# Appendix: Mathematics Review

A certain familiarity with certain mathematical concepts will help you when trying to analyze algorithms. This section is meant as a review for some commonly used mathematical concepts, notation, and methodology.  Where possible analogies between mathematical and programming concepts are drawn

## Mathematical Notations and Shorthands

### Shorthands

| shorthand             | meaning                                                                                      |    |                     |
| --------------------- | -------------------------------------------------------------------------------------------- | -- | ------------------- |
| iff                   | if and only if                                                                               |    |                     |
| $$\therefore$$        | therefore                                                                                    |    |                     |
| $$\approx$$           | approximately                                                                                |    |                     |
| $$ab$$                | a \* b                                                                                       |    |                     |
| $$a(b)$$              | a \* b                                                                                       |    |                     |
| $$                    | a                                                                                            | $$ | absolute value of a |
| $$\lceil{a}\rceil$$   | ceiling, round aa up to next biggest whole number. Example: $$\lceil{2.3}\rceil = 3$$        |    |                     |
| $$\lfloor{a}\rfloor$$ | floor, round aa down to the next smallest whole number. Example: $$\lfloor{2.9}\rfloor = 2$$ |    |                     |

### Variables

In math, like programming, we use variables. Variables can take on some numeric value and we use it as a short hand in a mathematical expression. Before using a variable, you should define what it means (like declaring a variable in a program)

#### For example:

"Let ***n*** represent the size of an array"

This means that the variable n is a shorthand for the size of an array in later statements.

### Functions

Similar to functions in programming, mathematics have a notation for functions. Mathematically speaking, a function has a single result for a given set of arguments. When writing out mathematical proof, we need to use the language of math which has its own syntax

As a function works with some argument, we first define what the arguments mean then what the function represents.

**For example:**

Let $$n$$ represent the size of the array  - (n is the name of the argument). 

Let $$T(n)$$ represent the number of operations needed to sort the array - T is the name of the function, and it accepts a single variable $$n$$ 

We pronounce $$T(n)$$ as "T at n". Later we will assoicate $$T(n)$$ with a mathematical expression that we can use to make some calculation. The expression will be a mathematical statement that can be used to calculate the number of operations needed to sort the array. If we supply the number 5, then $$T(5)$$ would be the number of operations needed to sort an array of size 5

**Summary**

$$T(n)$$ - read it as ***T at n***, we call the function $$T$$ .

$$T(n) = {n^2} + n + 2$$ means that $$T(n)$$  is the same as the mathematical expression $${n^2} + n + 2$$.  Think of $$T(n)$$ as being like the function prototype, and $${n^2} + n + 2$$ as being like the function definition

$$n$$ can take on any value (unless there are stated limitations) and result of a function given a specific value is calculated simply by replacing n with the value

$$T(5) = {5^2} + 5 + 2 = 32$$ ( we pronounce T(5)  as "T at 5")

{% hint style="danger" %}
When we talk about big-O notation (and related little-o, theta and omega notation) those are NOT functions.  For example O(n) is NOT a function named O that takes a variable n.  It's meaning is different. 
{% endhint %}

### Sigma Notation

Sigma notation is a shorthand for showing a sum. It is similar in nature to a for loop in programming.

**General summation notation.**

$$\sum\limits\_{i=1}^{n} t\_i = t\_1 + t\_2+ ... + t\_n$$ 

The above notation means that there are n terms and the summation notation adds each of them together.

Typically the terms $$t\_i$$,​​ is some sort of mathematical expression in terms of $$i$$ (think of it as a calculation you make with the loop counter). The $$i$$ is replaced with every value from the initial value of 1 (at the bottom of the $$\sum$$ ), going up by 1, to n (the value at the top of the \sum∑)

**Example:**

$$\sum\limits\_{i=1}^{5} i = 1 + 2 + 3 + 4 + 5$$ 

### Mathematical Definitions and Identities <a href="#mathematical-definitions-and-identities" id="mathematical-definitions-and-identities"></a>

Mathematical identities are expressions that are equivalent to each other. Thus, if you have a particular term, you can replace it with its mathematical identity.

### Exponents

#### **Definition**

​​ $${x^n}$$ means $$(x)(x)(x)...(x)$$ (n x's multiplied together)

#### I**dentities**

$${x^ax^b}={x^{a+b}}$$ <br>

$$\frac{x^a}{x^b} = {x^{a-b}}​$$ <br>

$${({x^a})^b} = {x^{ab}}$$

$${x^a}+{x^a} = 2{x^a} \neq {x^{2a}}$$ <br>

$${2^a}+{2^a} = 2({2^a}) = {2^{a+1}}$$&#x20;

### Logarithms

{% hint style="info" %}
In computer text books, unless otherwise stated $$log$$ means $$log\_2$$ ​​ as opposed to $$log\_{10}$$ ​​ like math text books
{% endhint %}

#### Definition

$${b^n} = a$$ iff $$log\_ba = n$$ In otherwords $$log\_ba$$ is the exponent you need to raise $$b$$ by in order to get $$a$$&#x20;

#### Identities

$$\log\_ba = \frac{\log\_ca}{\log\_cb}$$ , where $$c > 0$$ <br>

$$\log {ab} = \log a + \log b$$ <br>

$$\log (\frac{a}{b}) = \log a - \log b$$ <br>

$$\log {a^b} = {b}{\log a}$$&#x20;

$$\log x < x$$  for all $$x > 0$$&#x20;

$$\log 1 = 0$$&#x20;

$$\log 2 = 1$$&#x20;

### Series

A series is the sum of a sequence of values. We usually express this using sigma notation (see above).

#### I**dentities**

$$\sum\limits\_{i=0}^{n} c(f(i)) = c \sum\limits\_{i=0}^{n}f(i)$$ ​, where $$c$$ is a constant

$$\sum\limits\_{i=0}^{n} {2^i} = {2^{n+1}} - 1$$&#x20;

$$\sum\limits\_{i=0}^{n} {a^i} = \frac{a^{n+1}- 1}{ a - 1}​$$&#x20;

$$\sum\limits\_{i=0}^{n} {a^i} \leq \frac{1}{a-1}$$&#x20;

$$\sum\limits\_{i=1}^{n} {i} = \frac{n(n+1)}{2}​$$&#x20;

$$\sum\limits\_{i=1}^{n}{i^2} = \frac{n(n+1)(2n+1)}{6}$$&#x20;

$$\sum\limits\_{i=1}^{n}{f(n)} = nf(n)​$$&#x20;

$$\sum\limits\_{i=n\_0}^{n} f(i) = \sum\limits\_{i=1}^{n} f(i) - \sum\limits\_{i=1}^{n\_0 - 1} f(i)$$&#x20;


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