Infinity Definition
As stated earlier, infinity is used to describe something that cannot be counted. Infinity is an idea of unbounded space, time or any other quantity. Mathematically, infinity defines something that is greater than natural numbers or counting numbers. So, for something that is endless we use the term infinite. Do note that infinity is not a number but is used as a number for the count or measure of something.
Let’s suppose you place two mirrors parallel to each other. Then the number of images formed in each mirror is something you cannot count as it is limitless. So we say that there are an ‘infinite’ number of images formed. Just give this a try and check for yourselves whether you’re able to count the number of images when two mirrors face each other!
Infinity Symbol
The infinity symbol, ‘∞’ was invented in the year 1657 by an English mathematician, John Wallis. The symbol, ‘∞’ is read as infinity.
Properties of Infinity
The most significant property of infinity is, -∞ < x < ∞, where x is a real number.
If we apply the basic arithmetic operations on infinity we can derive the properties of infinity.
- Addition Property – If any number is added to infinity the sum is infinity.
\(a + \infty = 8 \)
\(\infty + \infty = \infty \)
\(a + \left( – \infty \right) = – \infty \)
\(\left( – \infty \right) + \left( – \infty \right) = – \infty \)
Here, a is any integer.
Let’s take a few examples here:
\(2 + \infty = \infty , 3 + \left( – \infty \right) = – \infty\text{ }and\text{ }100 + \infty = \infty .\)
- Subtraction Property
\(a – \left( – \infty \right) = \infty \)
\(a – \infty = – \text{ }\infty \)
So, the value of \(1999 – \infty\text{ }is\text{ }- \infty .\)
- Multiplication Property – If any number is multiplied by infinity the product is infinity.
\( \infty \times \infty = \infty \)
\(\left( -\infty \right) \times \left( – \infty \right) = \infty \)
\(\left( -\infty \right) \times \left( \infty \right)\text{ }= – \infty \)
\(a \times \infty = \infty \) [If a > 0]
Example: \(11 \times \infty = \infty \)
\(a \times \left( – \infty \right)\text{ }= – \infty \) [If a > 0]
Example: \(190 \times \left( – \infty \right) = – \infty \)
\(a \times \infty = – \infty \) [If a < 0]
Example: \(-12 \times \infty = – \infty \)
\(a \times \left( – \infty \right) = \infty \) [If a < 0]
Example: \(-1 \times \left( – \infty \right)\text{ }= \infty \)
Undefined Operations
The operations in the table are undefined operations.
Infinity is useful in many areas of math and physics to describe something that is endless and cannot be counted or measured.
There are two types of infinity, potential infinity and actual infinity.
‘Infinite number’ describes a set of endless or never ending numbers.
A sequence of natural numbers would mean 1, 2, 3, 4, ……, 100,101,……… and so on. We see here that the sequence does not end. So the sequence of natural numbers is an example of infinite numbers.
The value of ⅓ in decimal form is 0.3333333………. So the value of ⅓ is not infinite, but the digit 3 in its value in the decimal repeats infinitely.