What are real numbers?

What are real numbers? What are real numbers? This article will provide you with everything you need to know about real numbers.

 

Real numbers are continuous quantities that can represent a distance on a straight line, because real numbers include both rational and irrational numbers. Rational numbers occupy points at a finite distance, and irrational numbers fill the spaces between them, forming a real straight line. In other words, real numbers are numbers that can be represented on a real straight line.

Real numbers include rational numbers, which are positive integers, negative integers, fractions, and irrational numbers. Basically, any number we can think of is a real number.

Examples of rational numbers are 2, 3.5, 6/7, √5, 0.35, -67,
π
, e, etc.

In this article, we will discuss real numbers in detail, including their properties, representation on the number line and decimal expansion, and check whether 0 is a real number.

1. What are real numbers?

1. Real numbers are the set of rational and irrational numbers.
2. A set   is the notation for the set of real numbers, which consists of real numbers.

1. A rational number is a number that can be written as a fraction  (a, b ∈ Z, b ≠ 0). For example. 
2. The set of rational numbers is denoted by 
3. Irrational numbers are infinite, non-repeating decimal numbers. For example:
4. The set of irrational numbers is denoted by 

The set of real numbers 

  covers the entire number line.

For example:

2. The real number line

 

Each real number is represented by a point on the number line.

1. Conversely, each point on the number line represents a real number.
2. Only the set of real numbers can fill the number line.

3. Comparing real numbers

Method

1. For any two real numbers x and y, we always have x = y, or x < y, or x > y.
2. Real numbers greater than 0 are called positive real numbers, and real numbers less than 1 are called negative real numbers. Zero is neither a positive nor a negative real number.
3. Comparing positive real numbers is done similarly to comparing rational numbers.
4. With a and b being two positive real numbers, if a > b then .

Example: Fill in the square with an appropriate digit:

a) -7.5(.)8 > -7.513	b) -3.02 < -3,(.)1
c) -0.4(.)854 < -0.49826	d) -1,(.)0765 < -1,892

Solution guide

a) -7.5(0)8 > -7.513

b) -3.02 < -3.(0)1

c) -0.4(9)854 < -0.49826

d) -1,(9)0765 < -1,892

Example: Sort the real numbers:

in ascending order

Solution guide

Arrange the real numbers in ascending order:

Example: Prove that:

For a and b being two positive real numbers, if a > b then

Solution guide

If a > b then

Since a and b are two positive real numbers, a + b > 0.

If a > b then a – b > 0

Analysis

Therefore

,

a² ^– b² ^> 0

=> a ² > b ² => QED

3. Properties of the set of real numbers

In the set of rational numbers

, we also define the operations of addition, subtraction, multiplication, division, exponentiation, roots, etc. And in these operations, real numbers also have the same properties as the operations in the set of rational numbers.

In the set of real numbers, the operations with respect to multiplication have the following properties:

1. For all attributes :
2. Adding zero: 
3. Commutative property:  ;
4. Associative properties: 
5. Commutative property: a. b = b. a
6. Associative property: (a. b). c = a. (b. c)
7. Properties of multiplication by 1:
8. Distributive property of multiplication over addition: a. (b + c) = a. b + a. c
9. For every real number a ≠ 0, there exists an inverse such that

 

- This means that the above operations

also have the commutative and associative properties, just like other sets of numbers. And the same applies to subtraction, multiplication, division, etc.

The relationship between sets of numbers

Example: Perform the calculation:

Solution guide

Example: Find x, knowing that:

Solution guide

4. Absolute value of a real number

Definition: The distance from point a to point 0 on the number line is the absolute value of a number a (a is a real number). The absolute value of a non-negative number is itself, and the absolute value of a negative number is its opposite.

Overview:

1. If
2. If
3. If
4. If

Nature

1. The absolute value of every number is non-negative.
2. In general: for every a ∈ R

Specifically:

1. 
2. 

Some properties

- If two numbers are equal or opposite, then they have the same absolute value, and conversely, if two numbers have the same absolute value, then they are either equal or opposite.

Overview:

- Every number is greater than or equal to the opposite of its absolute value and at the same time less than or equal to its absolute value.

In general:

and

- Of two negative numbers, the smaller one has a larger absolute value.

In general: If

- Of two positive numbers, the smaller one has a smaller absolute value.

In general: If

- The absolute value of a product is equal to the product of its absolute values.

Overview:

- The absolute value of a quotient is equal to the quotient of the two absolute values.

Overview:

5. Example exercises on real numbers

Example 1: Fill in the appropriate sign ∈, ∉, or ⊂ in the blank (…):

3…. Q ; 3…. R ; 3… I ; -2.53…Q;

0.2(35) …. I ; N…. Z ; I …. R.

Instruct

a) 3 ∈ Q ; 3 ∈ R ; 3 ∉ I ; -2.53 ∈ Q

b) 0.2(35) ∉ I ; N ∈ Z ; I ⊂ R

Example 2: Find the sets

a) Q ∩ I ;
b) R ∩ I.

Instruct

a) Q ∩ I = Ø ;
b) R ∩ I = I.

Example 3: Fill in the appropriate digit in (…) 

a) – 3.02 < – 3, … 1

b) – 7.5 … 8 > – 7.513

c) – 0.4 … 854 < – 0.49826

d) -1, … 0765 < – 1,892

Instruct

a) – 3.02 < – 301
b) – 7,508 > – 7,513 ;
c) – 0.49854 < – 0.49826 ;
d) -1.90765 < – 1.892.

Example 4: Find x, knowing that:

3.2x + (-1.2x) + 2.7 = -4.9;

Instruct

3.2. x + (-1,2).x + 2,7 = -4,9

[3,2 + (-1,2)].x + 2,7 = -4,9.

2.x + 2.7 = – 4.9.

2.x = – 4.9 – 2.7

2.x = – 7.6

x = -7.6 : 2

x = -3.8

Besides real numbers, you can learn more about other definitions in mathematics such as perfect squares , irrational numbers, rational numbers , prime numbers , natural numbers , etc.