- Book is driven by a 'desire to present calculus not as a prelude, but a first real encounter with mathematics'.

- Precision and rigour shouldn't be deterrents - they are the natural medium for formulating mathematical questions

- 29 Chapters, with 2 'starred' as optional

- Presented as an *evolution of ideas*, rather than disjointed topics.

- 625 problems total (from 1st edition), additional 160 with 2nd edition, and even more with the third.

>Read: I will not be finishing this entire book in a semester with concurrent study...

#Chapter 1: Basic Properties of Real Numbers

**not a review, a condensation!**

Addition can be thought of as an operation performed on a pair of numbers.

(1)

we may construct all of addition, for any length sum (finite or infinite) by considering a pair of numbers being added at a time, and "piling them up". For example, we can define the sum five different ways;

Similarly, there are different possible ways of summing numbers...

(2)

(3)

>Or in other words, is important.

If a number satisfies , then .

\proof

\qed

Moreover, subtraction is simply addition rephrased:

(4)

This shouldn't be taken for granted for all operations. Indeed, not all are so

well behaved.

\likeremark{Example}

in general. In particular, iff . Curiously, we need

multiplication to fully explore this.

\qef

Now onwards to multiplication!

\likeremark{Remark}

(5)

(6)

and of course, .

\qef

>We prove that in abstract algebra actually. It's pretty easy, you just get a

contradiction somewhere.

A few more essential axioms for multiplication...

(7)

\likeremark{Remark}

We actually do need , since most importantly such that . This has very important consequences for

division, and further on, limits.

\qef

>and commutative multiplication, because we're not crazy.

(8)

Just as subtraction is rephrased addition, so to is division rephrased

multiplication: . Now as in our remark, since there

doesn't exist , we now see that cannot mean anything, by

definition!

Now from 7, we find the **cancellation law**,

Suppose , and ,

also, if , then it follows from the above that or .

>In abstract algebra, rings which have this property are known as Euclidean

domains, in which there are no zero divisors.

\proof

Suppose , and w/o loss, . Then

\qed

This is massively important for solving equations.

\begin{example}

Consider , then by the previous property, it must be the case

that either or , so we find or .

\end{example}

Now we're getting into the swing of things, we can begin combining addition and

multiplication with one more axiom...

(9)

Now we may truly prove when it is that !

\proof

Suppose . Then

\qed

Furthermore, we may now also assert that .

\proof

\qed

Yay! That's a useful thing to have isn't it?

We may also assert some reasoning about negative numbers.

\begin{theorem}

\end{theorem}

\proof

*** QuickLaTeX cannot compile formula: \begin{align*} (-a)\cdotb + a \cdot b &= [(-1)+a]\cdot b \\ &= 0 \cdot b \\ -(a \cdot b)+(-a) \cdot b + a \cdot b &= 0 - (a \cdot b) \\ (-a \cdot b) &= -(a \cdot b) \end{align*} *** Error message: Error: Cannot create dvi file

\qed

>and now, a drumroll...why two negatives make a positive!

\proof

and now add ,

Therefore,

\qed

>Hurrah! Now all those annoying questions from junior high school have been satisfied!

...just don't ask where we got those axioms k?

9, the distributive law, is key to factorization and multiplication of

arabic numerals.

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