Foundations of Mathematics · 3 min read
Order of Operations · Place Value and Base-Ten System · Factors and Multiples
The order of operations provides a standardized sequence for evaluating mathematical expressions. The acronym PEMDAS helps remember this sequence.
Parentheses
Operations within parentheses or other grouping symbols (brackets, braces, fraction bars, radical signs) are performed first. Work from innermost grouping outward when multiple levels exist.
Exponents
After resolving grouping symbols, evaluate exponents and roots. This includes squares, cubes, square roots, and all other powers.
Multiplication and Division
Perform multiplication and division from left to right in the order they appear. These operations have equal priority.
Addition and Subtraction
Perform addition and subtraction from left to right in the order they appear. These operations have equal priority.
Step 1: Parentheses first, evaluate exponent inside: 2³ = 8 Step 2: Continue parentheses: 8 - 6 = 2 Step 3: Multiplication: 4 × 2 = 8 Step 4: Division: 8 ÷ 2 = 4 Step 5: Addition: 3 + 4 = 7
The base-ten numeration system uses ten digits (0-9) and place value to represent numbers. Each position represents a power of ten.
Whole Number Place Values
Moving left from the ones place, each position represents the next power of ten: ones (10⁰), tens (10¹), hundreds (10²), thousands (10³), ten thousands (10⁴), hundred thousands (10⁵), millions (10⁶), and so forth.
Decimal Place Values
Moving right from the decimal point, each position represents a negative power of ten: tenths (10⁻¹ = 0.1), hundredths (10⁻² = 0.01), thousandths (10⁻³ = 0.001), and so forth.
Expanded Form
Expanded form expresses a number as the sum of each digit multiplied by its place value. For example, 3,542 = 3,000 + 500 + 40 + 2 = 3(1000) + 5(100) + 4(10) + 2(1).
Comparing and Ordering Numbers
To compare numbers, align place values and compare digits from left to right. The first position where digits differ determines which number is greater.
Factors
Factors of a number are integers that divide evenly into that number. To find all factors of a number, systematically test divisors starting from 1. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Prime and Composite Numbers
A prime number has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, and 23. A composite number has more than two factors. The number 1 is neither prime nor composite. The number 2 is the only even prime number.
Prime Factorization
Prime factorization expresses a composite number as a product of prime factors. Use factor trees or repeated division to find prime factorization. For example, 60 = 2² × 3 × 5. The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization.
Greatest Common Factor
The greatest common factor (GCF) of two or more numbers is the largest factor they share. To find the GCF, identify common prime factors and multiply them together using the lowest power of each. For 24 = 2³ × 3 and 36 = 2² × 3², the GCF = 2² × 3 = 12.
Least Common Multiple
The least common multiple (LCM) of two or more numbers is the smallest positive number divisible by all of them. To find the LCM, identify all prime factors and multiply using the highest power of each. For 24 = 2³ × 3 and 36 = 2² × 3², the LCM = 2³ × 3² = 72.
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