Number System Terms

Whole numbers and place value 9

• DecomposeDecompose means to break a number or shape into smaller parts that together make the whole. You can decompose 8 into 5 + 3, decompose 347 into 300 + 40 + 7, or decompose 3/4 into 1/4 + 2/4. Most numbers can be decomposed in many ways, and the parts always add back to the original.Kindergarten, Grade 1, Grade 2
• DigitA digit is one of the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 that we use to write every number. The number 507 has three digits: 5, 0 and 7. What a digit is worth depends on its place: in 507, the 5 stands for 5 hundreds, or 500.Grade 1, Grade 2, Grade 3
• Expanded FormExpanded form is a way of writing a number as the sum of the values of its digits. Each digit is multiplied by its place value, so 4072 = 4000 + 70 + 2, or 4 × 1000 + 7 × 10 + 2 × 1. It shows what every digit is worth. Zeros add nothing, so they are usually left out.Grade 2, Grade 3, Grade 4
• Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Grade 3, Grade 4, Grade 5
• RoundingRounding replaces a number with a nearby number that is easier to use, to a chosen place value. Look at the digit immediately to the right of that place: 5 or more rounds up, 4 or less rounds down. So 47 rounds to 50 to the nearest ten, and 3.68 rounds to 3.7 to the nearest tenth.Grade 3, Grade 4, Grade 5
• Skip CountingSkip counting is counting forward or backward by the same number each time instead of by 1, such as 2, 4, 6, 8 or 5, 10, 15, 20. Each jump adds the same amount, so skip counting by 5 six times reaches 6 × 5 = 30. It is a quick way to count groups and learn multiplication facts.Kindergarten, Grade 1, Grade 2
• Standard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Grade 2, Grade 3, Grade 4
• Word FormWord form is a way of writing a number using words instead of digits, as you would say it aloud: 4,306 is written "four thousand, three hundred six". Read each three-digit period, then say its name (million, thousand). For decimals, say "and" at the decimal point: 6.25 is "six and twenty-five hundredths".Grade 2, Grade 3, Grade 4
• ZeroZero (0) is the number that represents none or no quantity. It is neither positive nor negative and sits in the middle of the number line, between the negative and positive numbers. Adding zero to a number does not change it, as in 9 + 0 = 9, so zero is called the additive identity.Grade 3, Grade 4, Grade 5

Types of numbers 9

• Counting NumbersCounting numbers are the numbers you say when you count objects one by one: 1, 2, 3, 4, 5 and so on, without end. They start at 1, not 0, and include no fractions, decimals or negative numbers. Adding 0 to the counting numbers gives the whole numbers.Kindergarten, Grade 1, Grade 2
• Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Grade 2, Grade 3, Grade 4
• Irrational NumberAn irrational number cannot be written as a fraction of two integers. Its decimal form does not end and does not repeat in a pattern. Examples are square root of 2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Grade 8, Algebra 1
• Natural NumberA natural number is one of the numbers 1, 2, 3, 4 and so on, used for counting and ordering. Natural numbers are positive and whole, so they include no fractions, decimals, negatives or zero in most US school math. Some books also include 0, so check the convention. The set is written ℕ.Grade 5, Grade 6, Grade 7
• NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Kindergarten, Grade 1, Grade 2
• Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Grade 2, Grade 3, Grade 4
• Ordinal NumberAn ordinal number tells the position of something in an ordered list: first, second, third, fourth and so on. It answers "Which place?" rather than "How many?". Ordinals are written as words or as a numeral with an ending: 1st, 2nd, 3rd, 4th, 11th, 21st, 22nd.Kindergarten, Grade 1, Grade 2
• Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as square root of 2 and π). Square roots of negative numbers, like square root of −4, are not real.Grade 8, Algebra 1
• Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 3/4 and −3 are not. Every whole number is also an integer.Grade 3, Grade 4, Grade 5

Integers and absolute value 7

• Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Grade 6, Grade 7, Grade 8
• Additive InverseThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Grade 6, Grade 7, Grade 8
• IntegerAn integer is a whole number, its opposite, or zero: the counting numbers, their negatives, and zero. The integers are …, −3, −2, −1, 0, 1, 2, 3, … and go on forever in both directions. Integers have no fractional or decimal part, so 4.5 and 1/2 are not integers.Grade 6, Grade 7, Grade 8
• Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −2/3. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Grade 6, Grade 7
• Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Grade 2, Grade 3, Grade 6
• OppositeThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Grade 6, Grade 7, Grade 8
• Rational NumberA rational number is any number that can be written as a fraction of two integers, a/b, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 3/4, −5 = −5/1, 1.6 and 0.333… all qualify. Numbers such as square root of 2 and π are irrational.Grade 7, Grade 8, Algebra 1

Fractions 13

• Common DenominatorA common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 1/3 and 1/4, 12 works: 4/12 and 3/12. You need one to add, subtract or compare fractions with different denominators.Grade 4, Grade 5, Grade 6
• DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 3/5, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Grade 3, Grade 4, Grade 5
• Equivalent FractionsEquivalent fractions are fractions that name the same value, like 1/2, 2/4 and 3/6. To make one, multiply or divide the numerator and denominator by the same nonzero number. That multiplies the fraction by 1, so its value does not change, and the fractions land on the same point of a number line.Grade 3, Grade 4, Grade 5
• FractionA fraction is a number that names equal parts of a whole, written a/b. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 3/4 means 3 of 4 equal parts. Every fraction is also a point on the number line.Grade 3, Grade 4, Grade 5
• Improper FractionAn improper fraction has a numerator that is greater than or equal to its denominator, so its value is 1 or more. Examples are 7/4, 9/2 and 5/5. Every improper fraction can be rewritten as a whole number or a mixed number: 7/4 = 1 3/4. Improper does not mean wrong.Grade 4, Grade 5, Grade 6
• Least Common DenominatorThe least common denominator (LCD) of two or more fractions is the smallest number that every denominator divides into evenly. It is the least common multiple of the denominators. For 1/4 and 5/6, the LCD is 12, so you rewrite them as 3/12 and 10/12 before you add, subtract or compare.Grade 5, Grade 6, Grade 7
• Mixed NumberA mixed number has a whole number and a fraction together, such as 2 3/4, which means 2 + 3/4. It represents a value greater than one whole when the whole-number part is positive. Every mixed number can be rewritten as an improper fraction, and back: 2 3/4 = 11/4.Grade 4, Grade 5, Grade 6
• NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 3/5, the numerator is 3: three of five equal parts are taken.Grade 3, Grade 4, Grade 5
• Proper FractionA proper fraction is a fraction whose numerator is smaller than its denominator, such as 3/4 or 2/9. Because fewer parts are taken than make one whole, a positive proper fraction is always less than 1. If the numerator is equal to or greater than the denominator, the fraction is improper instead.Grade 3, Grade 4, Grade 5
• ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 3/4 is 4/3, and the reciprocal of 5 is 1/5. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Grade 5, Grade 6, Grade 7
• Simplest FormSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 6/8 = (6 ÷ 2)/(8 ÷ 2) = 3/4. The value stays the same; only the numbers get smaller. It is also called lowest terms.Grade 4, Grade 5, Grade 6
• SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 18/24 = 3/4. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Grade 4, Grade 5, Grade 6
• Unit FractionA unit fraction is a fraction with a numerator of 1, such as 1/2, 1/5 or 1/12. It names one equal part of a whole split into that many parts. The bigger the denominator, the smaller the unit fraction, and every fraction a/b is a copies of 1/b.Grade 3, Grade 4, Grade 5

Decimals and percents 5

• DecimalA decimal is a number written with a decimal point to show parts of a whole. Decimals use place value to represent tenths, hundredths, thousandths and more: in 3.14, the 1 means 1 tenth and the 4 means 4 hundredths. So 0.75 = 75/100, and 0.5 is one half.Grade 4, Grade 5, Grade 6
• Decimal PointA decimal point is the dot that separates the whole-number part of a number from its fractional part. Digits to its left are ones, tens and hundreds; digits to its right are tenths, hundredths and thousandths. In 46.25 the point sits between 6 ones and 2 tenths, and you read it aloud as "and".Grade 4, Grade 5, Grade 6
• HundredthsHundredths are the parts you get when one whole is split into 100 equal parts; one hundredth is 1/100 or 0.01. The hundredths place is the second digit to the right of the decimal point, so in 3.47 the 7 means 7 hundredths. One cent is one hundredth of a dollar.Grade 4, Grade 5, Grade 6
• PercentA percent is a number out of 100. Percent means per hundred, so 25% means 25 out of 100, or 25/100 = 0.25. To find a percent of a quantity, change the percent to a decimal and multiply: 20% of 80 is 0.20 × 80 = 16. The symbol for percent is %.Grade 5, Grade 6, Grade 7
• TenthsTenths are the parts you get when one whole is split into 10 equal parts; each part is one tenth, written 1/10 or 0.1. The tenths place is the first digit to the right of the decimal point, so in 4.7 the 7 means 7 tenths, or 0.7.Grade 3, Grade 4, Grade 5

Factors, multiples and primes 9

• Composite NumberA composite number is a whole number greater than 1 that has more than two factors. It can be divided evenly by a number other than 1 and itself: 12 is composite because 1, 2, 3, 4, 6 and 12 all divide it. Every whole number above 1 is either prime or composite; 1 is neither.Grade 4, Grade 5, Grade 6
• Divisibility RulesDivisibility rules are shortcuts that tell whether one whole number divides another with no remainder, without doing the division. A number is divisible by 2 if its last digit is even, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, and by 10 if it ends in 0.Grade 4, Grade 5, Grade 6
• FactorA factor is a number or expression that divides another number or expression evenly, with no remainder. Factors multiply together to make a product: 4 × 6 = 24, so 4 and 6 are factors of 24. Every whole number greater than 1 has at least two factors, 1 and itself.Grade 4, Grade 5, Grade 6
• Factor PairA factor pair is two whole numbers that multiply to give a certain number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. Listing factor pairs is a sure way to find every factor of a number: start at 1 and stop once the pairs begin to repeat.Grade 4, Grade 5, Grade 6
• Greatest Common FactorThe greatest common factor (GCF) is the largest factor shared by two or more numbers: the largest number that divides into both without a remainder. The GCF of 18 and 24 is 6. Find it by listing factors or by multiplying the prime factors the numbers share. The GCF is how you simplify fractions in one step.Grade 5, Grade 6, Grade 7
• Least Common MultipleThe least common multiple (LCM) is the smallest positive multiple shared by two or more numbers. The LCM of 6 and 9 is 18: it is the first number on both lists of multiples, 6, 12, 18, … and 9, 18, …. The LCM of the denominators is the least common denominator for adding fractions.Grade 5, Grade 6, Grade 7
• MultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Grade 4, Grade 5, Grade 6
• Prime FactorizationPrime factorization is writing a whole number as a product of prime numbers only, such as 72 = 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2. Find it with a factor tree or repeated division. Every whole number greater than 1 has exactly one prime factorization, apart from the order of the factors.Grade 4, Grade 5, Grade 6
• Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Grade 4, Grade 5, Grade 6

Exponents and roots 6

• Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written cube root of x. Because 4 × 4 × 4 = 64, cube root of 64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: cube root of −27 = −3.Grade 8, Algebra 1, Algebra 2
• Perfect SquareA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and square root of 49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Grade 6, Grade 7, Grade 8
• PowerA power is a number written as a base with an exponent, such as 5^3, or the value that results, 5^3 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or "5 to the third power". The second power is called the square and the third power the cube.Grade 5, Grade 6, Grade 7
• RadicalA radical is an expression that uses the root symbol square root of (), such as square root of 25 or cube root of x. The number under the symbol is the radicand, and the small number in the notch is the index, which tells which root to take. With no index, it is a square root: square root of 25 = 5.Grade 8, Algebra 1, Algebra 2
• Scientific NotationScientific notation writes a number as a value from 1 to less than 10 times a power of 10, such as 5.4 × 10^3 for 5,400. The exponent records how far the decimal point moved: positive for large numbers and negative for numbers between 0 and 1, as in 0.0062 = 6.2 × 10^(−3).Grade 8, Algebra 1
• Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Grade 8, Algebra 1

Complex numbers 2

• Complex NumberA complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = −1. The real part is a and the imaginary part is b. Examples: 3 + 2i and −4i. Complex numbers let you take square roots of negatives, such as square root of −9 = 3i.Algebra 2, Precalculus
• Imaginary NumberAn imaginary number is a real number multiplied by i, the imaginary unit, where i = square root of −1 and i^2 = −1. Examples are 3i, −1/2i and square root of −25 = 5i. Imaginary numbers give square roots of negative numbers a value, so equations like x^2 = −9 have solutions: x = ± 3i.Algebra 2, Precalculus

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Updated November 2026·Reviewed by the Effortless Math team