Real Numbers¶

1. Introduction to Real Numbers¶

Real numbers are the foundation of most of the mathematics we deal with in daily life. They encompass all the numbers that can be represented on a number line, including whole numbers, fractions, decimals, and irrational numbers. In fact, real numbers are divided into various subsets, each with its own properties.

The set of real numbers is denoted by \( \mathbb{R} \).

2. Subsets of Real Numbers¶

Real numbers can be classified into several important subsets:

2.1 Natural Numbers (Counting Numbers):¶

The set of natural numbers is:

\[ \mathbb{N} = \{ 1, 2, 3, 4, 5, \dots \} \]

These are the numbers we use for counting. Sometimes, 0 is included, depending on context.

2.2 Whole Numbers:¶

Whole numbers include all natural numbers plus 0:

\[ \mathbb{W} = \{ 0, 1, 2, 3, 4, 5, \dots \} \]

2.3 Integers:¶

Integers extend whole numbers to include negative numbers:

\[ \mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \} \]

They do not include fractions or decimals.

2.4 Rational Numbers:¶

A rational number is any number that can be expressed as the quotient of two integers, \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \). Rational numbers include terminating and repeating decimals.

For example:

\[ \frac{1}{2} = 0.5, \quad \frac{1}{3} = 0.\overline{3} \quad (\text{repeating decimal}) \]

The set of rational numbers is denoted by \( \mathbb{Q} \).

2.5 Irrational Numbers:¶

Irrational numbers cannot be expressed as a fraction of two integers. Their decimal representation is non-terminating and non-repeating.

Examples include:

\[ \sqrt{2}, \pi, e \]

These numbers extend the real number system beyond rationals, and the set of irrational numbers is denoted by \( \mathbb{I} \).

3. Properties of Real Numbers¶

The set of real numbers \( \mathbb{R} \) has several important properties:

3.1 Closure:¶

Real numbers are closed under addition, subtraction, multiplication, and division (except by zero). This means that if you perform any of these operations on two real numbers, the result is still a real number.

3.2 Commutativity:¶

For real numbers \( a \) and \( b \):

\( a + b = b + a \) (commutative property of addition)
\( a \cdot b = b \cdot a \) (commutative property of multiplication)

3.3 Associativity:¶

For real numbers \( a \), \( b \), and \( c \):

\( (a + b) + c = a + (b + c) \) (associative property of addition)
\( (a \cdot b) \cdot c = a \cdot (b \cdot c) \) (associative property of multiplication)

3.4 Distributivity:¶

Multiplication distributes over addition for real numbers. For example:

\[ a(b + c) = ab + ac \]

3.5 Identity Elements:¶

The identity element for addition is 0, because \( a + 0 = a \) for any real number \( a \). The identity element for multiplication is 1, since \( a \cdot 1 = a \).

3.6 Inverses:¶

Every real number \( a \) has an additive inverse \( -a \) such that:

\[ a + (-a) = 0 \]

Every non-zero real number \( a \) has a multiplicative inverse \( \frac{1}{a} \) such that:

\[ a \cdot \frac{1}{a} = 1 \]

4. The Number Line¶

Real numbers can be represented on a number line, which extends infinitely in both directions. Key points on the number line include:

\( 0 \), which is the origin and separates positive and negative numbers.
Positive numbers are to the right of 0, and negative numbers are to the left.
Any real number corresponds to a unique point on the number line.

Density of Real Numbers:¶

The real number line is continuous, meaning there is no "gap" between numbers. Between any two real numbers, no matter how close, there is always another real number. For example, between \( 1 \) and \( 2 \), you can find numbers like \( 1.5 \), \( 1.25 \), and \( 1.125 \).

5. Absolute Value¶

The absolute value of a real number \( a \), denoted by \( |a| \), is its distance from 0 on the number line, regardless of direction. The absolute value is always non-negative.

For example:

\[ |3| = 3, \quad |-5| = 5 \]

If \( a \geq 0 \), then \( |a| = a \). If \( a < 0 \), then \( |a| = -a \).

6. Operations with Real Numbers¶

Addition:¶

Adding two real numbers can be visualized on the number line. For example, \( 3 + (-2) \) means starting at 3 and moving 2 units to the left, resulting in 1.

Subtraction:¶

Subtracting a real number can be thought of as adding its negative. For instance, \( 5 - 3 = 5 + (-3) = 2 \).

Multiplication:¶

Multiplication can be seen as repeated addition. For example, \( 4 \times 3 \) means adding 4 to itself 3 times. If one of the numbers is negative, the result is negative. For instance:

\[ (-3) \times 4 = -12 \]

Division:¶

Division is the inverse of multiplication. Dividing a number by another can be thought of as finding how many times the divisor fits into the dividend. For example:

\[ \frac{10}{2} = 5 \]

7. Square Roots and Real Numbers¶

For any non-negative real number \( a \), the square root of \( a \), denoted \( \sqrt{a} \), is the number \( b \) such that:

\[ b^2 = a \]

For example:

\[ \sqrt{9} = 3 \quad \text{because} \quad 3^2 = 9 \]

However, square roots of negative numbers are not real numbers; they belong to the set of complex numbers.

8. Irrational Numbers¶

Irrational numbers cannot be expressed as fractions of integers. Famous examples include:

\( \pi \approx 3.14159 \), which is the ratio of the circumference of a circle to its diameter.
\( \sqrt{2} \), which is the length of the diagonal of a square with side length 1.

Irrational numbers have non-repeating, non-terminating decimal expansions. For example:

\[ \pi = 3.1415926535\ldots \]

9. Applications of Real Numbers¶

Real numbers are used in a wide variety of real-world applications:

Measurements: Distances, weights, and time intervals are represented by real numbers.
Finance: Money, interest rates, and other financial quantities are modeled using real numbers.
Science: Quantities like temperature, speed, and force are described using real numbers.
Geometry: Real numbers describe lengths, areas, and volumes in geometry.

Summary¶

Real numbers form the basis of much of arithmetic and algebra. They can be classified into subsets like natural numbers, integers, rational numbers, and irrational numbers. Their properties, including closure, commutativity, associativity, and the existence of inverses, make them fundamental to solving a wide range of problems in mathematics and science.

Here are 12 exercises to test knowledge of real numbers, focusing on algebraic concepts.

Exercises: Basic Arithmetic and Properties of Real Numbers¶

These exercises range from basic arithmetic operations and properties of real numbers to more advanced topics like solving equations and working with square roots and powers. Each problem reinforces core algebraic skills while focusing knowledge o real numbers.

Addition and Subtraction of Real Numbers: Simplify the expression:

\[ 7.5 - 3.2 + 8.6 - 1.4 \]

Show all steps.

Multiplication and Division of Real Numbers:

Evaluate:

\[ \frac{6.4 \times 2.5}{5} - 3 \]

Absolute Value:

Simplify the following:

\[ | -5 | + | 3.7 | - | -2.4 | \]

Commutative and Associative Properties:

Using the commutative and associative properties, rewrite and simplify:

\[ 4 + (7 + 2) + 3 \]

Solving Equations Involving Real Numbers¶

Solving Linear Equations:

Solve for \( x \):

\[ 3x - 5 = 2x + 7 \]

Solving Equations with Absolute Values:

Solve for \( x \):

\[ | 2x - 3 | = 7 \]

Give all possible solutions.

Operations with Fractions and Decimals¶

Addition and Subtraction of Fractions:

Simplify:

\[ \frac{5}{6} + \frac{7}{12} - \frac{2}{3} \]

Multiplying and Dividing Fractions:

Simplify the expression:

\[ \frac{3}{8} \times \frac{4}{5} \div \frac{2}{7} \]

Operations with Decimals:

Evaluate: [ 0.75 \times 2.6 - 1.3^2 ]

Square Roots and Powers¶

Simplifying Square Roots:

Simplify:

\[ \sqrt{144} - \sqrt{81} + \sqrt{25} \]

Operations with Powers:

Simplify the expression:

\[ 3^2 - 4^3 + 2^4 \]

Solving Quadratic Equations¶

Quadratic Equation (Factoring):

Solve for \( x \) by factoring:

\[ x^2 - 5x + 6 = 0 \]