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Real Numbers

In math, the real numbers contains both rational number and irrational number.
The rational number is in the form of $\frac{5}{3}$, $\frac{-23}{40}$. And the irrational numbers like pi, square root of two, any number in decimal representation like 2.4873.... where the digit is continuing in the way. Real number may be infinity long number line.
A real number may be rational or irrational number, either algebraic, positive number, negative number, or zero.

Is Zillion a Real Number?

No, zillion is not a real number. Because we cant square it or we cant even triangle it.

What is a Real Number?

What are real numbers, the numbers which can be represented in the number line are called real numbers.
Any number is a real number except the imaginary numbers.
All other numbers like natural number, fraction, decimals comes under real number. All positive and negative are real numbers. Once imaginary numbers were defined, they termed rest of numbers as real numbers.

Real Numbers Definition

The combination of both rational and irrational numbers together can be defined as real numbers, combined together is called a set of real numbers. It is denoted by R.

R = set of all rational numbers and irrational numbers ……
All the properties of rational of irrational numbers, that holds for real numbers, as it is an extended set of rational numbers.

All Real Numbers Symbol

Here we will discuss about the symbol for all real numbers.

Real numbers can be classified into two parts

The combination of rational and all irrational numbers together form the set of all Real Numbers. The set is denoted by R. The set of real numbers is also called the continuum.

The real numbers can be express in decimal notation, including those that require an infinite decimal expansion.

The real numbers includes all integers, positive value and negative value, all fractions; and the irrational numbers.

Real numbers symbols

The listing of common symbols found within all branches of mathematics.

= - Stand for equal to

≠ - Stand for not equal to

< - Less than

> - Greater than

≤ - Less than or equal

≥ - Greater than or equal

+ - Addition

− - Minus

× - Multiplication

÷ - Division

√ - Square root

Is 0 a Real Number?

Is zero a real number? Yes, zero is a real number.
All the numbers which exists in reality are real numbers. The square root of a negative radical is not a real number. It is imaginary and is denoted by i. A number without any imaginary part is a real number.

Is Pi a Real Number?

Yes, pi is also a real number. A number π (sometimes note down as pies) is a numerical constant whose value is the ratio of every circle's circumference to its diameter in Euclidean space, this is the like worth as the ratio of a circle's area to the four-sided figure of its radius.

Pi (π) is a real number. Diameter and circumference values are real.

Circumference = pi x diameter.

Pi = $\frac{c}{d}$

Example for numbers like pie (π) real number π (Pi) = 3.1415

Set of Real Numbers

Inside the set of real numbers there are three familiar sets of a numbers, which are natural numbers (1, 2, 3....), the integers ( ..-2, -1, 0, 1, 2....) and the rational numbers(fractions).

Solved Example

Question: The bag contains red and green balls, the ratio of red ball to green ball is 3:4. The bag contains 140 green balls, then how many red balls are there?
Solution:

Step 1: assign the variable

Let x = red balls

To write the ration as fraction

Red= $\frac{(3)}{(4)}$ = $\frac{(x)}{(140)}$

Step 2: solve the equation

Cross multiply

3 x 140 = 4 X x

420 = 4x

Isolate the variable x

X = $\frac{(420)}{(4)}$

Correct answer is 105 red balls

Subset of Real Numbers

The real numbers include both rational numbers, irrational numbers, such as 42 and $\frac{-23}{129}$, pi and the square root of two, or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way. Here we are going to learn about different type’s subset of the real number and its example problem.

Example: Find square root of 74 is which subset,

Solution: Square root of 74 is 8.60232526

So it is irrational number subset.

Operations on Real Number

The operations on real number are addition, subtraction, multiplication and division.

1. Add real numbers:

`(5/8)+(3/8)`

Step 1: The bottom numbers are same then go to step 2

Step 2: Add the top numbers $\frac{5}{8}$ + $\frac{3}{8}$ = $\frac{8}{8}$

Step 3: simplify fraction $\frac{8}{8}$ = 1

2. Subtract real numbers:

`(5/8)-(3/8)`

Step 1: The bottom numbers are same then go to step 2

Step 2: Subtract the top numbers $\frac{5}{8}$ - $\frac{3}{8}$ = $\frac{2}{8}$

Step 3: Simplify fraction $\frac{2}{8}$ = $\frac{1}{4}$

3. Multiply real numbers:

`(2/4)*(4/8)`

Step1: Multiply the top numbers 2 x 4 = 8

Step 2: Multiply the bottom numbers 4 x 8 = 32 therefore, $\frac{8}{32}$

Step 3: Simplify the fraction. $\frac{8}{32}$ = $\frac{1}{4}$

4. Divide real numbers:

`(1/2)/(4/2)`

Step 1: Turning the second fraction upside-down

`(4)/(2)` =>`(2)/(4)`

Step 2: Multiplying the first fraction by reciprocal

`(1)/(2)` x `(2)/(4)` = `(1*2)/(2*4)` = `(2)/(8)`

Step 3: Simplifying the fraction

= $\frac{2}{8}$

= $\frac{1}{4}$

Real Numbers Examples

Below are a few examples on real numbers -

Solved Examples

Question 1: A box contains blue and yellow balls, the ratio of blue balls to yellow balls is 4:5. If the bag contains 200 yellow balls, how many blue balls are there?
Solution:

Solving Blue balls,

Step 1: Assign variables:

Let 'y'= blue balls

Write the items in the ratio as a fraction.

$\frac{Blue}{yellow}$ = $\frac{4}{5}$ = $\frac{y}{200}$

Step 2: Solve the equation

Cross Multiply

4 × 200 = 5 × y

800 = 5y

Solving Isolate variable y

Y = $\frac{800}{5}$ = 160.

Correct answer is 160 blue balls

Question 2: To solve a bag of orange and brown toys, the ratio of orange toys to brown toys is 2:3. If the bag contains 90 brown toys, how many orange toys are there?
Solution:

Solving orange toys,

Step 1: Assign variables:

Let y = orange toys

Write the items in the ratio as a fraction.

$\frac{Orange}{brown}$ = $\frac{2}{3}$ = $\frac{y}{90}$

Step 2: Solve the equation

Cross Multiply

2 × 90 = 3 × y

180 = 3y

Solving Isolate variable y

y = 180 ÷ 3 = 60

Correct answer is 60 orange toys