# Category: Irrational Numbers

## Squares and Square Roots

To square a number, multiply it by itself.

For, example, square of `5` is `5\times5=25`.

When we talked about exponents and integers, we said that number `a` raised to `b`-th power is number `a` multiplied by itself `b` times: $$$\color{purple}{a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}}$$$.

## Perfect Square

Perfect square is a result of multiplying whole number by itself.

Whole number 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Perfect square (square of a whole number) 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225Perfect numbers are written on the main diagonal of the multiplication table.

## Cubes and Cube Roots

To cube a number, use it in multiplication three times.

For, example, cube of `5` is `5times5times5=125`.

When we talked about exponents and integers, we said that the number `a` raised to `b`-th power is the number `a` multiplied by itself `b` times: $$$\color{purple}{a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}}$$$.

## Perfect Cube

Perfect cube is a result of cubing whole number by itself.

Whole number 0 1 2 3 4 5 6 7 8 9 10 Perfect cube (cube of a whole number) 0 1 8 27 64 125 216 343 512 729 1000For more information, see cubes and cube roots.

## Nth Root

Similarly to square root and cube root, we can define nth root.

Nth root of a number `b` is such number `a`, that `a^n=b`.

Notation for nth root is following: `huge color(purple)(root(n)(b))`.

Square root is 2nd root, which is `sqrt(b)` (we just don't write 2).

## What is Irrational Number

Irrational number is a number, that is not rational.

What does that mean?

It means, that we can't represent irrational number as a fraction, decimal with finite number of digits, or repeating decimal.

## Irrational Numbers on a Number Line

Since each irrational number can be represented as infinite decimal, then we can proceed in the same way, as we did when placed decimals on a number line.

`pi~~3.14`, so it is slightly to the right of 3.

## Real Numbers

Real numbers are rational numbers plus irrational numbers.

Since rational numbers include integers and fractions, then real numbers include:

- integers {..., -5,-4,-3,-2,-1,0,1,2,3,4,5,...}
- fractions (proper, improper, mixed numbers)
- irrational numbers (like `sqrt(2)`, `root(3)(-15)`, `pi`, `e` etc.)

Set of real numbers, i.e. group of real numbers, is denoted by `RR`.