Notation is a symbolic system for the representation of mathematical items and concepts. Mathematics is a very precise language, and different forms of description are required for different aspects of reality. Mathematics’ reliance on notation is essential to the abstract concepts it explores.
For example, It is most appropriate to attempt to describe the lay of the land to someone who wants to find their way around places they are not familiar with by drawing a map instead of using text.
The concept of notation is designed so that specific symbols represent specific things so communication can be effective. Let’s take these two sentences as examples. ‘The number of ways is only 4!’ is very different from ‘There are only 4 ways!’. The first sentence could be misleading since it implies 4 factorial (4!).
Types of notation
Notation is mainly made of letters, symbols, figures, and signs. Notation can use symbols, letters only, numbers only, or a mixture like the factorial symbol n!. Let’s look at some basic notation.
Counting notation
While studying maths, you are likely to come across the notation n!. This represents the factorial.
n! = 1 if n = 0
Otherwise \(n! = n \cdot (n-1) \cdot (n-2) \cdot (n-3) \cdot ... \cdot 3 \cdot 2 \cdot 1\)
n! counts the number of ways to arrange n distinct objects. So it is intuitive to know that when you have zero (0) objects, there is only one way to arrange them – do nothing.
Related to factorials is the binomial coefficient notation \(\Bigg(\begin{array} n n \\ k \end{array}\Bigg)\).
\(\Bigg(\begin{array} n n \\ k \end{array}\Bigg) = {^n}C_k = \frac{n!}{(n-k)!k!}\)
The formula above is a way to express the number of k subsets in an n set. So here we think of n as a non-negative integer and k as a non-negative integer which is less than or equal to n.
Set notation
This system is used to define the elements and properties of sets using symbols. We write down our sets as elements inside curly brackets.
For example, S = {1, 2, 3} is used to declare that 1, 2, and 3 are elements inside a set (S), whose elements are listed in the curly brackets.
We can have another scenario where S = {1, 2, 3, ......, n}.
Or write the same thing as \(S = {x | 1 \leq x \leq n}\)
The first expression states that a group named S contains the number from 1 to n.
The second expression states that a group named S is equal to the elements x such that x exists between 1 to n. The second expression says nothing about the number progression. The variable x can be any number between 1 to n such as 1.5, while in the first, 1.5 is not a member as the list jumps from 1 to 2.
There are a few symbols below we use when describing sets. The symbols apply left to right as the equal symbol, so a ∈ A will read “member a exists or is an element or the group / set A”
symbol | Meaning |
∈ | “Is a member of” or “is an element of”. |
∉ | “Is not a member of” or “is not an element of”, for example, “a is not a member of the group A”, as a ∉ A. |
{} | Denotes a set. Everything between the curly brackets belongs to the set. |
| | “Such that” or “for which” |
: | “Such that” or “for which” |
⊆ | “Is a subset of”, for example, “group B is a subset / belongs to group A”, as B ⊆ A. |
⊂ | “Proper subset”, for example, “B is a proper subset of A”, as B ⊂ A. |
⊇ | “Is a superset of”, for example, “B is a superset of A”, as B ⊇ A. |
⊃ | Proper superset, for example, “B is a proper superset of A”, as B ⊃ A. |
∩ | “Intersection”, for example, “B set intersection A set”, as B ∩ A. |
∪ | “Union”, for example, “B set union A set”, as B ∪ A. |
Numbers are not the only things that qualify as elements in sets. Pretty anything you want to talk about can. For example, if A = {a, b, c}, it can be written to denote that a is an element of the set A as a ∈ A. Sets themselves can be elements in other sets. We can use the notation {a, b} ⊆ A to note that {a. B} is a subset of A.
Summation notation
Summation notation is a convenient form to express long sums. For example, 1 + 2 + 3 + 4 + 5 could also be written as \(\sum^5_{i=1}{i}\). This means that we are summing up all the values of i starting from i = 1 until we get to i = 5, which is where we stop.
\[3^2 + 4^2 +5^2+6^2+7^2+8^2+9^2+10^2 = \sum_{n=3}^{10} n^2\]
Notice that plugging in the values of n should give you the answer you are looking for.
Pi notation
Pi notation is used to indicate repeated multiplication. It is also called product notation. This notation is quite similar to summation notation. An example is given below.
\[\Pi^N_{n = 5}(n^2-1) = (5^2-1)(6^2-1)...(N^2-1)\]
This reads the products from n = 5 to N, where N is larger than n.
Pi notation is also used to define the factorial n!
\[n! = \Pi^n_{i=1}i = (1)(2)(3)(4)...(n-1)(n)\]
Index notation
This form of notation in mathematics is used to denote figures that multiply themselves a number of times.
Using index notation 3 · 3 can be written as 32 which is the same as 9. 32 can be read as three to the power of two. In the expression “the number that is raised to the power of X”, X is the number of times that the base number multiplies itself.
Index notation is also useful to express large numbers.
The number 360 can be written in indices as either \(2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5\) or \(2^3 \cdot 3^2 \cdot 5\). Any number raised to the power 0 equals 1.
Qualities of notations
For notations to function, they need to possess certain qualities. These are discussed below.
Expressiveness: this means the clarity of notation. Correct notation should contain all relevant information in the exact manner that it should be used. For example, an index notation can be expressed as 42 which is the same as 4 · 4. Writing the notation but leaving out the power doesn’t make it the same as 4 · 4.
Brevity and simplicity: Notations are as brief and straightforward as possible. There is a chance mistakes may be incurred while writing long ones and considering the nature of precision they require to be valid, they need to be easy to read, pronounce and write.
Notation - key takeaways
- Notation is a symbolic system for the representation of mathematical items and concepts.
- The concept of notation is designed so that specific symbols represent specific things and communication is effective.
- Index notation in mathematics is used to denote figures that multiply themselves a number of times.
- Notation contains all relevant information exactly as it should be used.
- Notations are mostly as simple as possible.
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