💯Math for Non-Math Majors Unit 4 – Number Systems and Calculations

What's the Deal with Number Systems?

• Number systems provide a way to represent quantities and perform arithmetic operations
• The most commonly used number system is the decimal system (base-10) which uses digits 0-9
• Other number systems include binary (base-2), octal (base-8), and hexadecimal (base-16)
  • Binary uses only digits 0 and 1 and is the foundation of digital computing
  • Octal uses digits 0-7 and was used in early computing systems
  • Hexadecimal uses digits 0-9 and letters A-F and is used in computer programming and color codes
• Number systems have a base which determines the number of digits used and the place value of each digit
• Converting between number systems involves understanding the place value and using division and remainder operations
• Understanding number systems is essential for fields like computer science, cryptography, and digital electronics

Counting and Place Value: The Basics

• Counting is the process of assigning a number to each item in a set to determine the total quantity
• Place value is the value of a digit based on its position in a number
  • In the decimal system, each place represents a power of 10 (ones, tens, hundreds, etc.)
  • The value of a number is the sum of the products of each digit and its place value
• The ones place is the rightmost digit, and the value of each place increases by a factor of 10 moving left
• Zero is used as a placeholder to indicate the absence of a value in a particular place
• Counting and place value form the foundation for performing arithmetic operations and understanding numbers

Whole Numbers: Adding, Subtracting, and More

• Whole numbers are positive integers and zero (0, 1, 2, 3, ...)
• Addition is the process of combining two or more numbers to find the total sum
  • The commutative property of addition states that the order of the addends does not affect the sum ($a + b = b + a$)
  • The associative property of addition states that grouping the addends differently does not affect the sum ($(a + b) + c = a + (b + c)$)
• Subtraction is the process of finding the difference between two numbers
  • The minuend is the number being subtracted from, and the subtrahend is the number being subtracted
• Multiplication is the process of repeated addition of a number to itself a specified number of times
  • The multiplicand is the number being multiplied, and the multiplier is the number of times it is added
  • The commutative and associative properties also apply to multiplication
• Division is the process of splitting a number into equal parts or finding how many times one number goes into another
  • The dividend is the number being divided, and the divisor is the number dividing it
  • The quotient is the result of the division, and the remainder is the amount left over

Fractions and Decimals: Not as Scary as They Look

• Fractions represent parts of a whole or a ratio between two quantities
  • The numerator is the top number and represents the count of the parts
  • The denominator is the bottom number and represents the total number of equal parts
• Proper fractions have a numerator smaller than the denominator ($\frac{3}{4}$), while improper fractions have a numerator greater than or equal to the denominator ($\frac{5}{3}$)
• Mixed numbers consist of a whole number and a proper fraction ($2\frac{1}{3}$)
• Equivalent fractions have the same value but different numerators and denominators ($\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$)
• Decimals represent fractions with denominators that are powers of 10 (tenths, hundredths, thousandths, etc.)
  • The decimal point separates the whole number part from the fractional part
  • Each place to the right of the decimal point represents a power of 10, with the exponent decreasing by 1 for each place
• Converting between fractions and decimals involves division or identifying the place value

Percentages: Making Sense of the World

• Percentages represent a fraction with a denominator of 100 or a ratio out of 100
  • The symbol % means "out of 100" or "per 100"
  • To convert a percentage to a decimal, divide by 100 (move the decimal point two places to the left)
  • To convert a decimal to a percentage, multiply by 100 (move the decimal point two places to the right)
• Percentages are used to express parts of a whole, change over time, and comparisons between quantities
• The three main types of percentage problems are finding the percentage of a number, finding the whole given a part and its percentage, and finding what percentage one number is of another
• Solving percentage problems involves setting up proportions or equations and using multiplication and division
• Percentages are commonly used in finance (interest rates, discounts), statistics (polling, demographics), and many other fields

Real-World Applications: Where This Stuff Actually Matters

• Budgeting and financial planning involve using whole numbers, decimals, and percentages to track income, expenses, and savings goals
• Cooking and baking require measuring ingredients using fractions and converting between units (cups, tablespoons, etc.)
• Home improvement projects often involve calculating areas, volumes, and percentages for materials and costs
• Understanding statistics and data in the news, such as polling results and demographic trends, requires knowledge of percentages and fractions
• Calculating tips, discounts, and sales tax involves using percentages and decimals
• Time management and scheduling rely on understanding fractions and decimals (half an hour, quarter past, etc.)
• Sports statistics, such as batting averages and shooting percentages, are expressed as decimals or fractions

Common Mistakes and How to Avoid Them

• Forgetting to line up the decimal points when adding or subtracting decimals
  • Always align the decimal points vertically and add zeros as placeholders if needed
• Misplacing the decimal point when multiplying or dividing decimals
  • Count the total number of decimal places in the factors and place the decimal point in the product or quotient accordingly
• Confusing the numerator and denominator when working with fractions
  • Remember that the numerator is the count of the parts, and the denominator is the total number of parts
• Incorrectly converting between percentages, decimals, and fractions
  • Double-check the placement of the decimal point and the denominator when converting
• Mixing up the order of operations (PEMDAS) when evaluating expressions
  • Follow the order of Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Not checking the reasonableness of the answer in the context of the problem
  • Estimate the expected range of the answer and make sure the calculated result makes sense

Study Hacks and Practice Tips

• Create flashcards for key concepts, formulas, and definitions to review regularly
• Practice converting between fractions, decimals, and percentages to build fluency
• Use mnemonics or memory tricks to remember the order of operations (PEMDAS) or other important rules
• Break down complex problems into smaller steps and tackle them one at a time
• Work through plenty of practice problems, focusing on areas that are challenging or confusing
• Explain concepts and problem-solving strategies out loud to yourself or a study partner to reinforce understanding
• Use online resources, such as interactive tutorials and practice quizzes, to supplement your learning
• Collaborate with classmates to share ideas, strategies, and support each other's learning