What Are Irrational Numbers?

Irrational numbers are numbers that cannot be expressed as a ratio (fraction) of two integers. Unlike rational numbers, they cannot be written as fractions where both the numerator and denominator are integers. These numbers are typically represented as decimal expansions that neither terminate nor repeat.

Definition of Irrational Numbers

Irrational numbers are numbers that are non-terminating and non-repeating when written in decimal form. This means they cannot be expressed as the ratio of two integers. Their infinite decimal representation lacks any repeating pattern or order, distinguishing them from rational numbers.

Examples of Irrational Numbers

1. √2:
   • The square root of 2 is approximately 1.414213….
   • It cannot be expressed as the ratio of two integers, making it irrational.
2. π (Pi):
   • Pi represents the ratio of a circle’s circumference to its diameter, with a value of approximately 3.141592653….
   • Since its decimal form neither terminates nor repeats, π is irrational.
3. e (Euler’s Number):
   • Euler’s number, used in natural logarithms and mathematical analysis, is approximately 2.718281828….
   • Like π, it is non-terminating and non-repeating, making it irrational.

Perfect Squares and Irrational Numbers

• Perfect Squares:
  The square root of a perfect square is always rational. Examples include:
  • √4 = 2
  • √9 = 3
• Non-Perfect Squares:
  The square root of a non-perfect square is always irrational. Examples include:
  • √3 ≈ 1.732…
  • √5 ≈ 2.236…

Square Root

In mathematics, the operation of square roots is the reverse of exponentiation. While the exponentiation of a number is written as, xⁿ the n-th root of x is expressed as

Here, n is referred to as the degree of the root. In the case of n=2, the degree is often omitted, but when n > 2, it is necessary to write it explicitly.

What Does Taking a Root Mean?

Root extraction is the reverse process of expressing a number in its exponential form. It involves determining which number, when raised to the power, results in the given number. Let’s explain this with some examples:

As we can see from these examples, if , then

In other words, the root operation is the inverse of exponentiation.

What we need to understand from this is that every radical expression can be converted into an exponential form. Therefore, if someone has fully grasped exponential numbers, they should not have difficulty understanding radical expressions.

*** For roots with an even degree, the radicand (the number inside the root) cannot be negative because, in the real number system, taking an even-degree root is only defined for positive numbers.

Simplification – Expansion Rules

Expressing a Rooted Expression as Another Rooted Expression

This rule allows us to rewrite a root expression with a different root degree while maintaining its value.

Expressing Roots in a Different Form

This rule helps express roots with different indices or exponents in an alternative format.

Writing a Rooted Expression as Exponent

Converting a Rooted Expression into an Exponential Expression

This rule is useful for converting root expressions into exponent notation.

When the Root’s Degree is Odd, the Value Remains the Same

For odd-degree roots, the value inside the root remains unchanged when simplified.

When the Root’s Degree is Even, the Absolute Value is Taken

For even-degree roots, the result is always positive, so we take the absolute value.

Taking a Number Out of the Radical

When taking a number out of the radical, use the following formula:

Putting a Number Inside the Radical

When putting a number into the radical, use this formula:

Addition and Subtraction in Rooted Expressions

If the roots have the same degree (index) and the same radicand (the expression inside the root), the coefficients can be added or subtracted. The general formula is:

Multiplication of Expressions with Equal Degrees

The product of radical expressions with the same degree can be written as the radical of the product of the radicands. Mathematically, this is expressed as:

This shows that the product of two radical expressions with the same degree is equal to the radical of the product of the radicands.

In these examples, the product of radical expressions with the same degree is calculated as the radical of the product of the radicands.

Division of Expressions with Equal Degrees

The quotient of radical expressions with the same degree can be written as the radical of the quotient of the radicands. Mathematically, this is expressed as:

This shows that the quotient of two radical expressions with the same degree is equal to the radical of the quotient of the radicands.

In these examples, the quotient of radical expressions with the same degree is calculated as the radical of the quotient of the radicands.

Conjugation in Rooted Expressions

Type 1 Conjugate

The conjugate of a single radical expression is the expression itself. 

Type 2 Conjugate

The conjugate of the sum of two radical expressions is the difference of those expressions.