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Basic Arithmetic

Numbers, operations, and properties — the foundation of all mathematics.

Number Systems and Their Symbols

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Mathematics organizes numbers into nested sets, each with its own special symbol. The natural numbers, denoted by a bold N, are the counting numbers starting from 1. The integers, denoted by a bold Z (from the German word Zahlen, meaning numbers), add zero and negative values. The rational numbers, written with a bold Q (for quotient), are numbers that can be expressed as a fraction of two integers. The real numbers, written with a bold R, fill the entire number line including irrational values like pi and the square root of two. Finally, the complex numbers, written with a bold C, extend the real numbers by including the imaginary unit i, where i squared equals negative one. The subset symbol (a rounded less-than sign) means every member of one set is also a member of another.
The symbol ⊂ means "is a subset of". N = natural numbers, Z = integers (Zahlen), Q = rationals (quotient), R = reals, C = complex numbers.

The Order of Operations (PEMDAS)

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When an expression contains multiple operations, mathematicians follow a strict convention called PEMDAS to avoid ambiguity. First, evaluate everything inside Parentheses. Second, compute Exponents, including powers and square roots. Third, perform Multiplication and Division working from left to right — these two have equal priority and are processed in the order they appear. Fourth, perform Addition and Subtraction, also from left to right. Without PEMDAS, the same mathematical expression would give different answers depending on who evaluated it. The popular mnemonic is "Please Excuse My Dear Aunt Sally."
PEMDAS demonstration: first exponent (2²=4), then multiplication (4×4=16), finally addition (3+16=19).

The Four Fundamental Properties of Arithmetic

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Arithmetic operations obey four axiomatic properties that govern how numbers behave. The commutative property states that the order of operands does not change the result for addition and multiplication — but subtraction and division are NOT commutative. The associative property says that grouping with parentheses does not matter. The identity property identifies zero as the additive identity (adding zero changes nothing) and one as the multiplicative identity (multiplying by one changes nothing). The distributive property connects multiplication with addition and is the single most important property for algebra. Every algebraic manipulation you will ever learn ultimately rests on the distributive property.
The distributive property. Multiply the term outside by each term inside the parentheses, then add the results.

The Four Properties in Detail

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  • Commutative: for addition, a plus b equals b plus a. For multiplication, a times b equals b times a. Subtraction and division are NOT commutative.
  • Associative: for addition, adding a to (b plus c) equals (a plus b) plus c. Same for multiplication. Grouping with parentheses does not change the result.
  • Distributive: a times (b plus c) equals a times b plus a times c. This property allows factoring, expanding, and all algebraic manipulation.
  • Identity: zero is the additive identity (a plus zero equals a). One is the multiplicative identity (a times one equals a). No other numbers have this effect.

Factors, Multiples, and Prime Numbers

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A factor of a number divides it evenly, leaving no remainder. For example, the factors of twelve are one, two, three, four, six, and twelve. A multiple is the product of a number and any integer. The prime numbers are the building blocks of all integers: each has exactly two factors, itself and one. The numbers two, three, five, seven, eleven, and thirteen are the first six primes. The Fundamental Theorem of Arithmetic states that every integer greater than one can be expressed as a unique product of prime numbers. The Greatest Common Divisor (GCD) of two numbers is the largest integer that divides both evenly. The Least Common Multiple (LCM) is the smallest positive integer that both numbers divide.
Prime factorization. For GCD, use the minimum exponent of each common prime factor.

Worked Examples

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Worked Examples
Example 1: Find all factors of 24.
The factor pairs: 1×24, 2×12, 3×8, 4×6. So factors: 1, 2, 3, 4, 6, 8, 12, 24.

Example 2: GCD of 48 and 18.
Prime factorizations: 48 = 2⁴×3. 18 = 2×3². Take min exponents: 2¹×3¹ = 6.

Example 3: Apply PEMDAS to 5×3 + 8÷2².
Exponent first: 2² = 4. Then multiplication: 5×3 = 15. Division: 8÷4 = 2. Finally: 15 + 2 = 17.

Number Line Explorer

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Visualize arithmetic operations as movements along the number line. Adjust the parameter to see how sums, differences, and multiples behave.