Basic Math Skills for Introductory Finance
Topics Covered
- Exponents
- Sigma Notation for Addition
- Order of Operations
- Solving Equations
Exponents
Multiplication can be seen as a shorthand for addition:
$ 5 \times 3 = 3+3+3+3+3 $
$=5+5+5$
Exponents can be seen as a shorthand for multiplication:
$ 5^3 = 5 \times 5 \times 5 $
Properties of Exponents
A negative exponent can be made positive by taking the reciprocal of the base:
$x^{-2} = \dfrac{1}{x^2}$
We can add exponents when common bases are multiplied together:
$x^{5} \times x^2 = x^{5+2} = x^7$
Division is similar:
$ \dfrac{x^7}{x^7} = x^{7-7}= x^0 $
$= 1$
Summary of Exponents
- Exponents are like a shorthand for multiplication
- Exponent changes sign with the reciprocal of the base
- $x^{-2} = \dfrac{1}{x^2}$
- Add exponents when multiplying common bases
- $x^2 \times x^2 = x^{2+2} = x^4$
- Anything to the power 0 equals 1
Sigma Notation for Addition
Sometimes we might have a series of numbers:
$ 1, 2, 3, 4, 5, ..., 100.$
If we wanted to add these all up, this would be hard to represent:
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+...
Sigma notation makes this much easier:
$\sum_{i=1}^{100} x_i = 1 + 2 + 3 + 4 + 5 + ... + 100$
$x_i$ is often replaced by a function of $x$
Example of a series summation:
$ f(x) = x^2$
$\sum_{i=1}^{5} f(x_i) = \sum_{i=1}^{5} x_i^2 $
$ = 1 + 4 + 9 + 16 + 25 = 55 $
Summary of Sigma Notation
- Sigma Notation is a short hand for lengthy addition
- $\sum_{i=1}^{100} x_i = 1 + 2 + 3 + 4 + 5 + ... + 100$
- We can even add a series where a function is applied to $x$
- $\sum_{i=1}^{5} f(x_i) = \sum_{i=1}^{5} x_i^2 =1 + 4 + 9 + 16 + 25 = 55$
Order of Operations
- Math statements are evaluated in a systematic way
- PEMDAS
- Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
- Parentheses ( ) are evaluated first
- Addition and subtraction are evaluated last
- Operations go from left to right
Example
$ P = \dfrac{F}{(1+r)^t} $
Suppose we know $F=12$, $r=0.1$, and $t=2$. How can we evaluate this?
Example
$ P = \dfrac{12}{(1+0.1)^2}$ $= \dfrac{12}{(1.1)^2} $
$=\dfrac{12}{1.21}$
$\approx 9.92$
Order of Operations Summary
- Math statements are evaluated in a systematic way
- PEMDAS
- Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
- No matter how complicated, there is a unique way to evaluate mathematical statements.
Solving Equations
- Equations are statements of equality with unkown variables
- Our objective is usually to solve for the unkown variables
- We can solve as long as the number of unique equations equals the number of unkowns
- We will discuss how to solve a single equation with a single unknown variable
- Since equations are equalities, we can do whatever we like as long as we do it to both sides
- Our objective is usually to isolate the unknown variable
- This is considered a solution.
Example
- $ P = \dfrac{F}{(1+r)^t} $
- Suppose we know $P=100$, $F=300$, and $t=12$
- $r=?$
- $ 100 = \dfrac{300}{(1+r)^{12}}$ [multiply both sides by $(1 + r)^{12}$]
- $ 100 \times (1 +r )^{12} = 300 $ [divide both sides by 100]
- $(1+r)^{12} = \dfrac{300}{100}$ $=3$
- $(1+r)^{12} = 3$ [Raise both sides to $(1/12)$]
- $1+r = 3^{1/12}$ [Subtract $1$]
- $r = 3^{1/12}-1$ $=0.0959$
Equation Solving Summary
- Follow PEMDAS
- Manipulate both sides of the equation equally
- Isolate the variable of interest
- Be creative - don't worry if it's not immediately obvious