As we are approaching the end of our Math Symbols and Their Meanings series, we are also wrapping up the difficult side of mathematical language. Last time we dealt with Linear and Multilinear Algebra which mostly applies to high-dimensional geometry. In this article, we return to numbers, more specifically to infinite numbers and their operations, part of set theory and calculus.
Infinity concept in mathematics
The concept of infinity “∞” in mathematics holds various meanings:
- Endless sets of numbers;
- An indefinitely great number or amount;
- A value greater than any finite numerical value;
- A distant ideal point at which two parallel lines are assumed to meet;
The history of this endless concept can be traced back to Pythagoras in geometry, however, it was approached as a more definite theory by Georg Cantor. He came up with the idea that infinity is much more complex than a neverending story: there are different sizes of infinity when it comes to the set theory of natural numbers. Before getting into this, let’s look at the fundamentals of infinite numbers.
Basic symbols for infinite number sets
You can see this symbol often in set theory. It means that the set of numbers continues indefinitely.
This symbol used independently indicates the sum of the number set, known as the Sigma notation.
These symbols indicate the sum of infinite arithmetic series which is known as the GP (Geometric progression)
These letters stand for Geometric Progression. It means that every number holds a constant ratio to the precedent number in a set. In order to determine the next number, we have to multiply with a common ratio. For example, we have the following set:
GP = 2,4,6,8,10,…
The common ratio would be number 2. This helps us determine the next numbers in the set. A common ratio can be any given number, negative or positive integer. Geometric Progression can be either finite or infinite.
The GP formula is: where a is the first number of the series, r is the common ratio and n is the unknown value.
Convergence refers to a property of a number or a series of numbers approaching a limit as a variable of the function or a number of a set increases or decreases.
A set or a series is the sum of numbers/terms in an infinite sequence of numbers and is noted with “S”.
A series is convergent if the sequence of its partial sums approaches a limit. For example, the sign of reciprocals (reciprocal means an expression which when multiplied by another expression, gives unity-1 as a result) of powers of 2 give a convergent series:
A divergent series is still an infinite series but where the infinite sequence of the partial sums of the series does not have a finite limit.
A transfinite number is an infinite number in the sense that is far larger than a number but it is not absolutely infinite. There are cardinal transfinite numbers and ordinal transfinite numbers. The concept of transfinity was invented in mathematics by Georg Cantor.
This symbol indicates the “aleph”, meaning the cardinality of any set that can be matched with the integers.
Variations of this symbol are: “ ℵ0 “, read as aleph-null which is standard for the cardinal number of ℕ, while “ 𐡠 ”aleph is sometimes used for the set of real numbers.
Formula: n < ℵ0 for each n ∊ ℕ and ℵ0 < ℵ.
“ ω+1 “
This symbol indicates hyperreal numbers and it is read like: infinity plus one.
There are also “surreal” numbers which indicate an ordered, proper class with real numbers including infinite and infinitesimal numbers, larger or smaller than any positive real number. But enough of confusing infinite numbers that grow and continue forever.
Let’s see how the two most prominent figures when it comes to infinite numbers, Aristotle and Georg Cantor, classified infinite series.
Aristotle’s infinite series
1) Convergent series by division
In Aristotle’s mathematical language, division means subtraction. Therefore, given the set of numbers:
, where the series approaches a point or a magnitude/value lesser than A.
2) Convergent series by addition
We have the sets A, B and C:
S = A, A0,… , An, … ,
S = Bo , …. ,Bn+1 , where Bn+1 is subtracted from each An,
Therefore, the S = Co, …., Cn, …. is convergent to A, but never exceeds it.
3) Non-convergent series by the addition which also includes the non-convergent series by division.
You can find a more detailed explanation of Aristotle’s concept of infinite series, here.
Georg Cantor’s infinite series
- Enumerable infinite series – defines a set of elements that can be paired with a natural number and form a countable set.
- Non-enumerable infinite series – defines the sets of rational and real numbers, because the sets of rational and real numbers will always be incomplete according to Georg Cantor’s findings.
If you want to dive deeper into Cantor’s interpretation of infinite series, check this out.
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