Real Numbers Lesson: Types, Properties, Number Line and Examples

Lesson Overview

• What Is A Real Number Line?
• Real Number Chart
• What Are the Rules of Real Numbers?

A real number is any number that can be found on the number line. This includes:

Type of Number	Description	Examples
Whole Numbers	Numbers starting from 0 and increasing without fractions.	0, 1, 2, 3
Negative Numbers	Numbers less than zero.	-1, -2, -3
Fractions	Numbers that represent parts of a whole.	1/2, 3/4
Decimals	Numbers with a decimal point.	0.5, 3.14
Irrational Numbers	Numbers that can't be written as fractions, with non-repeating decimals.	√2, π
Rational Numbers	Numbers that can be written as fractions.	1/2, 3, 0.75 Export to Sheets

Real numbers include both rational numbers (numbers that can be written as fractions) and irrational numbers (numbers that can't be written as fractions). All of these types of numbers make up the real number system. Represented by the letter ‘R', real numbers can be both positive and negative.

Comparison of Rational and Irrational Numbers

Property	Rational Numbers	Irrational Numbers
Definition	Can be expressed as p/q where p and q are integers and q≠0	Cannot be expressed as p/q
Decimal Expansion	Either terminating or repeating	Non-terminating and non-repeating
Examples	1/2, 3, -4, 0.75, 22/7	√2, π, e, √3, 1/3
Can it be written as a fraction?	Yes	No
Examples of Decimal	0.5 (terminates), 0.3333... (repeats)	3.14159... (non-repeating), 1.41421356... (non-terminating)

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What Is A Real Number Line?

The real number line is a straight line that represents all real numbers, extending infinitely in both the positive and negative directions. It's a way to visualize numbers, showing how they relate to each other.

Key Features of the Real Number Line:

Key Feature	Description
Zero (0) placement	Zero is placed at the center of the line.
Positive numbers	Positive numbers are to the right of 0 (e.g., 1, 2, 3, 4, etc.).
Negative numbers	Negative numbers are to the left of 0 (e.g., -1, -2, -3, -4, etc.).
Infinite extension	The line continues infinitely in both directions, meaning it has no end.
Correspondence of points	Each point on the line corresponds to a real number, and numbers increase to the right and decrease to the left.

The numbers on the line can be whole numbers (integers), fractions, decimals, and irrational numbers like √2 or π. The real number line is continuous, meaning between any two numbers, there's always another number. For example, between 1 and 2, there's 1.5, 1.25, and many more numbers.

The real number line helps us understand how numbers are ordered and spaced in a simple, visual way.

Example: Represent the Following Numbers on a Number Line

We are given the following numbers to represent on a number line:

1. 23/5
2. 6
3. -33/7
   
   

Step 2: Plot the numbers on a number line.

Here's how you can place them on a number line:

• 23/5 = 4.6 is slightly greater than 4, so it will be just a little to the right of 4.
  
• 6 is a whole number, so it will be exactly at 6.
  
• -33/7 ≈ −4.71 is slightly less than -4, so it will be just to the left of -4.

The number line will look like this:

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Is 0 A Real Number?

Yes, 0 is a real number.

• 0 is a whole number, an integer, and a rational number (since it can be expressed as 0/1.

It is also part of the set of real numbers, as the real number system includes all integers, fractions, and irrational numbers, and 0 is a valid integer.

Key Points:

0 is neither positive nor negative, but it is considered a real number.

It is the additive identity in the real number system, meaning any real number added to 0 will remain unchanged (for example - 5+0 =5).

Real Number Chart

A real number chart is a visual representation of the real number system. It typically shows the relationship between various sets of real numbers along a number line, helping to illustrate the different categories of numbers.

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The properties of a real number chart are key features that describe how real numbers behave on the number line. These properties include:

1. Order: Real numbers are ordered from least to greatest, with negative numbers on the left and positive numbers on the right. Zero is in the middle.
   

2. Density: Between any two real numbers, there are infinitely many other real numbers. This means that no matter how close two numbers are, you can always find more numbers between them.
   

3. Continuity: The real number line is continuous, meaning there are no gaps between numbers. Every point on the number line represents a real number.
   

4. Intervals: Real numbers can be grouped into intervals (e.g., (a, b), [a, b]), which represent a range of values between two numbers.
   

5. Addition and Multiplication: Real numbers follow certain properties for operations:

• Commutative: a + b = b + a, and a × b = b × a.

• Associative: (a + b) + c = a + (b + c), and (a × b) × c = a × (b × c).

• Distributive: a × (b + c) = (a × b) + (a × c).
  

6. Inverses: Every real number (except 0) has an additive inverse (e.g., -a for a) and a multiplicative inverse (e.g., 1/a for a ≠ 0).

These properties help define the behavior of real numbers on the number line.

What Are the Rules of Real Numbers?

The rules for real numbers define how real numbers behave during arithmetic operations like addition, subtraction, multiplication, and division. These rules are fundamental to understanding how numbers interact and ensure that operations are consistent. Here are the key rules, each illustrated with an example:

1. Closure Property:

• Rule: The sum or product of two real numbers is always a real number.

• Example: 2+3=5 (sum) and 2×3=6 (product).

2. Commutative Property:

• Rule: The order of addition or multiplication does not affect the result.

• Example: 3+4=4+3 and 2×5=5×2.

3. Associative Property:

• Rule: The grouping of numbers does not affect the sum or product.

• Example: (1+2)+3=1+(2+3) and (2×3)×4=2×(3×4).

4. Identity Property:

• Rule: Adding 0 or multiplying by 1 leaves a number unchanged.

• Example: 5+0=5 and 7×1=7.

5. Inverse Property:

• Rule: Every real number has an opposite (additive inverse) and a reciprocal (multiplicative inverse).

• Example: 5+(−5)=0 and 3×31​=1.

6. Distributive Property:

• Rule: Multiplication distributes over addition or subtraction.

• Example: 2×(3+4)=(2×3)+(2×4).

7. Zero Property of Multiplication:

• Rule: Any number multiplied by 0 equals 0.

• Example: 6×0=0.
  

These rules guide how real numbers behave during arithmetic operations.

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Exercise:

Add the following real numbers:

Step-by-Step Solution

Step 1: Identify the types of numbers.

• 5 is a natural number, and it's a rational number (since it can be written as 5/1)
  
• ⅔ is a rational number (a fraction). 
  

√2 is an irrational number (its decimal expansion is non-repeating and non-terminating).

Step 2: Add the rational numbers

First, add the rational numbers 5 and ⅔. To do this, express 5 as 15/3 so the denominators match:

So, the sum of the rational numbers is 17/3.

Step 3: Add the irrational number.

Now, we add the irrational number √2 to the result from Step 2:

This is the simplest form of the answer. The sum is a real number, but it can't be simplified further.

Final Answer: