Negative Infinity: The Bottomless Pit of Numbers
What Exactly is Negative Infinity?
Imagine you have a number line that stretches forever in both directions. To the right, the numbers get bigger and bigger: 10, 100, 1,000,000... this is positive infinity ($ +\infty $). Now, look to the left. The numbers get smaller and smaller: -10, -100, -1,000,000... Negative infinity ($ -\infty $) is the idea of a point on this line that is further left than any negative number you could ever name. It's not a specific number you can reach; it's a concept that represents an endless, bottomless decrease.
Think of it like a bottomless pit. No matter how deep you go, you can always go deeper. Negative infinity is the mathematical way of saying "this value decreases without any lower bound." It is crucial to remember that $ -\infty $ is not a real number. You cannot do normal arithmetic with it like you would with 5 or -3. It follows its own special set of rules.
Negative Infinity in Calculus and Limits
One of the most important places where negative infinity appears is in calculus, specifically when dealing with limits[1]. A limit describes the value that a function[2] approaches as its input approaches some value.
For example, consider the function $ f(x) = -x^2 $. What happens to $ f(x) $ as $ x $ gets larger and larger? Let's make a table to see the pattern:
| Value of $ x $ | Value of $ f(x) = -x^2 $ |
|---|---|
| 1 | -1 |
| 10 | -100 |
| 100 | -10,000 |
| 1,000,000 | -1,000,000,000,000 |
As $ x $ gets larger, $ f(x) $ becomes a larger and larger negative number. We say that the limit of $ f(x) $ as $ x $ approaches positive infinity is negative infinity. In mathematical notation, this is written as:
$ \lim_{x \to +\infty} (-x^2) = -\infty $
This is read as "the limit, as x approaches positive infinity, of negative x squared, is negative infinity."
Arithmetic with Negative Infinity
Since negative infinity is not a real number, you can't simply add, subtract, multiply, and divide it like one. However, in extended real number systems used in higher math, we define some specific rules to handle it. These rules are based on logic and the concept of unbounded behavior.
| Operation | Result | Explanation |
|---|---|---|
| $ -\infty + a $ (where $ a $ is a finite number) | $ -\infty $ | Adding any finite number to an infinitely large negative value is still infinitely negative. |
| $ -\infty \times a $ (where $ a > 0 $) | $ -\infty $ | Multiplying a large negative by a positive number makes it even more negative. |
| $ -\infty \times a $ (where $ a < 0 $) | $ +\infty $ | Multiplying a large negative by a negative number gives a large positive (a negative times a negative is positive). |
| $ -\infty + (-\infty) $ | $ -\infty $ | The sum of two infinitely negative values is still infinitely negative. |
| $ \frac{a}{-\infty} $ | 0 | Dividing any finite number by an infinitely large magnitude (positive or negative) gives a number infinitely close to zero. |
There are also operations that are undefined, like $ \infty - \infty $ or $ \frac{\infty}{\infty} $. These are called "indeterminate forms" because there isn't enough information to determine what the result should be.
Negative Infinity in the Real World
While you can't physically measure negative infinity, the concept helps us model and understand real-world situations where values can decrease without a known lower limit.
Example 1: Temperature
In theory, temperature can keep decreasing. Scientists have come very close to absolute zero ($ -273.15^\circ C $), but the concept of getting colder and colder without bound is analogous to approaching negative infinity on the temperature scale. If we imagine a theoretical scale where lower temperatures are possible without limit, we would be approaching negative infinity.
Example 2: Debt and Finances
Imagine a person who keeps spending more money than they have. Their bank account balance would go deeper and deeper into the negative. While in reality, there's a limit (like a credit limit), the idea of debt growing infinitely large is described by approaching negative infinity. If someone owed $ 1,000 $, then $ 10,000 $, then $ 1,000,000 $, and so on, we would say their net worth is approaching negative infinity.
Example 3: Elevation Below Sea Level
The deepest part of the ocean is the Mariana Trench, about 11,000 meters below sea level. We can represent that as -11,000 meters. Now, imagine a fictional story about a hole that goes endlessly deep into the Earth. The elevation of a point in that hole would be approaching negative infinity.
Common Mistakes and Important Questions
Q: Is negative infinity the same as zero?
No, they are completely different. Zero is a finite number that represents "nothing." Negative infinity represents an endless, unbounded decrease. For example, a temperature of $ 0^\circ F $ is cold, but a temperature approaching negative infinity is unimaginably colder than that.
Q: Can you have a number that is less than negative infinity?
By definition, no. Negative infinity is defined as being less than every finite number. If you think you have found something smaller, then that thing would actually be negative infinity, or our understanding of the context would need to change. It is the "bottom" of the conceptual number line.
Q: Why is $ \infty - \infty $ not zero?
Think of it this way. Imagine you have two piles of sand, both described as "endlessly large." If you take one endless pile away from the other, what's left? It could be another endless pile, or it could be nothing, or it could be a pile of any size! Without knowing exactly how the "endlessness" compares, the result is indeterminate. Similarly, $ \infty - \infty $ has no single, defined answer.
Footnote
[1] Limit: A fundamental concept in calculus that describes the value that a function or sequence "approaches" as the input or index approaches some value.
[2] Function: A relation between a set of inputs and a set of permissible outputs, where each input is related to exactly one output.