Powers, logs, exponentials and complex numbers
Background mathematics review David Miller
Powers, logs, exponentials and complex numbers
Powers
Background mathematics review David Miller
Powers
Remember from elementary algebra
which is 2 to the power 3The 3 here can also be called the
exponent the power to which the number 2
is raised
32 2 2 2
Powers
Multiplying by the same number raises the power
Dividing by the same number lowers the power
Following this logicand
Generalizingany number to the power zero is 1
3 42 2 2 3 22 / 2 2
2 12 / 2 2 2 1 02 / 2 2 1
0 1x
Powers
Continuing
and so on for further negative powers
Generalizing, for any number x and any power a
0 11/ 2 2 / 2 2 2
1 22
1 1 1 1 12 / 2 22 2 2 2 4
1aax x
Reciprocal
is called the reciprocal of xIt becomes arbitrarily large in
magnitude as x goes towards zeroLoosely, it explodes at the originRigorously, it is singular at the origin
Negative powers generally have this property
11 xx
5
5
-5
-5
0x
Squares and square roots
Multiplying a number x by itself
is calledtaking the square
Because it gives the area of a square of side x
x
x Area
2x x x 2x x x
Squares and square roots
For some number xthe number that, when multiplied by itself gives x
is called the square root of x
e.g.,
is the radical sign
Also
x
x x x
0 1 2 3 4 5
0.5
1
1.5
2
2.5
x
x4 2
1/2x x
Square root
Note thatIf
So also 2 is the square root of 4
So also -2 is the square root of 4Conventionally, we presume we mean the
positive square root unless otherwise statedBut we always have both positive and negative versions of the square root
x x x x x x
Distance and Pythagorass theorem
Pythagorass theorem gives
or equivalently
where we always take the positive square root
so r is a distanceand is always positive
x
yr
22 2r yx 22r yx
Quadratics and roots
For a quadratic equation of form
the solutions or roots are
For , , andSo
Note
2 1 0 1 2 3
321
1
2
3
x
2 2x x
2 0ax bx c 2 4
2b b acx
a
2 2 0x x 1a 1b 2c 1 1 8 1 3 2 1
2 2or x
2 2 2 1x x x x Example of a
parabola
Powers of powers
To raise a power to a powerMultiply the powers, e.g.,
Generalizing
23 62 2 2 2 2 2 2 2
cb bca a
Powers, logs, exponentials and complex numbers
Logarithms and exponentials
Background mathematics review David Miller
Powers and logarithms
The inverse operation of raising to a power is
taking the logarithm (log for short)
With logarithms, we need to specify the base of the logarithm
Our example was2 to the power 3 is 8
The logarithm to the base 2 of 8 is 3
32 2 2 2 8 2log 8 3
Powers and logarithms
Generalizinga chain of n numbers g multiplied together is
andn is the log to the base g of gn
ng g g g
log ngn g
Powers and logarithms
Though we created these ideas using integer numbers of multiplications n
we can generalize to non-integersFor some arbitrary (real, positive, and
non-zero) number b, we can write
which meanslogga b
ag b
Powers and logarithms
We note, for example, that
Note that in multiplying we have added the exponentsGeneralizing
Equivalently, if we writeso
Then
multiplying numbers is equivalent to adding their logarithms
2 3 5 2 32 2 2 2 2 2 2 32 2 2 a b a bg g g
aA g bB glogg A a logg B b
log log ( )a bg gA B g g log a bg g a b log logg gA B and so
Bases for logarithms
When logarithms are used for calculationsTypically base 10 is used
Base 10 logarithms are often used by engineers in expressing power ratios
in practice using decibels (abbreviated dB)
which are 10 times the logarithm of the ratio
Bases for logarithms
E.g., for an amplifier with an output power Pout that is 100 times larger than the input power Pin
Gain (in dB)=10 log10(Pout/Pin)i.e.,
Gain (in dB)=10 log10(100)=10x2=20dB
E.g., for an amplifier with a gain of 2Noting that log102 0.301
Gain of times 2 (in dB)=10 log102 3dB
Changing bases of logarithms
Supposei.e.,
Now, by definition
So
So
Generalizing, and dropping parentheses and
10log b a 10ab 2log 1010 2
22 log 10log 102 2a ab 2 2 2 10log log 10 log 10 logb a b
log log logc c db d b
Bases for logarithms
Sometimes log means log10e.g., on a calculator keyboard
Another common base is base 2 (i.e., log2)e.g., in computer science because of binary numbers
Fundamental physical science and mathematics almost always uses logs to the base e
e is the base of the natural logarithms2.71828 18284 59045 23536e
Notations with e
Logs to base e are called natural logarithmsloge (sometimes just log) or ln
letter l for logarithm and letter n for natural
To avoid confusion with other uses of e e.g., for the charge on an electron
And to avoid superscript characterswe use the exponential notation
Also means this can be referred to as the exponential function
exp xx e
Exponential and logarithm
Exponential function For larger negative arguments
Gets closer and closer (asymptotes) to the x axis
For larger positive argumentsGrows faster and faster
LogarithmFor smaller positive arguments
arbitrarily large and negative
1 0.5 0 0.5 1
1
2
3
x
exp x
0 1 2 3
321
12
xln x
Exponentials and logarithms
Note all the following formulasWhich follow from the discussions above
exp exp expa b a b
1 exp
expa
a
ln exp a a
ln ln lnab a b
exp ln a a
ln 1/ lna a
Powers, logs, exponentials and complex numbers
Imaginary and complex numbers
Background mathematics review David Miller
Square root of minus one
In ordinary real numbersno number multiplied by itself
gives a negative resultEquivalently
There is no (real) square root of a negative number
If, however, we choose to define an entity that we call the square root of minus one
We can write square roots of negative numbersWe obtain a very useful algebra
2 2 4 2 2 4
Square root of minus one
DefineAlso, common engineering notation is
Any number proportional to i is calledan imaginary number
e.g., , ,Common to put the i after numbers,
but before variables or constants Can write the square root of any negative
number using i
1i 1j
4i 3.74i i
4 1 4 1 4 2 2i i
so 2 1i and 2 1i
Complex numbers
A number that can be written
where a and b are both real numbersis called a complex number
a is called the real part of g
b is called the imaginary part of g
g a ib
Rea g Imb g
Complex conjugate and modulus
The complex conjugate has the sign of the imaginary part reversed
And is indicated by a superscript *
Multiplying g by gives a positive numberCalled the modulus squared of g
The (positive) square root of this is called the modulus of g
g a ib g
2g g g gg
2g g
Important complex number identities
Note for the modulus squared
i.e.,
and for the reciprocal
i.e.,
Still a sum of real and imaginary parts
2g gg
1gc id
2 2 2g a b
2 2 2 2
1 c dg ic id c d c d
a ib a ib 2 2 2a iab iba i b 2 2a b
1 c idc id c id
2 2c id
c d 2 2 2 2
c dic d c d
Powers, logs, exponentials and complex numbers
Eulers formula and the complex plane
Background mathematics review David Miller
Eulers formula
Eulers formula is the remarkable result
A major practical algebraic reason for use of complex numbers in engineering
Exponentials much easier to manipulate than sines and cosines
exp cos sini i
Some results from Eulers formula
Using Eulers formula
Note that
so
Also
so
exp expi i
exp exp expi i i i
2 2 2
cos sin cos sin
cos sin cos cos sin sin
i i
i i i
exp cos sini i
exp expi i cos sini cos sini
exp 0 1
2 2cos sin 1
Complex exponential or polar form
For any complex numberwe can write
which we can write in the form
so any complex number can be written in the form
g a ib a ibg g
g
a
b22r ba
cos sing g i
expg g i
2 2 2 2
a bg ia b a b
Complex plane
Propose a complex planehorizontal real axisvertical imaginary axis
Then any complex number
is a point on this planeSometimes called an Argand diagram
Sometimes is called the argument of this polar version of a complex number,
real axis
imaginary axis
exp( )g i
2 2g a b
cosa g
sinb g g
expg a ib g i
arg g
.
Multiplication in polar representation
In the polar representationTo multiply two numbers
Multiply the moduli and add the anglesI.e., with
expg g i
exph h i exp expg h g i h i
expg h i
nth roots of unity
Note that the number when raised to the nth power is 1 (unity)
Many different complex numbers when raised to the nth power can give 1
But this specific one is conventionally called the nth root of unity
exp 2 /i n
2 2exp exp exp 2 1n
i i n in n
1 exp 2 /n i n
Algebraic results for complex numbers
All the following useful algebraic identities are easily proved from the complex exponential form
gh g h 1 1gh g h
g gh h
Sine and cosine addition formulas
Sine and cosine sum and difference formulas are easily deduced from the complex exponential form
e.g.,
Equating real parts gives
Equating imaginary parts gives
exp 2 cos2 sin 2i i
2 2cos2 cos sin sin 2 2sin cos
exp expi i 2cos sini 2 2cos sin 2 sin cosi