Fundamentals of Arithmetic

• Arithmetic is a branch of mathematics dealing with numbers and basic operations.
• It serves as the foundation for more advanced mathematical concepts.

Basic Operations

• Addition involves combining numbers to find their sum.
• Subtraction finds the difference between two numbers.
• Multiplication is repeated addition, resulting in a product.
• Division splits a number into equal parts, yielding a quotient.

Numbers

• Natural Numbers are positive integers starting from 1 (1, 2, 3,...).
• Whole Numbers include natural numbers and 0 (0, 1, 2, 3,...).
• Integers consist of whole numbers and their negative counterparts (..., -2, -1, 0, 1, 2,...).
• Rational Numbers can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational Numbers cannot be expressed as a simple fraction (e.g., √2, π).
• Real Numbers comprise all rational and irrational numbers.
• Complex Numbers are in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).

Properties of Operations

• Commutative Property:
  • Addition: a + b = b + a
  • Multiplication: a * b = b * a
• Associative Property:
  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a * b) * c = a * (b * c)
• Distributive Property: a * (b + c) = a * b + a * c
• Identity Property:
  • Addition: a + 0 = a (0 is the additive identity)
  • Multiplication: a * 1 = a (1 is the multiplicative identity)
• Inverse Property:
  • Addition: a + (-a) = 0 (-a is the additive inverse of a) -Multiplication: a * (1/a) = 1 (1/a is the multiplicative inverse or reciprocal of a, a ≠ 0)

Order of Operations

• Parentheses/Brackets: Operations inside parentheses or brackets are performed first.
• Exponents: Exponents (powers and roots) are calculated next.
• Multiplication and Division: Multiplication and division are performed from left to right.
• Addition and Subtraction: Addition and subtraction are performed from left to right.
• Acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) help to recall the order.

Fractions

• Numerator: Indicates how many parts of the whole are being considered.
• Denominator: Indicates the total number of equal parts of the whole.
• Proper Fraction: Numerator is less than the denominator.
• Improper Fraction: Numerator is greater than or equal to the denominator.
• Mixed Number: Consists of an integer and a proper fraction.
• Equivalent Fractions: Represent the same value with different numerators and denominators.
• Simplifying Fractions: Divide the numerator and denominator by their greatest common divisor (GCD).
• Adding/Subtracting Fractions: Requires a common denominator, use the least common multiple (LCM).
• Multiplying Fractions: Multiply the numerators and denominators.
• Dividing Fractions: Invert the divisor and multiply.

Decimals

• Decimal Point: Separates the whole number part from the fractional part.
• Place Value: Digit value depends on its position relative to the decimal point.
• Converting Fractions to Decimals: Divide the numerator by the denominator.
• Converting Decimals to Fractions: Use a denominator of 10, 100, 1000, etc., then simplify.
• Adding/Subtracting Decimals: Align the decimal points.
• Multiplying Decimals: Multiply as whole numbers, then count and apply decimal places.
• Dividing Decimals: Adjust the decimal point in the divisor and dividend accordingly.

Percentages

• Definition: Expressing a number as a fraction of 100.
• Converting Percentages to Decimals: Divide the percentage by 100.
• Converting Decimals to Percentages: Multiply the decimal by 100.
• Converting Percentages to Fractions: Write the percentage over 100, then simplify.
• Finding a Percentage of a Number: Multiply the number by the percentage in decimal or fraction form.
• Percentage Increase/Decrease: Calculate the difference, divide by the original value, and multiply by 100.

Ratios and Proportions

• Ratio: Compares two or more quantities.
• Proportion: States that two ratios are equal.
• Solving Proportions: Use cross-multiplication.
• Direct Proportion: As one quantity increases, the other increases proportionally.
• Inverse Proportion: As one quantity increases, the other decreases proportionally.

Powers and Roots

• Exponent: Indicates how many times a base number is multiplied by itself.
• Base: The number being raised to a power.
• Power: The result of raising a base to an exponent.
• Square Root: A number that, when multiplied by itself, equals a given number.
• Cube Root: A number that, when multiplied by itself twice, equals a given number.
• nth Root: A number that, when multiplied by itself n-1 times, equals a given number.

Laws of Exponents

- Product of Powers: a^m * a^n = a^(m+n)
- Quotient of Powers: a^m / a^n = a^(m-n)
- Power of a Power: (a^m)^n = a^(m*n)
- Power of a Product: (a * b)^n = a^n * b^n
- Power of a Quotient: (a / b)^n = a^n / b^n
- Zero Exponent: a^0 = 1 (a ≠ 0)
- Negative Exponent: a^(-n) = 1 / a^n

Measurement

• Units of Measurement: Standard quantities to express object dimensions.
• Length: Includes meters, feet, inches, kilometers, and miles.
• Weight/Mass: Includes grams, kilograms, pounds, and ounces.
• Volume: Includes liters, milliliters, gallons, quarts, and cubic meters.
• Time: Includes seconds, minutes, hours, days, and years.
• Converting Units: Use conversion factors.

Averages

• Mean: Sum of values divided by the number of values.
• Median: The middle value in an ordered set.
• Mode: The most frequent value.
• Weighted Average: Values are assigned weights reflecting their importance.