A 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.
A number is the amount itself; 5, five, V and 2 + 3 are different names for it. Numbers count, order, measure and label.
Formal definition and notation
In mathematics, a number is an element of a number system, such as the whole numbers, the integers, the rational numbers or the real numbers, on which operations like addition and multiplication are defined. A numeral is a symbol or group of symbols that names a number, and a digit is one of the ten symbols 0 to 9 used to build numerals in base ten.
Symbol
Read it as
Meaning
0 1 2 3 4 5 6 7 8 9
the ten digits
the symbols used to write every number in base ten
ℕ, ℤ, ℚ, ℝ
natural numbers, integers, rationals, reals
names for the main sets of numbers
#
number
a shorthand for the word, as in “# of students”; not used in math expressions
Key facts
A number is an amount; a numeral is how it is written; a digit is one symbol in a numeral. 352 is one number written with three digits.
Numbers answer four kinds of questions: how many (count), which place (order), how much (measure) and which one (label).
Every number has a place on the number line, and a number to the right is greater than a number to the left.
Numbers come in families: counting numbers, whole numbers, integers, fractions and decimals, and more.
Label numbers, like a jersey number or a zip code, name things; adding them does not mean anything.
Try it: every number has a place
Interactive
Drag the two points anywhere, including between the whole numbers and below zero. The readout says which number is greater and how far each is from 0.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Number examples, step by step
Example 1: Digits versus number
Warm-up
How many digits are in the number 40,072, and what is the value of the 4?
The digits are 4, 0, 0, 7, 2: five digits.
The 4 is in the ten-thousands place, so it is worth 40,000.
Answer: 40,072 has 5 digits, and the 4 is worth 40,000.
Example 2: Many names for one number
Core
Write the number 12 in four different ways.
As a word: twelve.
As a sum: 10 + 2 (or 7 + 5).
As a product: 3 × 4.
As a fraction: 242.
Answer: Twelve, 10 + 2, 3 × 4 and 242 all name the number 12.
Example 3: Sorting numbers into families
Test-style
Which of these numbers are integers: −6, 0, 2.5, 13, 34?
Integers are whole numbers and their opposites: …, −2, −1, 0, 1, 2, …
−6, 0 and 13 qualify.
2.5 and 34 are between integers, so they are not integers.
Answer: The integers are −6, 0 and 13.
Common mistakes
✗ MistakeSaying 9 is greater than 10 because 9 is a bigger digit.
✓ Instead10 is greater than 9: 10 has 1 ten, while 9 has 0 tens.
A number’s size depends on place value, not on how big each digit looks.
✗ MistakeTreating a label as an amount: player 23 is better than player 7.
✓ InsteadJersey numbers are names, not amounts or scores.
Some numbers only identify things; comparing or adding them has no meaning.
✗ MistakeThinking only whole numbers are “real” numbers: 12 and −3 are not numbers.
✓ Instead12, −3 and 0.75 are all numbers with places on the number line.
Fractions, decimals and negatives are numbers too; they just belong to bigger families than the counting numbers.
A variable is a letter that stands for a number that can change or is unknown, like x in x + 3 = 10.
Quick check: 3 questions
Question 1 of 3
How many digits are used to write the number 3,050?
Show answer and explanation
Answer: C. The digits are 3, 0, 5, 0: four digits. Zeros count as digits. 2 and 3 skip one or both zeros, and 3,050 is the number itself.
Question 2 of 3
Which does NOT name the number 8?
Show answer and explanation
Answer: D.82 = 4, not 8. 2 × 4, the word eight and the Roman numeral VIII all name 8.
Question 3 of 3
Which number is greatest?
Show answer and explanation
Answer: D. On the number line 1 is farthest right. 0.9 and 78 = 0.875 are just under 1, and −10 is far to the left, even though 10 looks big.
Number: frequently asked questions
What is the difference between a number and a digit?
A digit is a single symbol, 0 through 9. A number is the amount that digits (or words) describe. The number 507 is written with three digits: 5, 0 and 7. Every one-digit number, like 6, is both a digit and a number, but 507 is a number, not a digit.
Is zero a number?
Yes. Zero is the number that tells how many objects there are when there are none. It is a whole number and an integer, and it sits between the negatives and positives on the number line. Zero is also a digit, used as a placeholder in numbers such as 205 and 1,000.
What are the main types of numbers?
Counting numbers (1, 2, 3, …), whole numbers (add 0), integers (add the negatives), rational numbers (add fractions and decimals that end or repeat), and real numbers (add irrational numbers like π). Each family sits inside the next, so 4 belongs to all of them, while 12 is rational and real but not an integer.
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, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5IntegerAn 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 12 are not integers.Number System · Grade 6, Grade 7, Grade 8Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1Real 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 √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4DigitA 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.Number System · Grade 1, Grade 2, Grade 3Number 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.Number System · Grade 2, Grade 3, Grade 6Counting 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.Number System · Kindergarten, Grade 1, Grade 2Place 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.Number System · Grade 3, Grade 4, Grade 5VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8Irrational 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 √2 = 1.41421… and π = 3.14159…. The square root of any whole number that is not a perfect square is irrational.Number System · Grade 8, Algebra 1FractionA fraction is a number that names equal parts of a whole, written ab. 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 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5DecimalA 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 = 75100, and 0.5 is one half.Number System · Grade 4, Grade 5, Grade 6Ordinal 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.Number System · Kindergarten, Grade 1, Grade 2