Arithmetic & Number formulas

All 20 formulas in this topic

Order of OperationsP→E→M,D→A,S\text{P} \to \text{E} \to \text{M},\text{D} \to \text{A},\text{S}Order of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Distributive Propertya(b+c)=ab+aca(b + c) = ab + acThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.Commutative Propertya+b=b+aab=baa + b = b + a \qquad ab = baThe Commutative Property says you can switch the order of numbers when adding or multiplying without changing the result.Associative Property(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)Grouping numbers differently in addition doesn't change the sum, so you can rearrange parentheses to make calculations easier or more convenient.Absolute Value∣a∣={aa≥0−aa<0|a| = \begin{cases} a & a \ge 0 \\ -a & a < 0 \end{cases}Absolute value measures the distance a number is from zero without regard to direction, so it is always positive or zero.Greatest Common Factorgcd⁡(a,b)=largest d with d∣a and d∣b\gcd(a,b) = \text{largest } d \text{ with } d \mid a \text{ and } d \mid bThe Greatest Common Factor finds the largest whole number that divides evenly into two or more numbers, helping you simplify fractions.Least Common Multiplelcm⁡(a,b)=∣ab∣gcd⁡(a,b)\operatorname{lcm}(a,b) = \frac{|ab|}{\gcd(a,b)}The least common multiple is the smallest positive number that all given numbers divide into evenly; use it when adding fractions with different denominators.Prime Factorizationn=p1e1p2e2⋯pkekn = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}Prime factorization breaks any whole number into a unique multiplication of primes—essential for finding common factors and simplifying fractions.Divisibility Rulesn≡0(modd)n \equiv 0 \pmod{d}Divisibility rules determine when one number divides into another evenly with no remainder — use them to quickly check if a number is a multiple.Scientific Notationx=a×10n,1≤∣a∣<10x = a \times 10^{n}, \quad 1 \le |a| < 10Scientific notation expresses very large or very small numbers as a coefficient times a power of ten, making them easier to work with.Rules of Signs(−a)(−b)=ab(−a)(b)=−ab(-a)(-b) = ab \qquad (-a)(b) = -abRules of Signs tell you whether the product of two numbers is positive or negative, which you need for solving equations with negative numbers.Square Roota=b  ⟺  b2=a,;b≥0\sqrt{a} = b \iff b^2 = a, ; b \ge 0Square root finds the non-negative number that, when multiplied by itself, gives your starting number; use it to undo squaring in equations.Cube Roota3=b  ⟺  b3=a\sqrt[3]{a} = b \iff b^3 = aCube root finds the number that, when multiplied by itself three times, equals your original number—the inverse of cubing.Modulo Operationa mod n=a−n⌊an⌋a \bmod n = a - n\left\lfloor \frac{a}{n} \right\rfloorThe modulo operation finds the remainder when one integer is divided by another, useful when tracking cycles and repetition patterns.Ratioa:b=aba : b = \frac{a}{b}A ratio compares two quantities to show their relationship, and can be written as a fraction or with a colon to help you scale or find missing values.Proportionab=cd  ⟺  ad=bc\frac{a}{b} = \frac{c}{d} \iff ad = bcUse Proportion when two ratios are equal to find a missing value by cross-multiplying to solve equations with fractions.Unit Rater=quantityunit of measurer = \frac{\text{quantity}}{\text{unit of measure}}A unit rate is a ratio that compares a quantity to exactly one unit of measure, helping you compare values or find the cost per item.Rounding Rulesround(x)=⌊x+12⌋\text{round}(x) = \left\lfloor x + \tfrac{1}{2} \right\rfloorRounding Rules teaches how to round any number to the nearest whole number using a simple mathematical formula for accurate estimation.Estimating with Roundingx≈x^x \approx \hat{x}Use rounding to simplify calculations and get quick estimates of what values should be roughly equal to, without needing exact arithmetic.Long Divisiona=qn+r,0≤r<na = qn + r, \quad 0 \le r < nLong Division breaks division into steps, giving you a whole-number quotient and a remainder, showing what divides evenly and what's left.

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