What is a real number?

What are real numbers? What numbers are in the set of real numbers? Please read the article below to better understand this important mathematical knowledge.

Real number

• 1. What is a real number?
• 2. Real number axis
•  3. Compare real numbers
• 3. Properties of the set of real numbers
• 4. Absolute value of a real number
• 5. Example exercises on real numbers

1. What is a real number?

• Real numbers are the set of rational numbers and irrational numbers.
• Set is the symbol for the set of real numbers, consisting of real numbers.

• A rational number is a number written as a fraction (a, b ∈ Z, b ≠ 0). For example 
• The set of rational numbers is denoted by 
• An irrational number is an infinite, non-repeating decimal. For example:
• The set of irrational numbers is denoted by 

The set of real numbers covers the number line.

For example:

2. Real number axis

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

• Conversely, every point on the number line represents a real number.
• Only the set of real numbers fills the number line.

 3. Compare real numbers

Method

• With any two real numbers x, y, we always have x = y or x < y or x > y
• Real numbers greater than 0 are called positive real numbers, real numbers less than 1 are called negative real numbers. The number 0 is neither a positive nor a negative real number.
• Comparing positive real numbers is similar to comparing rational numbers.
• With a, b being two positive real numbers, if a > b then .

Example: Fill in the appropriate digit in the square:

Solution guide

a) -7.5(0)8 > -7.513

b) -3.02 <>

c) -0.4(9)854 <>

d) -1,(9)0765 <>

Example: Arrange real numbers: in order from smallest to largest

Solution guide

Arrange the real numbers in order from smallest to largest:

For example: Prove that:

With a, b are two positive real numbers if a > b then

Solution guide

If a > b then

a, b are two positive real numbers so a + b > 0

If a > b then a – b > 0

Consider the product

Because a2 – b2 > 0

=> a2 > b2 => dpcm

3. Properties of the set of real numbers

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

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

• For all properties:
• Add 0: 
• Commutative property: ;
• Combined properties: 
• Commutative property: a. b = b. a
• Associative property: (a. b). c = a. (b. c)
• Properties of multiplication by number 1:
• Distributive property of multiplication over addition: a. (b + c) = a. b + a. c
• For every real number a ≠ 0, there is an inverse such that

- That is, the above calculations also have the same commutative and associative properties as other sets of numbers. And the same is true for subtraction, multiplication, division...

Relationship between sets of numbers

For example: Perform the calculation:

Solution guide

For example: Find x, knowing:

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, the absolute value of a negative number is its opposite.

Overview:

• If
• If
• If
• If

Nature

• The absolute value of every number is non-negative.
• General: for all a ∈ R

Specifically:

Some properties

- Two numbers that are equal or opposite have equal absolute value, and vice versa, two numbers that have equal absolute value are 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.

Overview: and

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

Overview: If

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

Overview: If

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

Overview:

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

Overview:

5. Example exercises on real numbers

Example 1: Fill in the blanks with appropriate symbols ∈, ∉, ⊂ (…):

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,="" …="">

b) – 7.5 … 8 > – 7.513

c) – 0.4 … 854 < –="">

d) -1, … 0765 < –="">

Instruct

a) – 3.02 < –="">
b) – 7,508 > – 7,513 ;
c) – 0.49854 < –="" 0.49826="">
d) -1.90765 < –="">

Example 4: Find x, knowing:

3.2.x + (-1.2).x +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

In addition to real numbers, you can learn more about other definitions in mathematics such as square numbers , irrational numbers, rational numbers , prime numbers , natural numbers ...