Madison, SD, USA
Dakota State University
Madison, SD, USA
Math Skills
This page provides a review of math skills needed to perform financial calculations. The list of topics will grow as student questions prompt the addition of new topics. Students should feel free to suggest topics to be added.
Exponents
Exponents appear frequently in time value calculations.
Example: FV = PV(1 + i/m)^mt
where FV = future value
PV = present value
i/m = interest rate per period
^ = indicates an exponent
t = time in years
m = number of compounding periods per year
If you are calculating the FV, then the exponent will be
positive because the PV will grow with time, i.e., you are
compounding forward.
If you are calculating the PV, then the formula can be
rearranged to:
FV(1 + i/m)^(-mt) = PV
and the exponent will be negative because you are
calculating the starting value (PV) which will grow to become
the final value (FV), i.e., you are discounting
back to the present.
Consider the following series:
10^3 = 1000 or 10 x 10 x 10 = 1000
10^2 = 100 or 10 x 10 = 100
10^1 = 10 or 10 = 10
10^0 = 1 or 10/10 = 1
10^(-1) = .1 or 1/10 = .1
10^(-2) = .01 or 1/(10x10) = .01
10^(-3) = .001 or 1/(10x10x10) = .001
So the exponent indicates the number of times the base (in
this case the 10) should be used as a factor, either
muliplying or dividing by it.
(The exponent 0 can be thought of as 1-1 or two exponents,
one saying to multiply by the factor and one saying to divide
by the factor. Therefore any number raised to the
0 power = 1.)
In time value problems compounding examples would be:
a) 8.5% for two years compounded annually:
(1.085)^2 = 1.177225 [an interest factor]
b) 6.0% for three years compounded monthly:
(1+ .06/12)^12*3 = (1.005)^36 = 1.196680525
The latter calculation is done easily on a calculator using
the y^x button: Enter 1.005, push the y^x button, enter
36, then press the = button.
Notice that compounding forward (using a positive exponent)
produces an interest factor greater than one.
In time value problems discounting examples would be:
a) discounting back two years at 8.5% compounded annually:
(1.085)^(-2) = .849455287 [an interest or discount factor]
b) discounting back three years at 6.0% compounded monthly:
(1+ .06/12)^(-12*3) = (1.005)^(-36) = .835644919
Negative exponents can be used with the y^x button
on calculators as easily as positive exponents.
Notice that discounting backwards (using a negative exponent)
produces an interest factor (discount factor) less than one.
Numbers
Imaginary numbers are numbers that cannot be plotted on a number line. They have as a factor the square root of minus one, i.
Integers include whole numbers and the negatives of the natural numbers, e.g., ... -3, -2, -1, 0, 1, 2, 3, ...
Irrational numbers are numbers that cannot be expressed as the quotient of two integers. Pi and the square root of two are examples. Irrational numbers have decimal equivalents that are nonterminating and nonrepeating.
Natural numbers are the counting numbers that people would come up with naturally, e.g., 1, 2, 3, 4, ...
A number line consists of a plot of all real numbers. A number is larger than any number to its left on a number line.
Rational numbers are numbers that can be expressed as the quotient of two integers. Rational numbers have decimal equivalents that are terminating or repeating, e.g., 1/2 and 1/3 are both rational, the former having a terminating decimal equivalent and the latter a repeating decimal equivalent.
Real numbers are numbers that can be plotted on a number line.
Whole numbers are the natural numbers plus the number 0. Zero is a much later invention than the natural numbers.
Significant
Digits
Significant digits are used to indicate the precision* (as opposed to accuracy*) with which a number has been calculated. Very commonly financial calculations are rounded off to the nearest cent or to, say, three significant digits. The following are some guidelines for determining the number of significant digits in a number. (How many significant digits to use in a calculation on a test is usually indicated in the instructions for the test.) 1. Non-zero digits are significant. 2. Leading zeros in decimal fractions are not significant; they are simply being used to indicate the order of magnitude of the number and not the precision of the number. Zeros which disappear when the number is written in scientific notation are not significant. 3. Zeros between non-zero digits are significant. 4. Trailing zeros are significant if a decimal point is used. They are not significant if a decimal point is not used; they are being used to indicate order of magnitude. Here are some examples with the number of significant digits indicated: One 4 One 40 One .4 One .04 One .0000004 Two 42 Two 40. Two .40 Two .040 Two .00000040 Three 425 Three 400. Three .400 Three .000000400 Four 4001 Four 4000. Four .4000 Four .0000004000 Five 12.005 Five 12.050 One $4,000,000,000,000,000 Sixteen $4,000,000,000,000,000. *Precision is an indication of how closely repeated measurements (calculations) of the same value fall to each other. Accuracy is an indication of how closely repeated measurements of the same value fall to the true value. A measurement can be very precise but very inaccurate. So, if you say your company earned $4,000,000 last quarter, you are also saying give or take a $million. If you say your company earned $4,000,000.00 last quarter, you are saying give or take a penny. And if you want to express the earnings to the nearest thousand dollars express it as $4.000 million.
College of Business & Information Systems
Dakota State University
Madison, SD 57042
Page Manager: Jim Janke
Contact at: jim.janke@dsu.edu
URL of this page: http://courses.dsu.edu/finance/math/mathskls.htm
Last update: August 2, 2000