Unit 1 Algebra Basics Homework 1 The Real Numbers

Unit 1: Algebra Basics – Homework 1: The Real Numbers

This comprehensive guide delves into the world of real numbers, providing a thorough understanding of their properties and applications, crucial for success in algebra and beyond. We'll explore various subsets of real numbers, their representations, and essential operations, equipping you with the knowledge to tackle your homework and excel in your studies. This guide is designed to be more than just an answer key; it aims to foster a deep understanding of the concepts involved.

Understanding the Real Number System

The real number system is the foundation upon which much of mathematics is built. It encompasses all numbers that can be plotted on a number line, extending infinitely in both positive and negative directions. This includes several key subsets:

1. Natural Numbers (Counting Numbers):

These are the numbers we use to count: 1, 2, 3, 4, and so on. They are also known as positive integers. Symbol: ℕ

Example: The number of apples in a basket.

2. Whole Numbers:

This set includes all natural numbers and zero: 0, 1, 2, 3, 4… Symbol: ℤ₀ or 𝕎

Example: The number of cars in a parking lot (it could be zero).

3. Integers:

Integers include all whole numbers and their negative counterparts: …, -3, -2, -1, 0, 1, 2, 3, … Symbol: ℤ

Example: The temperature in degrees Celsius.

4. Rational Numbers:

Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers, and q is not zero. These numbers can be expressed as terminating or repeating decimals. Symbol: ℚ

Examples: 1/2, -3/4, 0.75 (which is 3/4), 0.333… (which is 1/3).

5. Irrational Numbers:

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

Examples: π (pi), √2, e (Euler's number).

6. Real Numbers:

The real number system encompasses all rational and irrational numbers. It represents all points on the number line. Symbol: ℝ

Example: Any measurement – length, weight, temperature, etc.

Representing Real Numbers

Real numbers can be represented in several ways:

• Decimal Form: This is the most common representation, using a decimal point to separate the whole number part from the fractional part.

• Fraction Form: Useful for representing rational numbers and simplifying calculations.

• Scientific Notation: A convenient way to represent very large or very small numbers using powers of 10. For example, 6.022 x 10²³ (Avogadro's number).

• Set Notation: Used to define and group numbers with shared properties. For example, {x | x ∈ ℝ and x > 5} represents the set of all real numbers greater than 5.

Operations on Real Numbers

Understanding how to perform basic operations on real numbers is fundamental to algebra. These include:

1. Addition (+):

Combining two or more numbers to find their sum. The order of addition doesn't matter (commutative property).

Example: 5 + 3 = 8; -2 + 7 = 5

2. Subtraction (-):

Finding the difference between two numbers. Subtraction is the inverse operation of addition.

Example: 10 - 4 = 6; -1 - 5 = -6

3. Multiplication (× or ·):

Repeated addition. The order of multiplication doesn't matter (commutative property).

Example: 6 × 4 = 24; -3 × 2 = -6

4. Division (÷ or /):

The inverse operation of multiplication. Division by zero is undefined.

Example: 12 ÷ 3 = 4; -8 ÷ 2 = -4

Properties of Real Numbers

Several important properties govern how real numbers behave under these operations:

• Commutative Property: The order of numbers doesn't affect the result for addition and multiplication. a + b = b + a; a × b = b × a

• Associative Property: The grouping of numbers doesn't affect the result for addition and multiplication. (a + b) + c = a + (b + c); (a × b) × c = a × (b × c)

• Distributive Property: Multiplication distributes over addition. a × (b + c) = (a × b) + (a × c)

• Identity Property: Adding zero to a number doesn't change it (additive identity). Multiplying a number by one doesn't change it (multiplicative identity). a + 0 = a; a × 1 = a

• Inverse Property: Every number has an additive inverse (opposite) and a multiplicative inverse (reciprocal), except for zero which has no multiplicative inverse. a + (-a) = 0; a × (1/a) = 1 (for a ≠ 0)

Order of Operations (PEMDAS/BODMAS)

When dealing with expressions involving multiple operations, it's crucial to follow the order of operations:

• Parentheses/ Brackets
• Exponents/ Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Absolute Value

The absolute value of a number is its distance from zero on the number line. It's always non-negative. Represented by | |.

Example: |5| = 5; |-5| = 5

Solving Equations and Inequalities

A crucial application of real numbers is solving equations and inequalities. An equation states that two expressions are equal. An inequality compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

Example Equation: 2x + 5 = 11 (Solution: x = 3)

Example Inequality: 3x - 2 > 7 (Solution: x > 3)

Real-World Applications

Real numbers are fundamental to countless real-world applications:

• Engineering: Designing structures, calculating forces, and modeling systems.

• Finance: Managing money, calculating interest, and analyzing investments.

• Physics: Measuring quantities like velocity, acceleration, and energy.

• Computer Science: Representing data, performing calculations, and creating algorithms.

• Medicine: Measuring dosages, analyzing test results, and modeling biological processes.

Practice Problems

To solidify your understanding, try solving these problems:

1. Classify the following numbers as natural, whole, integer, rational, irrational, or real: -5, 0, 1/3, √7, 2.5, π.

2. Perform the following operations: -8 + 12, 15 - (-3), (-4) × 6, 24 ÷ (-6).

3. Simplify the expression: 3(2x + 4) - 5x.

4. Solve the equation: 4x - 7 = 9.

5. Solve the inequality: -2x + 1 ≤ 5.

This comprehensive guide provides a strong foundation in understanding real numbers. Remember to practice regularly to build proficiency and confidence in tackling more complex algebraic problems. Consistent practice and a thorough grasp of these fundamental concepts will set you up for success in your algebra studies and beyond. Remember to consult your textbook and teacher for further assistance and clarification. Good luck!