Undefined 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.
In Spanish
indefinido (no definido)
Also called
undefined value, not defined, division by zero, no value
Every division can be checked by multiplying back. 0 ÷ 7 passes the check; 7 ÷ 0 cannot, because anything times 0 is 0. That is why it is undefined.
Formal definition and notation
An expression is undefined when the operations it names cannot be carried out in the number system being used. For real numbers, a0 is undefined for every a: if a0 = q, then q × 0 = a, which fails for a ≠ 0 and holds for every q when a = 0, so no single value exists. Likewise √−4 is undefined as a real number, and a function is undefined at inputs outside its domain.
Symbol
Read it as
Meaning
a0
a divided by zero
undefined for every real number a
x ≠ 2
x is not equal to 2
how a restriction is written for 3x − 2
Key facts
Dividing by zero is undefined; dividing into zero is fine: 05 = 0.
Even 00 has no single value, so it is not defined either.
A slope is undefined for a vertical line, because the run is 0. A horizontal line has slope 0, which is defined.
A fraction with a variable is undefined where its denominator equals 0: x + 1x2 − 9 is undefined at x = 3 and x = −3.
zero is a defined number. Undefined means there is no number at all. A horizontal line has slope 0; a vertical line has an undefined slope.
Undefinedvs.No solution
An equation with no solution, like x + 1 = x + 2, is still made of defined expressions; no value of x makes it true. Undefined is about a single expression having no value.
Undefinedvs.Infinity
10 is not infinity. Infinity is not a real number, and 1x grows toward positive or negative infinity depending on the side, so no value can be assigned.
Quick check: 3 questions
Question 1 of 3
Which expression is undefined?
Show answer and explanation
Answer: C.8 − 8 = 0 in the denominator makes 20 undefined. A and B equal 0 (zero on top is fine), and D equals 4.
Question 2 of 3
Which line has an undefined slope?
Show answer and explanation
Answer: B.x = −2 is vertical, so the run between any two points is 0. y = −2 is horizontal with slope 0, and C and D have slopes −2 and 1.
Question 3 of 3
For what value of x is 52x + 6 undefined?
Show answer and explanation
Answer: A.2x + 6 = 0 gives x = −3. At x = 3 the denominator is 12, at x = 0 it is 6, and at x = −6 it is −6.
Undefined: frequently asked questions
Why is dividing by zero undefined?
Division is checked by multiplication: 12 ÷ 4 = 3 because 3 × 4 = 12. For 12 ÷ 0 you would need a number that gives 12 when multiplied by 0, but every number times 0 is 0. No answer exists, so mathematicians leave division by zero undefined rather than give it a value that breaks the rules of arithmetic.
Is 0 divided by 0 undefined?
Yes. Here the problem is the opposite: every number q satisfies q × 0 = 0, so there is no single answer. Because 00 could be anything, it is left undefined. In calculus it is called an indeterminate form, and limits are used to study expressions that approach it.
What is the difference between undefined slope and zero slope?
Zero slope belongs to a horizontal line: the rise is 0, and 0run = 0. Undefined slope belongs to a vertical line: the run is 0, and dividing the rise by 0 has no value. So y = 5 has slope 0, and x = 5 has an undefined slope.
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