**19 The Natural Logarithm**

**19.1.** The irrational number

e = 2.7182818284590451 ...

is most often used as the base for the exponential function. The exponential

function base e is called the **natural exponential **and the inverse
function

is called the **natural logarithm**. We shall explain what is natural about

the natural logarithm in two ways.

**19.2. **First we note that the average rate of change
(see Section** 14**) for the

natural exponential on the interval [x, x + h] is given by

It is proved in calculus that the number e is
characterized by the fact that

the ratio is very close to one when h is
small. This means that

the average rate of change over a short interval [x, x + h] (the instantaneous

rate of change) is the value e^{x }of the natural exponential.

Now consider a quantity N governed by an exponential law

N = N_{0}a^{t}

and let r = ln(a) be the natural logarithm of the base a. Then a = e^{r} so

and the exponential law takes the form

N = N_{0}e^{rt}

Now consider the average rate of change of N over a short time interval

[t, t+ Δt] of duration Δt. Using the above formula with x = rt and h = r Δt

we get

which says that the instantaneous rate of change is Nr.
For example, if r =

0.06 this says that the instantaneous rate of change is 6% of N. (Remember:

"per" means divide, "of" means times, and "cent" means 100 so 6 % means

6/100 and 6% of N is 0.06N.) In summary

The instantaneous rate of change of a quantity
governed by the exponential law N = N _{0}e^{rt} is 100r% (of N). |

**19.3. **The second way of understanding what is natural
about the natural

exponential function is to consider a bank account which earns a nominal

interest rate of (say) 6% per year. The interest rate is called nominal because

the actual amount of interest paid depends on how often the bank pays the

interest. An interest rate of 6% per year is the same as an interest rate of

6%/12 = 0.5% per month but if the bank pays the interest every month, the

account will grow by slightly more than 6% in a year. Here is how a few

different banks might pay interest.

(i) The Alaska Bank pays interest compounded annually. This means that

every year it looks at the balance in an account and adds 6%. Thus an initial

deposit of B_{0} dollars grows to

B = B_{0}(1.06)^{t}

dollars after t years (assuming no other deposits are made during this period).

(ii) The Minnesota Bank pays interest compounded monthly. This means

that every month it looks at the balance in an account and adds 6/12% =

0.5%. Thus an initial deposit of B_{0} dollars grows to

B = B_{0}(1.005)^{12t}

dollars after 12t months = t years (assuming no other deposits are made

during this period.)

(iii) The Wisconsin Bank pays interest compounded weekly. This means

that every week it looks at the balance in an account and adds 6/52% =

0.001154%. Thus an initial deposit of B_{0} dollars grows to

dollars after 52t weeks = t years

.

(iv) The Delaware Bank pays interest compounded daily. This means that

every day it looks at the balance in an account and adds 6/365%. Thus an

initial deposit of B_{0} dollars grows to

dollars after 365t days = t years.

The general formula for the balance B in a bank account
after t years if

the balance is initially B_{0}, the interest rate is r per year, and the interest
is

**compounded** (i.e. paid) m times per year is

In all cases the formula is of the form

B = B_{0}a^{t
}

where a = (1 + r/m)^{m} an the compounding period is 1/m years i.e. the

interest is compounded m times per year, When m becomes infinite we say

that the interest is **compounded continuously** and in calculus it is proved

that this formula works with a = e^{r} so the balance after t years is given by

B = B_{0}e^{rt}

The following table shows the values of for
t = 2, r = 0.05,

n = mt, and various values of the number m of compounding periods per

year.

Thus an account with a starting value of $1000 at 5% per
year will, in two

years, earn $102.50 in interest if the interest is compounded annually and

$105.17 in interest if the interest is compounded continuously.

**Remark 19.4. **Presumably in the old days the interest was credited to the

account at the the end of each compounding period and if the account was

closed in the middle of a compounding period the amount of
interest earned

since the beginning of the compounding period was forfeited. Today most

banks would use the formula
for all values of t not just

when mt is an integer. Of course, the contract signed when the account was

opened will make this precise.

**19.5. **The doubling time of a quantity N = N_{0}e^{rt} which is increasing

exponentially is the time t such that N = 2N_{0}. Since

2N_{0} = N_{0}e^{rt} => 2 = e^{rt} => ln 2 = rt

the doubling time is t = (ln 2)/r. Similarly, the **half life** of a quantity N =

N_{0}e^{rt} which is decreasing exponentially is the time t such that N = N_{0}/2,

i.e. t = -(ln 2)/r.

**20 Complex Numbers**

**Definition 20.1.** The **complex numbers** are those numbers of form

z = x + iy

where x and y are real numbers and i is a special new number called the

**imaginary unit** which has the property that i^{2} = -1:The real number x

is called the **real part** of z and the real number y is called the **imaginary**

**part **of z. Two complex numbers are equal iff their real parts are equal and

their imaginary parts are equal. The complex number

is called the **conjugate** of z.

**20.2.** The arithmetic operations are performed by treating i as a variable

and then replacing i^{2} by -1 if necessary. This makes clear that the com-

mutative, associative, and distributive laws hold, that the additive inverse

is so that the law z + (-z) = 0, and that the

rule for multiplying complex numbers is

zw = (x + iy)(u + iv)

= xu + (xv + yu)i + yvi^{2}

= xu + (xv + yu)i - yv

= (xu - yv) + i(xv + yu)

where z = x + iy and w = u + iv and x y, u, v are real.
Using the algebraic

law (a + b)(a - b) = a^{2} - b^{2} we see that the product of a complex number

and its conjugate is

which is a positive real number if z ≠ 0. Hence the multiplication inverse is

defined by

**20.3.** Conjugation satisfies the following laws.

1. A complex number is real if and only if it is equal to its conjugate:

x + iy = x - iy y = 0
z = x.

2. The conjugate of the conjugate is the original number.

3. Doing an arithmetic operation and then taking the conjugate gives the

same result as taking the conjugates first and then doing the operation:

These are all easy to prove. For example if
,

then

**21 Division of Polynomials**

**Theorem 21.1 (Division Algorithm)**. Let p(x) and d(x) be polynomi-

als and assume that d(x) is not the zero polynomial. Then the are unique

polynomialsq(x) and r(x) such that

p(x) = d(x) · q(x) + r(x)

and either r(x) is the zero polynomial or the degree of the remainder r(x) is

less than the degree of the divisor d(x).

Proof. Guided Exercise.

**Remark 21.2. **An analogous statement holds for integers. Let p and d be

be integers with d > 0. Then there are unique integers q and r with

p = d · q + r,

0 ≤ r < d.

For example,

23 = 7 · 3 + 2,

0 ≤ 2 < 7.

In the case of rational numbers (as opposed to rational functions) the condi-

tion that the degree of r(x) is less than the degree of d(x) is replaced by the

condition that the remainder is less than the divisor. In fact there is a very

strong analogy between integers and rational numbers on the one hand and

polynomials and rational functions on the other.

**22 The Fundamental Theorem of Algebra
Theorem 22.1 (Fundamental Theorem of Algebra).** A polynomial equa-

tion (of positive degree) has a complex root.

and is not ordinarily taught in undergraduate courses. The theorem is true

even if the coefficients in the polynomial are complex. In this section we shall

see what the theorem means for

real coefficients.

by (x - r), the remainder is f(r)

Proof. By the Division Algorithm (Theorem

f(x) = (x - r)q(x) + c (* )

where c is a polynomial of degree less than the degree of the divisor (x - r).

But the degree of the divisor (x - r) is one so the degree of c is zero, i.e. c

is a constant. Now plug in x = r to get f(r) = (r - r)q(r) + c = 0 + c = c

so (* ) becomes f(x) = (x - r)q(x) + f(r).

then (x - r) evenly divides f(x), i.e. f(x) = (x - r)q(x).

Proof. By the Remainder Theorem just proved we have

f(x) = (x - r)q(x) + f(r).

Hence if f(r) = 0, then f(x) = (x - r)q(x).

**Theorem 22.5 (Complete Factorization).** Let f(x) be a polynomial of

degree n. Then

where c is a constant and are the roots of f(x), i.e. the
solutions

of f(x) = 0.

Proof. By the Fundamental Theorem of Algebra, there is a complex number

such that so by the Factor Theorem just proved there is a

polynomial such that

Now is a polynomial of degree n - 1 so by the same argument there is

a complex number and a polynomial of
with

and hence

Repeating n times gives

where has degree 0, i.e. c is constant.

**Corollary 22.6.** Every real polynomial may be written as a product of real

linear polynomials and real quadratic polynomials with negative discriminant.

Proof. If the coefficients of f(x) are real and r is a complex but non real root

of f(x) then so the conjugate
is also a root. Let p and

q bet the real and imaginary parts of r so r = p + qi and
. The

and = p^{2} + q^{2}. The quadratic polynomial

has negative discriminant b^{2} - 4ac = 4p^{2} - 4(p^{2} + q^{2}) = -q^{2}. Thus the non

real complex roots occur in pairs in the complete factorization and may be

combined to give the desired factorization into real (quadratic and linear)

polynomials.