Infinity is the idea of something without any end or bound, written with the symbol ∞. It is not a real number, so you cannot add, subtract or divide with it like an ordinary number. Math uses it for endless sets, quantities that grow without limit, and intervals such as (3, ∞).
Infinity is never a place you reach. It is the arrow that keeps going: the number line, the ray x > 3 and the counting numbers all have no last point.
Formal definition and notation
Infinity, ∞, names an unbounded quantity, larger than every real number; it is not itself an element of the real numbers. Writing x → ∞ means x increases without bound, and limx → ∞f(x) = L describes what f(x) approaches as x does so. In interval notation, ∞ and −∞ mark a side with no endpoint, so that side always takes a parenthesis. A set is infinite when it has no finite number of elements, like the whole numbers.
Symbol
Read it as
Meaning
∞
infinity
without bound in the positive direction
−∞
negative infinity
without bound in the negative direction
x → ∞
x approaches infinity
x keeps growing larger than any number you name
(a, ∞)
the interval from a to infinity
all real numbers greater than a
…
and so on
a list that continues forever, such as 1, 2, 3, …
Key facts
∞ is a symbol for an idea, not a number on the number line, so ∞ − ∞ and ∞∞ have no set value.
In interval notation, always use a parenthesis next to ∞ or −∞: [2, ∞), never [2, ∞].
10 is undefined, not infinity. But 1x grows without bound as x gets close to 0 from the right.
As x → ∞, 1x → 0: 0.1, 0.01, 0.001, …
The sets of whole numbers, integers, fractions and real numbers are all infinite.
An equation such as x + 3 = x + 3 has infinitely many solutions: every real number works.
Where the name comes from. The English mathematician John Wallis proposed the symbol ∞ for infinity in 1655.
Try it: intervals that never end
Interactive
The shaded ray shows x > 2, the interval (2, ∞). Drag the endpoint, switch between open and closed, or send an end to infinity and back.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Undefined means an expression has no value at all, like 70. Infinity describes growth without bound, like 7x as x shrinks toward 0 from the right.
Infinityvs.Very large number
A googol, 10100, is huge but finite: you can add 1 to it. Infinity is larger than every number, so no number is close to it.
Quick check: 3 questions
Question 1 of 3
Which is the interval notation for x < 5?
Show answer and explanation
Answer: B. All numbers less than 5 run without end to the left, and 5 is not included: (−∞, 5). A bracket on −∞ is never allowed, (5, ∞) is x > 5, and (−∞, 5] includes 5.
Question 2 of 3
As x → ∞, what does 1x approach?
Show answer and explanation
Answer: D. Bigger denominators make smaller fractions: 0.1, 0.01, 0.001. So 1x approaches 0. It never grows, never reaches 1, and is defined for every positive x.
Question 3 of 3
Which statement about infinity is true?
Show answer and explanation
Answer: C. Infinity is not a real number, so no interval can include it. There is no largest real number, ∞ − ∞ is indeterminate, and 10 is undefined.
Infinity: frequently asked questions
Is infinity a number?
Not in ordinary arithmetic. Infinity is not on the number line and is not a real number, so ∞ + 1 or ∞ ÷ ∞ have no normal value. It is an idea or a symbol for going on without end. Some advanced branches of math do treat infinite sizes as numbers, but with their own special rules.
Is 1 divided by 0 equal to infinity?
No. 10 is undefined, because no number times 0 gives 1. What is true is that 1x gets larger and larger as x moves toward 0 from the positive side, and more and more negative from the negative side. Infinity describes that growth, not the value at 0.
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 √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1Interval NotationInterval notation is a short way to write a set of numbers between two endpoints, using brackets and parentheses. A bracket, [ or ], means the endpoint is included; a parenthesis, ( or ), means it is not. So −2 ≤ x < 3 is written [−2, 3), and x > 4 is written (4, ∞).Symbols and Notation · Algebra 1, Algebra 2, PrecalculusSideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2NumberA 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.Number System · Kindergarten, Grade 1, Grade 2Whole 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 5Number 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 6UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1Greater Than or Equal ToGreater than or equal to means a value is either bigger than a number or exactly equal to it, and its symbol is ≥. The inequality x ≥ 3 is true for 3 and every number above it. It means “at least 3”. On a number line it is graphed with a filled dot, because the endpoint is included.Symbols and Notation · Grade 6, Grade 7, Algebra 1DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5LimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusAsymptoteAn asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1x has both: x = 0 and y = 0.Functions · Algebra 2, Precalculus, CalculusEnd BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, PrecalculusInequalityAn inequality compares values using symbols such as <, >, ≤ or ≥. Instead of saying two expressions are equal, it says one is greater, less, or at least/at most another, as in x > 5. An inequality with a variable usually has infinitely many solutions, shown as a ray on a number line.Algebra · Grade 6, Grade 7, Grade 8