Infimum

Infimum is the greatest lower bound. A common example of infimum would be the largest negative number in a given series of numbers. For example, the infimum in 5, 6, -5,-8,-3 is -8. The word comes from ancient Greek and translates to “no less.”

The concept of infinity has continued to fascinate mathematicians for centuries because it can’t be solved by normal numerical processes and thus remains an unsolved problem. But what about a number that’s less than infinity? This paradoxical idea was first introduced by Zeno in his paradoxes (around 450 BC). Infimum notes will help in understanding the concept of Infimum along with its significance and examples.

What is Infimum?

Infimum can be simply defined as the lowest possible value in a set. It is a generalisation for any infinitesimal value that lies lower than any other, existing or non-existing in the set. The word, “Infimum” is derived from Latin for “bottommost”.

Definitions and Examples:

Earlier we defined 0 as an infinite series of natural numbers and an infinite number is a quantity that can be positive or negative.

Example: Let us consider the sequence of numbers 4, -4, 3, -3,-2,-1, 0 that is given below: -4 is the infimum of the above sequence. The sequence of negative numbers is called the natural numbers.

Definition: The smallest number which is strictly greater than any given number or expressions denoted by “x” (such as “x + 1”), is known as Infimum (INF).

How does Infimum relate to other concepts?

Infinite sets are always assumed to have an infinite number of values from which at least one Infimum can be found. In this case, it often appears that there are no finite values within the given set of variables. This assumption may not always hold true as some infinitesimal values may exist outside of an infinite set’s range and would not necessarily contribute to an Infimum. For example, a variable set can consist of all the positive integers from 1 – 10. The smallest value in this set would be 1. The largest value would be 10. The concept of Infimum is not applicable here as there are no values lower than 1 or higher than 10. 

Infimum is also related to the concept of Minimum and Maximum, which are two extreme points within a set that are connected by an imaginary line. The line forms a boundary along which values exist at either end with no finite points in between. However, an Infimum can exist in infinite quantities with no boundaries or lines separating them from one another and without requiring any imaginary lines to connect them with other extreme points within the same set.

Use of Infimum:

Mathematicians often use Infimum to calculate the lowest natural value or largest negative value in a given set. It is also useful in solving certain technical problems that require determination of extreme points within a specific range.

The most common application of Infimums is probably in calculating the area under a curve using Calculus, which requires that we first find the lowest and highest points on the function before being able to calculate the area beneath it.

Mathematicians also use Infimums to test one another in examination papers. A student who knows the concept of Infimum can guess the lowest and highest values in a group of numbers that require finding the infimum and is likely to fail without getting any right.

How Infimum is different from Supremum?

Infimum and supremum are similar concepts that have a different meaning in mathematics but they are still “named” after the same Greek word. Supremum is the greatest higher bound. In 6, there is no infimum because its highest value is 6. Infimum is the lowest value possible in a given set of numbers and is thus the largest lower bound. Example below. 

Thus, contrary to popular belief, infinitesimals are not infinitely small quantities while supremums are not infinitely large quantities. Further, there can be more supremums than infinitesimals. Infinitesimals and supremums are both infimums, just without any infinitesimal to limit the number of values. The names cannot help but be confusing. However, it is easier to think of them as actual numbers (infinitesimals) with a limited number of extreme points (supremums).

Infinity:

Infinity denotes a quantity that is greater than or equal to “c” and less than “d”. Infinity does not exist in the positive real numbers since there is no value greater than or equal to the positive infinity. The value of infinity can never be calculated since we do not know what it’s actually worth. It may have no meaning at all except that we are measuring something infinitely long.

Conclusion:

Infimum is a generalisation of Infinitesimal, which we have discussed earlier in this article. Infinitesimals are finite quantities, but they can still be thought of as infimums. The same system still applies to infinitesimals and supremums – the exact value that you choose to use defines which one you are thinking of. Infimum is the lowest possible number in a given series of numbers or expressions denoted by “x” (such as “x + 1”). In this definition, “x” can be any expression containing a finite set of values, such as natural numbers. A common example is to use 1 for the least value and 6 for the greatest value.