Basic Math Concepts: Addition to Division

Basic Math Study Notes

Addition

• Definition: Combining two or more numbers to get a sum.
• Symbol: +
• Properties:
  • Commutative: a + b = b + a
  • Associative: (a + b) + c = a + (b + c)
  • Identity: a + 0 = a

Subtraction

• Definition: Finding the difference between two numbers.
• Symbol: −
• Properties:
  • Non-commutative: a - b ≠ b - a
  • a - 0 = a
  • a - a = 0
• Can be seen as "adding the opposite": a - b = a + (-b)

Multiplication

• Definition: Repeated addition of a number.
• Symbol: × or *
• Properties:
  • Commutative: a × b = b × a
  • Associative: (a × b) × c = a × (b × c)
  • Distributive: a × (b + c) = (a × b) + (a × c)
  • Identity: a × 1 = a

Division

• Definition: Splitting a number into equal parts or finding how many times one number is contained in another.
• Symbol: ÷ or /
• Properties:
  • Non-commutative: a ÷ b ≠ b ÷ a
  • Division by 1: a ÷ 1 = a
  • Division by itself: a ÷ a = 1 (where a ≠ 0)

Fractions

• Definition: A part of a whole expressed as a numerator over a denominator (e.g., ½).
• Types:
  • Proper: Numerator < Denominator (e.g., 3/4)
  • Improper: Numerator ≥ Denominator (e.g., 5/4)
  • Mixed: Combination of whole number and fraction (e.g., 1 ½)
• Operations:
  • Addition/Subtraction: Common denominators required.
  • Multiplication: Multiply numerators and denominators.
  • Division: Multiply by the reciprocal of the divisor.

Percentage

• Definition: A fraction out of 100, expressed with the symbol %.
• Conversion:
  • From fraction: (numerator/denominator) × 100%
  • From decimal: Decimal × 100%
• Applications:
  • Finding discounts, interest rates, and comparing ratios.
• Key calculations:
  • To find a percentage of a number: (Percentage/100) × Total
  • To increase/decrease a number by a percentage: Original Value ± (Original Value × Percentage/100)

Addition

• Combines two or more numbers to produce a total sum.
• Represented by the symbol "+".
• Commutative property allows rearranging terms without changing the result: a + b = b + a.
• Associative property enables grouping terms differently: (a + b) + c = a + (b + c).
• Identity property states that adding zero leaves the number unchanged: a + 0 = a.

Subtraction

• Calculates the difference between two numbers.
• Represented by the symbol "−".
• Non-commutative: a - b does not equal b - a.
• Subtracting zero does not change the number: a - 0 = a.
• Subtracting a number from itself results in zero: a - a = 0.
• Can be viewed as adding the opposite: a - b = a + (-b).

Multiplication

• Represents repeated addition of a number.
• Symbols used include "×" or "*".
• Commutative property allows terms to be swapped: a × b = b × a.
• Associative property permits different grouping of terms: (a × b) × c = a × (b × c).
• Distributive property combines multiplication with addition: a × (b + c) = (a × b) + (a × c).
• Identity property states multiplying by one leaves the number unchanged: a × 1 = a.

Division

• Splits a number into equal parts or determines how many times one number fits into another.
• Symbols used include "÷" or "/".
• Non-commutative: a ÷ b does not equal b ÷ a.
• Dividing by one keeps the number the same: a ÷ 1 = a.
• Dividing a number by itself results in one, with the exception that the number cannot be zero: a ÷ a = 1 (where a ≠ 0).

Fractions

• Represents a portion of a whole, written as a numerator over a denominator (e.g., ½).
• Types include:
  • Proper fractions, where the numerator is less than the denominator (e.g., 3/4).
  • Improper fractions, where the numerator is equal to or greater than the denominator (e.g., 5/4).
  • Mixed numbers, combining a whole number and a fraction (e.g., 1 ½).
• Operations involve:
  • Addition/Subtraction requiring common denominators.
  • Multiplication achieved by multiplying numerators and denominators.
  • Division involves multiplying by the reciprocal of the divisor.

Percentage

• Represents a fraction of 100, denoted by the symbol "%".
• Conversion methods include:
  • From fraction: (numerator/denominator) × 100%.
  • From decimal: Decimal × 100%.
• Common applications include calculating discounts, interest rates, and comparing ratios.
• Key calculations:
  • To find a percentage of a number: (Percentage/100) × Total.
  • To adjust a number by a percentage: Original Value ± (Original Value × Percentage/100).

This quiz covers fundamental math concepts including addition, subtraction, multiplication, and division. Each section includes definitions, symbols, and key properties of these operations. Test your understanding of these essential math skills.