[Maths Class Notes] on Infinity Pdf for Exam

What is the Meaning of Infinity

In mathematics, “infinity” is the idea of describing something which is larger than what we know as natural numbers and is normally referred to as something with no limit. This concept of infinity is principally used in the domain of physics and maths. A practical example of the infinity concept is that no matter how long you count how precisely you count, you can never reach the end of all numbers, and hence you can’t even reach the end of the limitless number system. This kind of infinity is determined as potential infinity by Aristotle. It’s definitely present but you won’t be able to reach it.  

What is infinity?

Infinity is the concept of something that has no limits, endless. In general, infinity definition is something which has no boundation for a destination or having uncertain and endless state in the term of time, space, or other numerical concepts. 

There are Two Kinds of Infinity in Math:

• Potential infinity

• Actual infinity

Infinity Meaning

A set of numbers can be interpreted as infinite if there remains a 1-to-1 resemblance between the set and a proper subset of itself. 

Let’s take two parallel lines in the figure. 

We can clearly state that the meeting of the line is endless i.e. – an uncertain event whose final destination is not certain to us.

Similarly, we can represent in terms of 1/x, when x→0

The symbol x→ indicates that x increases without bound, and x→-∞ indicates that x decreases without bound.

More examples:   

{1, 2, 4……….} the sequence of the natural numbers is endless and is infinite.

1/3  when you divide the fraction you will find the repetition of the number which never ends i.e. 0.333333333………

The most influential thing about infinity is that:

  -∞

Where x is a real number

What Is The Value of infinity

Infinity is not a real number. As far as value is concerned, the infinity has no value. This is because a value needs to be defined and must be specific which is not possible in case of infinity. The truth is that infinity is not specific or certain.

Just consider all real numbers. You will find it quite interesting. We can say after 0 the next number is 0.1 or 0.000001. Hence it is not possible to count all the numbers between 0 and 1 because there will be infinite numbers in between them. After 0, we usually go on to 0.000…1. Now we can keep adding zero and never get to the next number after 0, as the number of real numbers in between 0 and 1 is infinite and uncountable. Hence, we can conclude that the set of all real numbers is one of uncountable infinity.

Examples Of Infinity

1. Value of ℼ or pi: ℼ= 3.1415926535……………………so on.

2. A set of all point on a number line

3. The set of leaves on a tree 

4. The set of an integer

5. A set of a whole number W= {0, 1, 2, 3, 4, 5, 6…….} 

Some More Properties of Infinity

          -∞ + -∞ = -∞

Eg-        ∞ × ∞ = ∞

             -∞ × -∞ =∞

             -∞ × ∞ = -∞

X + ∞ = ∞

X + (-∞) = -∞

X – ∞ = -∞

X – (-∞) = ∞

For x˃0: 

X × ∞ = ∞

X × (-∞) = -∞

For x ˂0: 

X × ∞ = ∞

X × (-∞) = ∞

0×∞

0×-∞

∞ + -∞

∞ – ∞

∞ / ∞

∞°

1∞

What is the Infinity Symbol        

John Wallis in the 17th century was the person who depicts the symbol of infinity in 1655 for the 1st time. In fact, this type of similar symbol was used by the Romans to represent a large number. Like 1000 was written in this form CIƆ which means “many”.

The infinity symbol ∞ is often called lemniscate and is a mathematical symbol that constitutes the concept of infinity. Sometimes it is used to represent a potential infinity, instead of representing the actual infinite number.

 

Fun Facts About the Infinity Symbol

1. In the early period, many theories and ideas are put forward for the contemplation of the symbol infinity but it was considered as a philosophical notion.

2. In mathematics, Calculus, Leibniz derived infinite numbers and their use in mathematics.

3. In real analysis, also the symbol infinity is an element to denote an unbounded limit, etc.

4. If we subtract infinity with another infinity, the answer can never be zero.