page 8
e |
= 2.7182818 |
|
1 |
n |
= natural logarithm base |
= lim 1 + -- |
|
||||
|
|
n → ∞ |
n |
|
|
π |
= 3.1415927 |
= pi |
|
|
|
γ |
= 0.57721566 |
= Eulers constant |
|||
1radian = 57.29578° |
|
|
|
||
2.1.2 Basic Operations
•These operations are generally universal, and are described in sufficient detail for our use.
•Basic properties include,
commutative |
a + b = b + a |
|
distributive |
a(b + c ) = ab + ac |
|
associative |
a(bc ) = (ab )c |
a + (b + c ) = (a + b )+ c |
2.1.2.1 - Factorial
• A compact representation of a series of increasing multiples. n! = 1 2 3 4 … n
0! = 1
2.1.3 Exponents and Logarithms
• The basic properties of exponents are so important they demand some sort of mention
page 9
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
(xn )(xm ) = xn + m |
x0 |
|
|
|
|
|
|
|
|
|
-- |
|
|
|
|
|||||
= 1 , if x is not 0 |
xn |
= n x |
|
|||||||||||||||||
|
|
|
|
|
|
–p |
|
1 |
|
|
|
|
|
m |
|
|
|
|
||
|
n |
|
|
|
x |
= |
|
|
|
|
|
--- |
|
n |
|
m |
||||
(x |
) |
|
n – m |
|
|
---- |
|
|
|
|
x |
n |
= |
x |
||||||
|
= x |
|
|
|
|
x |
p |
|
|
|
|
|
|
|
||||||
---------- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
(xm ) |
|
|
|
|
|
n |
|
|
n |
|
n |
|
|
x |
|
n x |
||||
|
|
|
|
|
|
|
|
= (x |
)(y |
) |
|
= |
||||||||
|
|
m |
= xn m |
|
(xy ) |
|
|
n |
- |
------ |
||||||||||
(xn ) |
|
|
|
|
|
|
|
|
|
|
|
y |
|
n y |
||||||
• Logarithms also have a few basic properties of use, |
|
|
|
The basic base 10 logarithm: |
|
|
|
|
log x = y |
x |
= 10y |
The basic base n logarithm: |
|
|
|
|
lognx = y |
x |
= ny |
The basic natural logarithm (e is a constant with a value found near the start of this section: |
|||
|
ln x = loge x = y |
x |
= ey |
• All logarithms observe a basic set of rules for their application, |
|||
logn(xy ) = logn(x )+ logn(y ) |
|
logn(n ) = 1 |
|
x |
|
|
logn(1 ) = 0 |
|
|
||
logn - |
= logn(x )–logn(y ) |
|
|
y |
|
|
|
logn(xy ) = ylogn(x ) |
|
|
|
|
logm(x ) |
|
|
logn(x ) = ------------------ |
|
|
|
|
logm(n ) |
|
|
ln (A θ ) = ln (A )+ (θ + 2π k )j |
k I |
||