July 19, 2021July 19, 2021

Complex Numbers are Real

Complex Numbers are Beautiful Too

Real Analysis Study Help for Baby Rudin, Part 1.7

Complex numbers can be represented either using Cartesian coordinates or polar coordinates. The geometric interpretation of complex multiplication is very beautiful using polar coordinates.

Are imaginary numbers real? After all, why are they be called “imaginary”? Doesn’t this name imply that they don’t exist? To

… Read the rest

June 14, 2021July 1, 2021

Existence of nth Roots of Positive Reals

The Completeness Axiom helps us prove the existence of nth roots of positive real numbers, but the proof is quite challenging.

Real Analysis Study Help for Baby Rudin, Part 1.6

The existence of nth roots, including the square root of two, is ultimately based on the continuity of power functions with positive integer exponents. — Ultimately, the continuity of $f(x)=x^{2}$ at $x=\sqrt{2}$ allows us to prove the existence of $\sqrt{2}$ . Later, the argument is based on the Intermediate Value Theorem. In this article, the argument is

f(x)=x^{2} — Ultimately, the continuity of $f(x)=x^{2}$ at $x=\sqrt{2}$ allows us to prove the existence of $\sqrt{2}$ . Later, the argument is based on the Intermediate Value Theorem. In this article, the argument is

… Read the rest

June 7, 2021June 14, 2021

The Archimedean Property

The Completeness Axiom is a sufficient condition to prove the Archimedean Property, but is it necessary?

Real Analysis Study Help for Baby Rudin, Part 1.5

A Visualization of the Archimedean Property in the case where $x=8$ and $y=70$ . The smallest positive integer value of $n$ that makes $nx>y$ is $n=9$ .

x=8 — A Visualization of the Archimedean Property in the case where $x=8$ and $y=70$ . The smallest positive integer value of $n$ that makes $nx>y$ is $n=9$ .

Does the sequence $\left(\frac{1}{n}\right)_{n=1}^{\infty}$ approach zero as $n\rightarrow \infty$ ? Sure! After all, given any number

… Read the rest

April 1, 2021May 7, 2021

Properties of the Supremum

The supremum and infimum of a bounded set of real numbers have many interesting properties

Study Help for Baby Rudin, Part 1.4

One of the properties of the supremum is that it is additive as a set function on sets of real numbers which are bounded above. — The supremum is additive as a set function on sets of real numbers which are bounded above.

The Completeness Axiom for the real number system is intimately tied to the concept of the supremum of a set of

… Read the rest

February 10, 2021February 10, 2021

Least Upper Bound (Supremum) in an Ordered Set

Suprema and Infima (Sups and Infs) Lie at the Heart of Real Analysis. The Range of Cos(n) is an Interesting Example.

Study Help for Baby Rudin, Part 1.3

For an ordered set $S$ and $M\subseteq S$ which is bounded above, the supremum of $M$ , when it exists in $S$ , is also called the least upper bound of $M$ in $S$ .

Which property separates the

… Read the rest

January 19, 2021June 3, 2021

Definitions of Ordered Set and Ordered Field

And an Ordered Field That Can Be Ordered in More Than One Way

Study Help for Baby Rudin, Part 1.2

A visual for a totally ordered set with three elements. — A graphic representation of the totally ordered set of three points $\left\{a,b,c\right\}$ with $a<b<c$ . The directed line segments are not part of the set. They only indicate the order.

\left\{a,b,c\right\} — A graphic representation of the totally ordered set of three points $\left\{a,b,c\right\}$ with $a<b<c$ . The directed line segments are not part of the set. They only indicate the order.

The set of real numbers $\Bbb{R}$ is more than just a set. As

… Read the rest

January 7, 2021January 24, 2021

Baby Rudin: Let Me Help You Understand It!

The First in a Series of Blog Posts on Baby Rudin

Study Help for Baby Rudin, Part 1.1

Bill Kinney's personal copy of Baby Rudin (Principles of Mathematical Analysis, 3rd edition) — My personal copy of Baby Rudin (Principles of Mathematical Analysis, 3rd edition, by Walter Rudin)

The textbook “Principles of Mathematical Analysis”, by Walter Rudin, is considered a “classic” text on real analysis, the subject that uses rigorous deductive

… Read the rest

October 29, 2019October 29, 2019

Differentiable Functions and Local Linearity

Calculus 1, Lectures 12 through 15B

An infinitely oscillating continuous and differentiable function. — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

x=0 — This function is both continuous and smooth at $x=0$ , in spite of oscillating infinitely often in any neighborhood of $x=0$ .

In Steven Strogatz’s excellent book, Infinite Powers, there is a big emphasis on an idea he calls The Infinity Principle.

The concepts of a continuous function and of a differentiable function are … Read the rest

January 28, 2019July 26, 2019

Does the Square Root of Two Exist?

The proof that the square root of two is an irrational number is considered to be one of the most elegant in all of mathematics.

As a formal “If…, then…” statement, it reads: If $x^{2}=2,$ then $x$ is not a rational number. A more informal statement of this is “the square root of two is irrational”.

Rational Numbers

But what is a … Read the rest

January 22, 2019July 18, 2019

Deconstructing the Mean Value Theorem, Part 3

The Mean Value Theorem implies that there is at least one tangent line with the same slope as the secant line.

Pete Brady strolled out of the U.S. Postal Service office just off Weaver Lake Road in Maple Grove, Minnesota at exactly 10 am.

Coffee and important package in hand, he set off on his 161 mile (259 kilometer) journey … Read the rest