Maths 101 revision - Lecture 9

Math 101 – Lecture 8

Basic Arithmetic

The set of real numbers R has the following properties a+b = b+a  R for a, b  R  a+ 0 = 0 + a, the additive identity.

For each a ( 0) R, there exists b  R such that ab = 1 where 1 is the multiplicative identity a*1 = 1*a = a.

Order for any two real numbers a, b  R  only one of the following is true:

a > b or

a = b or

a < b

If a, b, c  R and a > c, c > b then a > b

            If a, b, c  R and a > b then a + c > b + c

If a, b, c  R and a > b, if c > 0 then ac > bc

From these properties, together with the deankind axiom all the familiar results for real numbers.

Eg Index Laws a^m a . a . a .... a, m  IN

                        {          m times  }

            Extend to a^-m = 1/(am) = 1(a . a . a .... a) =              1/a . 1/a . 1/a .... 1/a

                                                            {          m times  }        {          m times  }

            Extend to a^(p/q), p, q   Z

Eventually extend to a^-σ , σ   R

Absolute value – a ‘distance measure’ on R

|a| = {a, a  0

            -a, for a < 0

Theorem (properties of absolute values)

1.      |a|  0 for all a   R, note  |a| = 0 if any only if a = 0

2.      |ab| = |a| * |b|, for all a, b  R

Cases

(i)                 |ab| = ab = |a| |b|     ie ab > 0 for [(i) a, b > 0

(ii) a, b < 0

(ii)               |ab| = ab = (-a)(-b)

3.      |a|^2 = a^2

4.      |a+b|  |a| + |b|, a, b  R

Proof: If a+b<0 then |a + b| = -(a + b)

                                                = (-a)  b),

Now – a  |a|, for any a  R

If a + b > 0 then |a + b| = a + b

But a  |a| and b  |b|

 |a + b|  |a| + |b|

5.      |a – b|  |a| - |b|, for any a, b   R

-||-

Bounded Sets

Infinite sets : 1) Q

                  2) Prime numbers

                  3) Z

      4) {x   Q: -1  x  1}

4) is a subset of an interval of the real number line.

[a, b] = {x^(R): a  x  b} – a closed interval.

[a, b] = { x^(R): a  x  b }

Also have [a, b] and [a, b]

Bounded sets: - A set S  R is bounded above if thre is K  R, such that K  for all x    S

      S is said to be bounded below.

                  If there is k    R such that x   k for all x    S

                        S is bounded if it bounded above and below.

*The least upper bound (supremum) for a bounded set S  R is K  R such that K  x for some x    S and any  > 0

*The greatest lower bound (infinum) for is K  R such that [ k  x, x  S

                                                                                                      K +  > x for some x  S and any  > 0