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
    • $x^3=x\times x \times x$
  • 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
    • $x^0 = 1$

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