider variety to choose from.