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not a number

Infinity: $\infty $

Infinity is not a real number.

This course presents a study of real numbers and real-valued functions. That doesn’t include $\infty $.

$\blacktriangleright $ You CANNOT perform arithmetic with $\infty $.

Addition, subtraction, multiplication, and division are operations for real numbers. Since $\infty $ is not a real number, it CANNOT be involved with any of these operations

$\infty + \infty \ne \infty $
$\infty - \infty \ne 0$
$\infty \cdot \infty \ne \infty $
$\infty \div \infty \ne 1$
$\infty ^{\infty } \ne \infty $

Our operations are strictly for real numbers, only.

$\blacktriangleright $ You CANNOT form fractions with $\infty $.

$\frac {\infty }{\infty } \ne \infty $
$\frac {\infty }{\infty } \ne 1$
$\frac {\infty }{\infty } \ne 0$
$\frac {1}{\infty } \ne 0$
$\frac {\infty }{1} \ne \infty $

$\blacktriangleright $ You CANNOT evaluate functions at $\infty $.

Let $f(x) = 3x^2 + 5x + 9$, then $f(\infty ) \ne \infty $

Let $g(x) = \frac {6x + 5}{3x - 1}$, then $g(\infty ) \ne 2$

Let $h(x) = \frac {4}{7x + 2}$, then $h(\infty ) \ne 0$

$\blacktriangleright $ $\infty $ CANNOT be a function value.

Let $k(x) = \frac {8}{3x + 6}$, then $k(-2) \ne \infty $

$\infty $ is not a real number and cannot be treated like a real number in any way.

Infinity: What is it?

If $\infty $ is not a real number, then what is it?

The problem here is the question itself. The question presupposes that $\infty $ is a mathematical object. For us, it isn’t.

For us, $\infty $ is simply shorthand communication. It is the shorthand symbol we use to let readers know that we have encountered an unbounded situation. Unbounded meaning there is no number greater than all of the values we are examining.

If you keep studying mathematics, especially logic, then $\infty $ might become an object, perhaps with its own operations.

However, for us, it is shorthand communication that describes the SIZE of a set of values we are analyzing.

The interval $(5, \infty )$ describes the set of all real numbers greater than $5$. $\infty $ tells us that there is no number greater than all of the numbers in this set.
The interval $(-\infty , 4)$ describes the set of all real numbers less than $4$. $-\infty $ tells us that there is no number less than all of the numbers in this set.

In interval notation, $\infty $ gets a parenthesis because it is not a real number and cannot be included in any set of numbers.

\[ \infty \in \mathbb {R} \]

True False

\[ \infty \notin \mathbb {R} \]

True False

\[ \infty + \infty = \infty \]

True False

\[ \infty \cdot \infty = \infty \]

True False

\[ \infty ^2 = \infty \]

True False

\[ \frac {1}{\infty } = 0 \]

True False

A Peek Ahead

With functions, we will investigate the relationships between sets of information (domain and range). We will be particularly interested in how the values in the range change compared to how domain values change.

One aspect of this is how the function values change as the domain values become unbounded and “approach” $\infty $.

We will need language to talk algebaically and rigorously about $\infty $. We have such language.

Our language is called limits and we will use limits extensively.

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more examples can be found by following this link
More Examples of Real-Valued Functions

2025-09-27 14:13:15