Understanding Arithmetic: Basic Operations and Numbers

Arithmetic is a branch of mathematics focused on numbers and their basic operations.

• Arithmetic provides a foundation for advanced mathematics.
• Key operations include addition, subtraction, multiplication and division.

Basic Operations

• Addition combines numbers to find a total, using symbol "+".
• Subtraction finds the difference between numbers, using symbol "-".
• Multiplication is repeated addition, using symbol "×" or "*".
• Division splits a number into equal parts, using symbol "÷" or "/".

Numbers

• Natural numbers: positive integers starting from 1 (1, 2, 3,...).
• Whole numbers: natural numbers including zero (0, 1, 2, 3,...).
• Integers: positive and negative whole numbers (... -2, -1, 0, 1, 2,...).
• Rational numbers: expressible as a fraction p/q, where p and q are integers and q ≠ 0.
• Irrational numbers: cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions (e.g., √2, π).
• Real numbers: include all rational and irrational numbers.

Properties of Operations

• Commutative property: order doesn't affect result in addition/multiplication (a + b = b + a, a × b = b × a).
• Associative property: grouping doesn't affect result in addition/multiplication ((a + b) + c = a + (b + c), (a × b) × c = a × (b × c)).
• Distributive property: multiplication distributes over addition (a × (b + c) = a × b + a × c).
• Identity property: adding 0 doesn't change a number (a + 0 = a), multiplying by 1 doesn't change a number (a × 1 = a).
• Inverse property: For any number a, a + (-a) = 0 (additive inverse); for any non-zero a, a × (1/a) = 1 (multiplicative inverse).

Order of Operations

• Operations should be performed in a specific sequence.
• PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Fractions

• Represents a part of a whole, written as p/q (numerator/denominator).
• Equivalent fractions: same value, different numerator and denominator (e.g., 1/2 = 2/4 = 3/6).
• Add/subtract fractions: must have a common denominator.
• Multiply fractions: multiply numerators and denominators separately (a/b × c/d = (a × c) / (b × d)).
• Divide fractions: multiply by the reciprocal of the divisor (a/b ÷ c/d = a/b × d/c = (a × d) / (b × c)).

Decimals

• Represent fractions with a base of 10.
• Digits to the right of the decimal point represent fractions with a denominator that is a power of 10 (e.g., 0.1 = 1/10, 0.01 = 1/100).
• Add/subtract decimals: align decimal points.
• Multiply decimals: multiply as whole numbers, then place the decimal point (number of decimal places equals the sum of the decimal places in the factors).
• Divide decimals: make the divisor a whole number (multiply both divisor and dividend by the same power of 10).

Percentages

• Way of expressing a number as a fraction of 100, using symbol "%".
• Convert fraction to percentage: multiply by 100 (e.g., 1/2 = 0.5 × 100 = 50%).
• Convert percentage to fraction: divide by 100 (e.g., 75% = 75/100 = 3/4).
• Find a percentage of a number: multiply the number by the percentage expressed as a decimal (e.g., 20% of 50 = 0.20 × 50 = 10).

Ratios and Proportions

• Ratio: a comparison of two or more quantities, written as a:b or a/b.
• Proportion: equation stating two ratios are equal (a/b = c/d).
• In a proportion, cross products are equal (a × d = b × c).

Powers and Roots

• Power: represents repeated multiplication of a number by itself, written as a^n (a is the base, n is the exponent).
• Root: the inverse operation of a power.
• The nth root of a number a is written as n√a.
• Square root (√a): the root when n=2.

Divisibility Rules

• Divisibility rules: shortcuts to determine if a number is divisible by another without division.
• Divisible by 2: last digit is even.
• Divisible by 3: sum of digits is divisible by 3.
• Divisible by 4: number formed by last two digits is divisible by 4.
• Divisible by 5: last digit is 0 or 5.
• Divisible by 6: divisible by both 2 and 3.
• Divisible by 9: sum of digits is divisible by 9.
• Divisible by 10: last digit is 0.

Prime Numbers and Composite Numbers

• Prime number: natural number greater than 1 with only two positive divisors: 1 and itself (e.g., 2, 3, 5, 7, 11).
• Composite number: natural number greater than 1 with more than two positive divisors (e.g., 4, 6, 8, 9, 10).
• The number 1 is neither prime nor composite.

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

• GCD: largest number that divides evenly into all numbers.
• LCM: smallest number that is a multiple of all numbers.

Absolute Value

• Distance from zero on the number line, written as |a|.
• Absolute value of a positive number: the number itself (|5| = 5).
• Absolute value of a negative number: the opposite of the number (|-5| = 5).
• Absolute value of zero: zero (|0| = 0).