Infinity subtracted by any number is still infinity.
While this might seem counterintuitive, it's important to remember that infinity is not a number in the traditional sense. It's a concept representing something without any bound or limit. Therefore, subtracting a finite number from an unbounded quantity doesn't change its unbounded nature.
Understanding Infinity
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Not a Number: Infinity (often denoted as ∞) is not a real number you can perform standard arithmetic operations on like addition, subtraction, multiplication, or division.
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Represents Unboundedness: It represents a quantity that continues without end, greater than any finite number.
Why Infinity Minus a Number is Still Infinity
Imagine you have an infinitely long line. If you remove a segment of that line, even a very long segment, the line is still infinitely long. The segment you removed, however large, is finite compared to the infinite length of the line.
Example:
Let's say you have infinity (∞) and you subtract the number 1,000,000 from it. You would think you would have a value significantly lower than infinity. However, the result is still infinity:
∞ - 1,000,000 = ∞
This is because infinity is a concept of endlessness. Removing any finite number from something that is endless doesn't change its endless nature.
Important Considerations
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Different Infinities: There are different "sizes" of infinity, a concept explored in set theory. For example, the infinity of real numbers is "larger" than the infinity of natural numbers. However, these distinctions don't change the basic principle that subtracting a finite number from any infinity results in that same infinity.
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Limits: In calculus, limits involving infinity are used to describe the behavior of functions as their input grows without bound. These limits often involve indeterminate forms like ∞ - ∞, which require more sophisticated techniques to evaluate. The simple rule above (∞ - a finite number = ∞) doesn't apply directly to indeterminate forms.
In conclusion, when dealing with infinity in its basic conceptual form, subtracting any finite number from it leaves you with infinity. The boundless nature of infinity remains unchanged by removing something finite.
