Relearn Math as Adult A-Z(1): Numbers
Summary
This post provides a comprehensive review of numbers, covering different types, their properties, basic operations, and practical examples. We aim to strengthen your understanding of these foundational concepts, essential for tackling more advanced mathematical challenges.
Introduction: Understanding the Number System
Numbers are the fundamental units of mathematics, used to represent quantities and perform calculations. Mastering the basics of numbers is essential for building a solid mathematical foundation.
Types of Numbers: A Detailed Overview
Let's explore the different types of numbers that form the foundation of mathematics:
Natural Numbers (Counting Numbers)
Natural numbers are the numbers we use for counting.
Example
1, 2, 3, 4, 5, ...
Info
Natural numbers start at 1 and extend infinitely. They are positive and whole.
Summary
Natural numbers are positive whole numbers used for counting.
Whole Numbers
Whole numbers include all natural numbers and zero.
Example
0, 1, 2, 3, 4, 5, ...
Info
Whole numbers start at 0 and continue infinitely.
Summary
Whole numbers are natural numbers plus zero.
Integers
Integers encompass all whole numbers and their negative counterparts.
Example
..., -3, -2, -1, 0, 1, 2, 3, ...
Info
Integers include positive and negative whole numbers, and zero.
Summary
Integers are whole numbers and their negatives.
Rational Numbers
Rational numbers can be expressed as a fraction \(\frac{p}{q}\), where p and q are integers, and q is not equal to zero (\(q \neq 0\)).
Example
\(\frac{1}{2}\), \(-\frac{3}{4}\), 5, 0.75 (which can be written as \(\frac{3}{4}\))
Info
Rational numbers include fractions, terminating decimals, and repeating decimals.
Summary
Rational numbers can be written as a fraction of two integers.
Irrational Numbers
Irrational numbers cannot be expressed as a fraction \(\frac{p}{q}\). They have infinite, non-repeating decimal representations.
Example
\(\pi\) (pi), \(\sqrt{2}\) (square root of 2)
Info
Irrational numbers have infinite, non-repeating decimal expansions.
Summary
Irrational numbers cannot be expressed as a fraction of two integers.
Real Numbers
Real numbers encompass all rational and irrational numbers. They can be represented on a number line.
Example
All numbers mentioned above (natural, whole, integers, rational, irrational) are real numbers.
Info
Real numbers include all numbers that can be plotted on a number line.
Summary
Real numbers are the union of rational and irrational numbers.
Complex Numbers
Complex numbers are in the form \(a + bi\), where a and b are real numbers, and 'i' is the imaginary unit (\(\sqrt{-1}\)).
Example
\(2 + 3i\), \(-1 - i\), \(5i\)
Info
Complex numbers extend the real number system by including an imaginary component.
Summary
Complex numbers are in the form \(a + bi\), where 'i' is the imaginary unit.
Basic Operations: Reviewing the Fundamentals
A quick review of the four basic operations:
- Addition (+): Combining numbers.
- Subtraction (-): Finding the difference between numbers.
- Multiplication (× or *): Repeated addition.
- Division (÷ or /): Splitting a number into equal parts.
Example
- Addition: \(5 + 3 = 8\)
- Subtraction: \(10 - 4 = 6\)
- Multiplication: \(2 \times 6 = 12\)
- Division: \(15 \div 3 = 5\)
Info
Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is crucial for correct calculations.
Summary
Mastering basic operations is essential for all further mathematical studies.
Properties: Simplifying/Expanding Calculations
Understanding these properties can significantly simplify calculations and problem-solving.
- Commutative Property: The order of operands doesn't matter for addition and multiplication.
- Addition: \(a + b = b + a\)
- Multiplication: \(a \times b = b \times a\)
- Associative Property: The grouping of operands doesn't matter for addition and multiplication.
- Addition: \((a + b) + c = a + (b + c)\)
- Multiplication: \((a \times b) \times c = a \times (b \times c)\)
- Distributive Property: Multiplying a sum by a number is the same as multiplying each addend separately and then adding the products.
- \(a \times (b + c) = (a \times b) + (a \times c)\)
- Identity Property:
- Addition: \(a + 0 = a\) (0 is the additive identity)
- Multiplication: \(a \times 1 = a\) (1 is the multiplicative identity)
- Inverse Property:
- Addition: \(a + (-a) = 0\) (-a is the additive inverse of a)
- Multiplication: \(a \times (\frac{1}{a}) = 1\) (\(\frac{1}{a}\) is the multiplicative inverse of a, where \(a \neq 0\))
Practice Questions
Let's test your understanding with a few practice questions.
Question
Classify the following numbers: -5, \(\frac{2}{3}\), \(\sqrt{7}\), 0, 12
Answer
- -5: Integer, Rational, Real
- \(\frac{2}{3}\): Rational, Real
- \(\sqrt{7}\): Irrational, Real
- 0: Whole, Integer, Rational, Real
- 12: Natural, Whole, Integer, Rational, Real
Question
What is the additive inverse of 7?
Answer
-7
Question
What is the multiplicative inverse of \(\frac{1}{4}\)?
Answer
4
Question
Simplify the expression using the distributive property: \(3 \times (2 + 4)\)
Answer
\(3 \times (2 + 4) = (3 \times 2) + (3 \times 4) = 6 + 12 = 18\)
Question
Is the number 3.14159 rational or irrational? Explain your answer.
Answer
While 3.14159 is a good approximation of \(\pi\), it's a terminating decimal. If the number continues infinitely without repeating, then it is irrational. If this is just 3.14159 it is a rational number because it terminates. It can be written as 314159/100000
Conclusion: Building Your Mathematical Foundation
This post provides a foundational review of numbers, their types, properties, and basic operations. A solid understanding of these concepts is crucial for further mathematical studies. Continue practising and exploring to strengthen your mathematical skills! The next post will delve into fractions, decimals, and percentages.
Book Recommendation
A Mind For Numbers: How to Excel at Math and Science (Even If You Flunked Algebra) - Amazon
Bonus: Number Systems Beyond Decimal
While the decimal system (base-10) is most common, other number systems are important, especially in computer science.
- Decimal (Base-10): Uses digits 0-9. Each position represents a power of 10.
- Binary (Base-2): Uses digits 0 and 1. Each position represents a power of 2. Fundamental to computers.
- Octal (Base-8): Uses digits 0-7. Each position represents a power of 8.
- Hexadecimal (Base-16): Uses digits 0-9 and letters A-F (A=10, B=11, C=12, D=13, E=14, F=15). Each position represents a power of 16.
Here's a table summarizing the number systems:
| Number System | Base | Digits/Symbols | Use Cases |
|---|---|---|---|
| Decimal | 10 | 0-9 | Everyday calculations |
| Binary | 2 | 0, 1 | Computer science |
| Octal | 8 | 0-7 | Some computer applications |
| Hexadecimal | 16 | 0-9, A-F | Computer programming, color codes |