List of Notes - 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 81 100 121 144 169 196 225

Perfect 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 1000

For 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`.