Algorithms and Discrete Mathematics 20082009

1
Algorithms and Discrete Mathematics 2008/2009

• Lecture 4
• Exponentiation and logarithms I

Ioannis Ivrissimtzis
03-Nov-2008
2
Overview of the strand

• Combinatorics
• Counting principles
• Factorials
• Permutations
• Combinations
• Binomial theorem, Pascals triangle
• Exponentials and logarithms
• Big-O notation
• Induction and recursion

3
Overview of the lecture

• Exponentiation
• Integer powers
• Rational powers
• Exponential functions

4
Positive integer powers

• Recall that if b is a real number and n a
  positive integer, we have
• The number b is called the base and the n is
  called the exponent or
• the power.
• Example 4.1

5
Positive integer powers

• Proposition 4.1 If b is a real number and n,m
  are positive integers,
• we have
• Indeed,

6
Positive integer powers

• Proposition 4.2 If b is a real number and n,m
  are positive integers,
• we have

7
Positive integer powers

• Indeed,

8
Positive integer powers

• Propositions 4.1 and 4.2 are intrinsic properties
  of positive integer
• powers.
• Aim generalize exponentiation in a way that
  Propositions 4.1 and 4.2
• still hold.

9
Zero power

• For a real number b?0 we define
• Notice that we did not define 00.

10
Negative integer powers

• Let b?0 be a real number and let n be a positive
  integer. We define
• This definition is a result of Proposition 4.1.
  Indeed, assuming
• Proposition 4.1, we have
• giving,

11
Negative integer powers

• Example 4.2

12
Overview of the lecture

• Exponentiation
• Integer powers
• Rational powers
• Exponential functions

13
Rational powers

• Let b0 be a real number and let n be a positive
  integer. We define
• as the n-th root of b. That is, is a
  real number x with the property
• We can also write instead of .
• This definition is a result of Proposition 4.2.
  Indeed, assuming
• Proposition 4.2, we have

14
Rational powers

• Example 4.3
• because
• because

15
Rational powers

• When bgt0 and n is even, the equation
• may have more than one real solution.
• For example, the equation
• has solutions 5 and -5.
• By convention we consider the positive solution
  as the value of the n-th
• root of b. E.g.

16
Rational powers

• Notice that we also assumed b0.
• If n is an odd integer we can extend the
  definition of the n-th root to
• negative bases b because the equation
• still has real solutions.
• For example,
• because

17
Rational powers

• Going one sep further we define rational
  exponents.
• If bgt0 is a real number and n,m are positive
  integers we have
• In some cases, the definition can be extended to
  include negative and
• zero values of b,n or m. Classifying all the
  possible cases is a
• straightforward process. It all depends on the
  real solutions of the
• equation

18
Rational powers

• Example 4.3
• Notice that even though we defined rational
  powers, we didnt give an
• algorithm to compute them.

19
Overview of the lecture

• Exponentiation
• Integer powers
• Rational powers
• Exponential functions

20
Exponential functions

• Because the set of rational numbers is a dense
  subset of the real
• numbers we can also define real exponents. That
  is, we can define bx
• for any positive real number b and any real x.
• The formal technique to do this is by taking the
  limit.
• That means that for any real exponent x, we can
  always find a rational
• n/m very close to x, so that bn/m is also very
  close to bx.

21
Exponential functions

• For a fixed positive real number b the function
• is called the exponential function for base b.
• The exponential functions are defined over the
  set of real numbers.
• Their values are positive real numbers.
• Remark Do not confuse exponential functions like
  2x , 3x with
• polynomials like x2 and x3.

22
Exponential functions

• The exponential functions
• are everywhere positive
• At zero their value is 1.
• For bgt1 they increase
• monotonically. They grow
• fast.

1
0
23
Exponential functions

• Proposition 4.3 Let a,b,x,y be real numbers,
  with a,bgt0. We have

24
Exponential functions

• Proposition 4.4 Let x,y,b be real numbers, with
  bgt1. We have
• We say that the exponential function with bgt1
  increases monotonically.

25
Exponential functions

• Example 4.4 We want to compare the growth rate
  of the functions
• The first function is exponential while the
  second is a polynomial. The
• table shows the values of the two functions for
  some small integers.

26
Exponential functions

• However, as x becomes larger the exponential
  function will grow faster
• than the polynomial and finally it will overtake
  it.
• Indeed,