Math 2 Powers, Logs, Exponentials and Complex Numbers · PDF filePowers, logs, exponentials...

Powers, logs, exponentials and complex numbers

Background mathematics review David Miller


Powers


Powers

Remember from elementary algebra

which is “2 to the power 3”The “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 22 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

21 2

2

1 1 1 1 12 / 2 22 2 2 2 4

1aax

x

Reciprocal

is called the “reciprocal of x”It becomes arbitrarily large in

magnitude as x goes towards zeroLoosely, it “explodes at the origin”Rigorously, 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 called“taking 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 Pythagoras’s theorem

Pythagoras’s 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

3

2

1

1

2

3

x

2 2x x

2 0ax bx c

2 42

b b acxa

2 2 0x x 1a 1b 2c

1 1 8 1 3 2 12 2

or 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


Logarithms and exponentials


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 was“2 to the power 3” is 8

The “logarithm to the base 2 of 8” is 3

32 2 2 2 8

2log 8 3


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


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


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 b

g 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


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


Sometimes “log” means “log10”e.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 logarithms”2.71828 18284 59045 23536e

Notations with “e”

Logs to base “e” are called “natural logarithms”“loge” (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 expexp

aa

ln exp a a

ln ln lnab a b

exp ln a a

ln 1/ lna a


Imaginary and complex numbers


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 2

c idc d

2 2 2 2

c dic d c d


Euler’s formula and the complex plane


Euler’s formula

Euler’s 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 Euler’s formula

Using Euler’s 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 gg

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 plane”horizontal “real axis”vertical “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

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