Real numbers and their properties
Algebra is arithmetic with letters standing in for numbers. Before solving equations, it helps to know the kinds of numbers we work with and the rules they always obey.
Sets of numbers
The natural numbers are 1, 2, 3, and so on; adding 0 gives the whole numbers; including negatives gives the integers. Rational numbers are fractions \(\frac{p}{q}\) of integers with \(q \neq 0\), such as \(\frac{3}{4}\), -2, and 0.125. Their decimals either stop or repeat, as in \(\frac{1}{3} = 0.333\ldots\)
Irrational and real numbers
Irrational numbers cannot be written as fractions; their decimals never stop or repeat. Examples are \(\sqrt{2} \approx 1.414\), \(\pi \approx 3.14159\), and \(e \approx 2.718\). The rational and irrational numbers together make up the real numbers, every point on the number line.
Order of operations
To evaluate expressions consistently, work inside grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. So \(3 + 4 \times 2^2 = 3 + 4 \times 4 = 19\), while \((3 + 4) \times 2^2 = 28\). The memory aid PEMDAS lists the same order.
Properties of operations
Addition and multiplication are commutative, \(a + b = b + a\), and associative, \((a + b) + c = a + (b + c)\). The distributive property links them: \(a(b + c) = ab + ac\), so \(3(x + 5) = 3x + 15\). Zero is the additive identity and 1 the multiplicative identity; every number has an opposite, and every non-zero number has a reciprocal.
Absolute value
The absolute value \(|a|\) is a number's distance from zero on the number line, so it is never negative: \(|-7| = 7\) and \(|7| = 7\). The distance between two numbers a and b is \(|a - b|\): the distance between -3 and 5 is \(|-3 - 5| = 8\).
Variables and expressions
A variable is a letter that stands for a number. An algebraic expression combines variables and numbers with operations, such as \(2x^2 - 5x + 1\). Evaluating means substituting a value: when x = 3, this gives \(2(9) - 15 + 1 = 4\). Terms with the same variable parts, such as 4x and -9x, are like terms and can be combined to -5x.
Simplifying expressions
To simplify, distribute, then combine like terms. For example, \(4(2x - 3) - (x - 5) = 8x - 12 - x + 5 = 7x - 7\). A common mistake is forgetting that the minus sign in front of the brackets changes the sign of every term inside.
Worked example
A phone plan costs $20 a month plus $0.10 per text, so the monthly cost for t texts is \(20 + 0.10t\). For 150 texts, the cost is \(20 + 0.10(150) = 20 + 15 = 35\) dollars. Writing the situation as an expression lets you compare plans for any number of texts at once.
Key takeaways
The real numbers include natural numbers, whole numbers, integers, rationals (fractions, with stopping or repeating decimals), and irrationals such as the square root of 2 and pi. Follow the order of operations: grouping, exponents, multiplication and division, then addition and subtraction, each left to right. The commutative, associative, and distributive properties let you rearrange and expand expressions. Absolute value is distance from zero and is never negative. Simplify by distributing and combining like terms, taking care with minus signs before brackets.
Flashcard
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Check your understanding
What is the value of \(3 + 4 \times 2^2\)?