💯Math for Non-Math Majors Unit 3 – Number Systems and Theory

What's This Unit All About?

• Number systems and theory explores the fundamental properties and relationships of numbers
• Covers various types of number systems (natural numbers, integers, rational numbers, real numbers, complex numbers)
• Investigates the rules and patterns that govern arithmetic operations within each number system
• Introduces the concept of base or radix, which determines the number of unique digits used in a number system
  • Most commonly used base is base-10 or decimal system (digits 0-9)
  • Other bases include binary (base-2), octal (base-8), and hexadecimal (base-16)
• Examines the properties of numbers, such as even and odd, prime and composite, and divisibility rules
• Delves into the relationships between numbers, including factors, multiples, and prime factorization
• Explores the idea of infinity and the different types of infinity (countable and uncountable)

Key Concepts to Know

• Natural numbers: positive integers starting from 1 (1, 2, 3, ...)
• Whole numbers: non-negative integers starting from 0 (0, 1, 2, 3, ...)
• Integers: positive and negative whole numbers, including 0 (..., -3, -2, -1, 0, 1, 2, 3, ...)
• Rational numbers: numbers that can be expressed as a ratio of two integers (fractions, decimals that terminate or repeat)
• Irrational numbers: numbers that cannot be expressed as a ratio of two integers (non-repeating, non-terminating decimals like $\sqrt{2}$ and $\pi$)
• Real numbers: the union of rational and irrational numbers, representing all points on a number line
• Complex numbers: numbers in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$)
  • Real part: the $a$ value in a complex number
  • Imaginary part: the $b$ value in a complex number

Different Number Systems Explained

• Decimal (base-10): the most common number system, using digits 0-9
  • Each place value represents a power of 10 (ones, tens, hundreds, etc.)
• Binary (base-2): a number system using only digits 0 and 1
  • Each place value represents a power of 2 (1, 2, 4, 8, 16, etc.)
  • Used extensively in computer science and digital electronics
• Octal (base-8): a number system using digits 0-7
  • Each place value represents a power of 8 (1, 8, 64, 512, etc.)
  • Often used in computing as a more compact representation than binary
• Hexadecimal (base-16): a number system using digits 0-9 and letters A-F (representing values 10-15)
  • Each place value represents a power of 16 (1, 16, 256, 4096, etc.)
  • Commonly used in computer programming and color representation (RGB color codes)
• Converting between number systems involves understanding the place values and applying arithmetic operations

Real-World Applications

• Computer science and digital electronics heavily rely on binary and hexadecimal number systems
  • Binary is used to represent data and instructions in computers (machine code)
  • Hexadecimal is often used to represent memory addresses and color codes (web design, graphic design)
• Cryptography and coding theory utilize number systems and theory to develop secure communication methods
  • Prime numbers and their properties are essential in creating encryption algorithms (RSA encryption)
• Music theory and tuning systems are based on ratios and intervals, which are related to rational numbers
  • The equal temperament tuning system divides an octave into 12 equal intervals (semitones)
• Fractals and chaos theory, which have applications in various fields (art, biology, physics), are built upon number systems and theory
  • The Mandelbrot set, a famous fractal, is generated using complex numbers and iterative functions

Common Pitfalls and How to Avoid Them

• Confusing the properties and rules of different number systems
  • Ensure you understand the specific characteristics of each number system (natural, whole, integer, rational, irrational, real, complex)
• Misapplying arithmetic operations when working with different number systems
  • Be aware of the rules for addition, subtraction, multiplication, and division within each number system
• Incorrectly converting between number systems (decimal, binary, octal, hexadecimal)
  • Double-check your place values and calculations when converting
  • Use online converters or calculators to verify your results
• Forgetting to consider the properties of numbers in problem-solving
  • Keep in mind concepts like even/odd, prime/composite, and divisibility rules when applicable
• Overlooking the importance of precision and rounding in real-world applications
  • Be mindful of the required level of accuracy in your calculations and results

Formulas and Tricks to Remember

• Divisibility rules for numbers (divisible by 2 if even, by 3 if digit sum is divisible by 3, by 5 if ends in 0 or 5, etc.)
• Prime factorization: expressing a number as a product of prime numbers ($24 = 2^3 \times 3$)
• Converting between fractions and decimals: divide the numerator by the denominator to obtain the decimal representation
• Binary to decimal conversion: multiply each binary digit by its place value (power of 2) and sum the results
  • Example: $1011_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 8 + 0 + 2 + 1 = 11_{10}$
• Hexadecimal to decimal conversion: multiply each hexadecimal digit by its place value (power of 16) and sum the results
  • Example: $A3F_{16} = (10 \times 16^2) + (3 \times 16^1) + (15 \times 16^0) = 2560 + 48 + 15 = 2623_{10}$

Practice Problems and Solutions

1. Convert the binary number $101101_2$ to decimal. Solution: $101101_2 = (1 \times 2^5) + (0 \times 2^4) + (1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 32 + 0 + 8 + 4 + 0 + 1 = 45_{10}$

2. Express the decimal number $60$ in hexadecimal. Solution: $60 \div 16 = 3$ remainder $12$ (C in hexadecimal), so $60_{10} = 3C_{16}$

3. Find the prime factorization of $180$. Solution: $180 = 2^2 \times 3^2 \times 5$

4. Determine the sum of the first 10 positive odd integers. Solution: The sum of the first $n$ odd integers is $n^2$, so the sum of the first 10 odd integers is $10^2 = 100$

5. Identify the next prime number after $47$. Solution: The next prime number after $47$ is $53$

How This Connects to Other Math Topics

• Number systems and theory form the foundation for algebra, as variables and equations rely on the properties of numbers
• Geometry and trigonometry utilize real numbers and their properties in calculations and proofs
  • The Pythagorean theorem ($a^2 + b^2 = c^2$) relates the lengths of the sides of a right triangle
  • Trigonometric functions (sine, cosine, tangent) are defined using ratios of side lengths in a right triangle
• Calculus builds upon the real number system to study rates of change and accumulation
  • Limits, derivatives, and integrals are defined using the properties of real numbers
• Probability and statistics rely on the understanding of rational numbers and their properties
  • Probability is expressed as a ratio of favorable outcomes to total outcomes (a rational number between 0 and 1)
• Number theory itself is a branch of mathematics that delves deeper into the properties and relationships of integers
  • It has applications in cryptography, coding theory, and computer science