Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Number Systems ↓


Real Numbers


Real numbers are an essential part of mathematics and are taught at an early age. They serve as the foundation for understanding more complex mathematical concepts later on. To fully understand the concept of real numbers, one must study them in depth. It is beneficial to understand them, from defining them to visualizing and using them in different mathematical contexts.

Introduction to real numbers

Real numbers are a collection of numbers that includes both rational and irrational numbers. Together, they form a comprehensive set of values that represent all the possible sizes and scales of measurement we encounter in everyday life.

Real numbers can be plotted continuously on the number line, covering integers, fractions, and decimals. Here is a simple number line showing some real numbers:

-3 -2 -1 0 1 2 3 -----|----|----|----|----|----|----|
-3 -2 -1 0 1 2 3 -----|----|----|----|----|----|----|

In mathematics, real numbers are used to measure distances, quantities, and prices, making them applicable in a variety of real-world scenarios.

Types of real numbers

Real numbers can be rational or irrational. Let's look at these types more closely:

Rational numbers

A rational number is any number that can be expressed as a quotient or fraction a/b where a and b are integers and b ≠ 0 Rational numbers include integers, fractions, and finite or repeating decimals.

Examples of rational numbers include:

  • 3 (This can be written as 3/1)
  • -7 (can be written as -7/1)
  • 1/2
  • 4.5 (This can be written as 9/2)
  • 0.333... (can be written as 1/3)

Irrational numbers

Irrational numbers cannot be expressed as simple fractions. Irrational numbers have non-terminating, non-repeating decimal parts. They fill the gaps between rational numbers on the number line.

Examples of irrational numbers include:

  • π (pi), which is approximately 3.14159...
  • √2 (the square root of 2), which is approximately 1.41421...
  • e (Euler's number), which is approximately 2.71828...

... √2 ... π ... e ... -----|----|----|----|----|----|----|
... √2 ... π ... e ... -----|----|----|----|----|----|----|

Visual representation of real numbers

Real numbers can be represented on a number line, a visual tool that helps us understand the continuity and order of these numbers. You can imagine the number line as extending infinitely in both directions, which includes all rational and irrational numbers.

... -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 ... -----|----|----|----|----|----|----|----|----|----|----|----|----
... -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 ... -----|----|----|----|----|----|----|----|----|----|----|----|----

Properties of real numbers

Real numbers obey several fundamental arithmetic properties that make them predictable and manipulable in mathematical expressions and equations. Here are the main properties:

Closing assets

The set of real numbers is closed under addition, subtraction, multiplication and division (except division by zero). This means that the result of any operation between two real numbers is also a real number. For example:

  • Addition: 2 + 3 = 5
  • Subtraction: 5 - 3 = 2
  • Multiplication: 4 × 2 = 8
  • Division: 6 / 2 = 3

Commutative property

The commutative property states that the order in which you add or multiply two numbers does not change the result. For example:

  • Addition: a + b = b + a
  • Multiplication: a × b = b × a

Associative property

The associative property states that when adding or multiplying, the way the numbers are grouped does not affect the result. For example:

  • Sum: (a + b) + c = a + (b + c)
  • Multiplication: (a × b) × c = a × (b × c)

Distributive property

The distributive property involves both addition and multiplication. According to this property, multiplying a number by a sum is the same as doing each multiplication separately. For example:

a × (b + c) = a × b + a × c
a × (b + c) = a × b + a × c

Operations with real numbers

You can operate on real numbers using the basic arithmetic operations: addition, subtraction, multiplication, and division. Let's look at some examples to illustrate these operations with different types of real numbers:

Add

When adding real numbers, align the decimal points and add the numbers column by column from right to left. Remember to add like terms, such as whole numbers and decimal places.

12.35 + 3.62 -------- 15.97
12.35 + 3.62 -------- 15.97

Subtraction

Subtraction also involves lining up the decimal points. Borrow when necessary:

12.35 - 3.62 -------- 8.73
12.35 - 3.62 -------- 8.73

Multiplication

Multiply as if the numbers were whole numbers. Count the number of decimal places in both factors and place the decimal in the product accordingly.

1.2 × 3.4 ------ 48 (this is from 12×4) +36 (this is from 12×3.shifted one position left) ------ 4.08 (Total - Place the decimal point, 2 decimal places)
1.2 × 3.4 ------ 48 (this is from 12×4) +36 (this is from 12×3.shifted one position left) ------ 4.08 (Total - Place the decimal point, 2 decimal places)

Division

When dividing, move the decimal point in the divisor and increase the quotient to remove the decimal in the divisor, then proceed with the long division.

12.35 ÷ 3.62 = 3.41 (approximately)
12.35 ÷ 3.62 = 3.41 (approximately)

Real-world applications of real numbers

Real numbers exist everywhere in the real world - from natural phenomena to engineering, finance, and beyond. Here are some examples:

Science

In science, real numbers help measure quantities such as temperature, mass, and speed. For example, temperature can be measured like this:

37°C
37°C

Finance

Real numbers are important in accounting and finance to represent money and to manage investments, budgets, and expenditures:

$123.75
$123.75

Engineering

Engineers use real numbers in calculations to construct buildings, manufacture components, and design machines. Accuracy in these numbers is important for safety and efficiency.

Conclusion

In conclusion, real numbers are the cornerstone of many mathematical concepts and practical applications that we encounter every day. From understanding basic phenomena in science to controlling our financial endeavors, real numbers continue to play a vital role. This concept is called real numbers. Having a thorough understanding from the start will help us tackle more complex mathematical problems in higher education and professional life.


Grade 8 → 1.3


U
username
0%
completed in Grade 8

← Prev (1.2)
Irrational Numbers
Next (1.4) →
Properties of Real Numbers

Comments 



Real Numbers

https://www.buddymath.com/g8/1.3?bmal=en