Notes for September 3
Real Analysis

1. Integers

2. Rational numbers

3. Irrational numbers

4. Algebraic numbers

5. Transcendental numbers

6. Real numbers

7. Countable sets

8. Denumerable sets

a. Countability of the rationals

9. Uncountable sets

a. Uncountability of the reals

b. Uncountability of the irrationals

c. Countability of the algebraic numbers

d. Uncountability of the transcendental numbers

10. The complex numbers

a. The complex exponential function

Complex exponentials:

Proof 1: Use Taylor’s series to show

Proof 2: Using calculus

Define the (possibly complex) function , of real variable x, as

.

The derivative of ƒ(x), according to the product rule, is:

Therefore, must be a constant function in x. Because we know, then constant that equals for all real x is also known. Thus,

Multiplying both sides by e^ix and using

,

it follows that

.

b. Roots of unity

11. The complex plane

a. The algebraic construction of the complex numbers

12. Hamiltonians and quaternions () – a non-commutative division algebra over the reals.

13. Cayley numbers, or octonions: consider either a basis of (1, i, j, k, l, il, jl, kl) with and

1	i	j	k	l	il	jl	kl
i	–1	k	–j	il	–l	–kl	jl
j	–k	–1	i	jl	kl	-l	–il
k	j	–i	–1	kl	–jl	il	–l
l	–il	–jl	–kl	–1	i	j	k
il	l	–kl	jl	–i	–1	–k	j
jl	kl	l	–il	–j	k	–1	–i
kl	–jl	il	l	–k	–j	i	–1

Or as pairs of quaternions (a,b) and (c,d) with (a,b)(c,d) = (ac-d*,da+bc*). This is a non-associative algebra over the reals.

14. Sedonions – pairs of octonions – weaker operation than the octonions.