Math explainers · 28 pages
Arithmetic explainers
Clear, citable explainers for the foundational concepts of arithmetic — numbers, fractions, operations, GCF, LCM, and order of operations.
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All 28 arithmetic explainers pages
Multiplication chart
A multiplication chart is a mathematical grid that displays the product of two numbers. One set of numbers is written along the top row, and another set down the left-hand column, with their products filling the grid where the row and column intersect. It is widely used to teach basic multiplication and reveal arithmetic patterns.
DefinitionSquare root
A square root of a number x is a number y such that y² = x. For example, the square root of 9 is 3, because 3 × 3 = 9. While every positive number has both a positive and a negative square root (such as 3 and −3 for 9), the radical symbol √ denotes the principal (positive) square root.
DefinitionPrime number
A prime number is a positive integer greater than 1 that cannot be formed by multiplying two smaller natural numbers. Its only divisors are 1 and the number itself. The first five prime numbers are 2, 3, 5, 7, and 11. Numbers with more than two factors are composite.
DefinitionPEMDAS
PEMDAS is an alphabetical mnemonic that dictates the correct sequence for solving mathematical equations. Operations are performed in this order: Parentheses first, then Exponents, followed by Multiplication and Division (processed left to right), and finally Addition and Subtraction (also processed left to right).
DefinitionIntegers
Integers are numbers that do not have fractional or decimal parts. The set includes positive whole numbers (1, 2, 3...), their negative opposites (−1, −2, −3...), and zero. Mathematically, the set of integers is denoted by the capital letter Z.
DefinitionPerfect squares
A perfect square is a number obtained by squaring an integer. For example, 9 is a perfect square because it equals 3 × 3 (or 3²). The first few non-zero perfect squares are 1, 4, 9, 16, 25, 36, and 49. Because negative integers multiplied by themselves also yield positive results, perfect squares are always non-negative.
DefinitionHow to multiply fractions
Multiplying fractions is a simple process: multiply the top numbers (numerators) to get the new top number, and multiply the bottom numbers (denominators) to get the new bottom number. The mathematical formula is (a/b) × (c/d) = (ac)/(bd). Finally, simplify the resulting fraction if necessary.
DefinitionDistributive property
The distributive property is a fundamental algebraic law expressed by the formula a(b + c) = ab + ac. It states that multiplying a number (a) by a group of numbers added together (b + c) gives the same result as multiplying the number by each addend individually (ab and ac) and then adding the products.
DefinitionRational numbers
A rational number is a number that can be written as a fraction where both the numerator and denominator are integers (e.g. 3/4). Whole numbers, terminating decimals (like 0.5), and repeating decimals (like 0.333...) are all rational because they can be converted to fractions. The set of rational numbers is denoted by Q.
DefinitionOrder of operations
The order of operations is the mathematical convention used to evaluate expressions unambiguously. Without a standard order, an equation like 3 + 4 × 2 could equal 14 or 11. The correct order requires solving parentheses (brackets) first, exponents (indices) second, multiplication and division third (left to right), and addition and subtraction last (left to right).
DefinitionLong division
Long division is an arithmetic process for dividing large numbers. It involves dividing the dividend (the number being divided) by the divisor (the number dividing it) to produce a quotient (the answer) and sometimes a remainder. The algorithm relies on a repeating cycle: Divide, Multiply, Subtract, and Bring Down.
DefinitionNumber line
A number line is a geometric diagram that represents real numbers as points along a straight line. Typically, integers are marked at equal intervals, with positive numbers extending to the right of zero, negative numbers extending to the left, and arrows at each end showing that the line goes on infinitely.
DefinitionIrrational numbers
An irrational number is a real number that cannot be written in the form a/b, where a and b are integers. In decimal form, irrational numbers go on forever without terminating and without repeating a pattern. Famous examples include the square root of 2 (√2), pi (π), and the golden ratio (φ).
DefinitionHow to add fractions
Adding fractions requires a common denominator. If denominators are the same, simply add the numerators together and place the sum over the shared denominator (e.g. 1/5 + 2/5 = 3/5). If they differ, you must first convert the fractions to equivalent forms with a common denominator.
DefinitionAbsolute value
Absolute value measures how far a number is from zero. It is denoted by vertical bars around the number, such as |x|. Because distance cannot be negative, the absolute value is always positive or zero. For example, |5| = 5, and |−5| = 5.
DefinitionFraction to decimal
Converting a fraction to a decimal is achieved by division. The fraction line acts as a division operator, so a/b is solved as a ÷ b. For example, to convert 3/4 to a decimal, divide 3 by 4, which yields 0.75. The division can be done by hand using long division or with a calculator.
DefinitionComposite numbers
A composite number is a whole number that can be formed by multiplying two smaller positive integers. Unlike prime numbers, which only have two factors (1 and the number itself), composite numbers have three or more factors. For example, 4 is composite because its factors are 1, 2, and 4.
DefinitionProportions
A proportion is an equation that sets two ratios equal, written as a/b = c/d. It states that the relationship between two quantities is equivalent to the relationship between another two. For example, because 1/2 and 5/10 are equal, they form a proportion.
DefinitionGreatest common factor
The greatest common factor (also known as the highest common factor, or HCF) of two or more numbers is the largest whole number that is a factor of all the numbers. For example, the GCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without a remainder.
DefinitionLeast common multiple
The least common multiple of a set of numbers is the smallest whole number that is a multiple of all the numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest positive integer that both 4 and 6 divide into evenly.
How-toHow to learn times tables
To learn times tables effectively, begin with simple patterns (such as the 2s, 10s, and 5s) before moving to harder tables. Practice regularly using visual grids, flashcards, and online games. Focus on understanding the concept of repeated addition first, and then build speed through active recall and spaced repetition.
DefinitionFractions
A fraction is a mathematical representation of a part of a whole. It consists of a top number called the numerator (representing how many parts we have) and a bottom number called the denominator (representing the total number of equal parts the whole is divided into). The horizontal line separating them is called the vinculum, which denotes division.
DefinitionMultiplication
Multiplication is one of the four basic operations of arithmetic, representing the process of combining equal groups. It is written using the symbols ×, ·, or *. The numbers being multiplied are called factors (specifically, the multiplicand and the multiplier), and the final result is called the product. Multiplication is commutative, associative, and distributive over addition.
DefinitionPercentages
A percentage is a dimensionless ratio that expresses a fraction of 100. The word "percent" comes from the Latin "per centum," meaning "by the hundred." For example, 45% represents 45 out of 100, which is equivalent to the fraction 45/100 or the decimal 0.45. Percentages are widely used to calculate discounts, interest rates, tax rates, and statistical proportions.
DefinitionDecimals
A decimal number is a value in the base-10 system that includes a decimal point to separate the whole number portion from the fractional portion. Each digit to the right of the decimal point represents a place value that is a power of one-tenth (tenths, hundredths, thousandths, etc.). Decimals provide a precise way to represent real numbers, including both rational and irrational values.
DefinitionProbability
Probability is the mathematical measure of the likelihood that a specific event will occur. It is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. A probability of 0 indicates an impossible event, a probability of 1 indicates a certain event, and values in between represent varying degrees of likelihood.
DefinitionPermutations
A permutation is a mathematical calculation of the number of ways a specific set of objects can be arranged where order is important. For example, the arrangements ABC and BCA are two distinct permutations of the same letters. The formula for the number of permutations of n items taken r at a time is ⁿPᵣ = n! / (n - r)!, where exclamation marks denote factorials.
DefinitionCombinations
A combination is a mathematical technique that determines the number of possible selections that can be made from a larger set of items where order is not important. For example, selecting a group of two students from a class of three is a combination: choosing Alice and Bob is identical to choosing Bob and Alice. The formula is ⁿCᵣ = n! / (r! × (n - r)!).
