Is 2i less than 3i? (Index)

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on why it doesn’t make sense to say one complex number is less than another complex number (using the usual definition of “less than”).

Part 1: A sketch of a direct proof based on the order axioms proving that $i$ is not an element of a number system that has the usual definition of inequality.

Part 2: An indirect proof.

Part 3: Discussion about the lexicographic ordering, which almost works.

Part 4: Two other partial orderings which almost work.

Lessons from teaching gifted elementary school students: Index (updated)

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on various lessons I’ve learned while trying to answer the questions posed by gifted elementary school students. (This is updated from my previous index.)

Part 1: A surprising pattern in some consecutive perfect squares.

Part 2: Calculating 2 to a very large exponent.

Part 3a: Calculating 2 to an even larger exponent.

Part 3b: An analysis of just how large this number actually is.

Part 4a: The chance of winning at BINGO in only four turns.

Part 4b: Pedagogical thoughts on one step of the calculation.

Part 4c: A complicated follow-up question.

Part 5a: Exponentiation is multiplication as multiplication is to addition. So, multiplication is to addition as addition is to what? (I offered the answer of incrementation, but it was rejected: addition requires two inputs, while incrementation only requires one.)

Part 5b: Why there is no binary operation that completes the above analogy.

Part 5c: Knuth’s up-arrow notation for writing very big numbers.

Part 5d: Graham’s number, reputed to be the largest number ever to appear in a mathematical proof.

Proving theorems and special cases: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on attempting to prove theorems by looking at special cases.

Famous propositions that are true for the first few cases (or many cases) before failing.

Part 1: The proposition “Is $n^2 -n + 41$ always a prime number?” is true for the first 40 cases but fails with $n = 41$ .

Part 2: The Polya conjecture is true for over 900 million cases before failing.

Part 3: A conjecture about the distribution of prime numbers is true for the first $10^{318}$ cases before failing.

Mathematical induction

Part 4: Pedagogical thoughts on mathematical induction.

Part 5: More pedagogical thoughts on mathematical induction.

Famous unsolved problems in mathematics

Part 6: The Goldbach conjecture.

Part 7: The twin prime conjecture.

Part 8: The Collatz conjecture.

Part 9: The Riemann hypothesis.

Theorems in secondary mathematics that can be proven using special cases

Part 10: The sum of the angles in a convex polygon with $n$ sides is $180n$ degrees.

Part 11: The Law of Cosines.

Part 12: Trig identities for $\sin(\alpha \pm \beta)$ and $\cos(\alpha \pm \beta)$ .

Part 13: Uniqueness of logarithms.

Part 14: The Power Law for derivatives, or $\displaystyle \frac{d}{dx} \left(x^n \right) = nx^{n-1}$ .

Part 15: The Mean Value Theorem.

Part 16: An old calculus problem of mine.

Part 17: Ending thoughts.

How to Avoid Thinking in Math Class: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. Recently, Math With Bad Drawings had a terrific series on how students try to avoid thinking in math class.

Part 1: Introduction: “In teaching math, I’ve come across a whole taxonomy of insidious strategies for avoiding thinking. Albeit for understandable reasons, kids employ an arsenal of time-tested ways to short-circuit the learning process, to jump to right answers and good test scores without putting in the cognitive heavy lifting. I hope to classify and illustrate these academic maladies: their symptoms, their root causes, and (with any luck) their cures.”

Part 2: Students’ natural desire to mindlessly plug numbers into a formula without conceptual understanding.

Part 3: The importance of both computational proficiency and conceptual understanding.

Part 4: Fears of word problems.

Part 5: What happens when students get stuck getting started on a problem.

Part 6: Is only getting the right answer important?

Lessons from teaching gifted elementary school students: Index

Part 1: A surprising pattern in some consecutive perfect squares.

Part 2: Calculating 2 to a very large exponent.

Part 3a: Calculating 2 to an even larger exponent.

Part 3b: An analysis of just how large this number actually is.

Part 4a: The chance of winning at BINGO in only four turns.

Part 4b: Pedagogical thoughts on one step of the calculation.

Part 4c: A complicated follow-up question.

Too many significant digits: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my short series on what happens when too many significant digits are reported.

Part 1: An analysis of the report of this Nike app:

Part 2: Links to resources that discuss how many significant digits should be reported.

The number of digits of n!: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on computing the number of digits in $n!$ .

Part 1: Introduction – my own childhood explorations.

Part 2: Why a power-law fit is inappropriate.

Part 3: The correct answer, using Stirling’s formula.

Part 4: An elementary derivation of the first three significant terms of Stirling’s formula.

Arithmetic with big numbers: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on doing basic arithmetic with very large numbers that exceed the character displays of most calculators.

Part 1: Addition

Part 2: Multiplication

Part 3: Division

Inverse Functions: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the different definitions on inverse functions that appear in Precalculus and Calculus.

Square Roots, nth Roots, and Rational Exponents

Part 1: Simplifying $\sqrt{x^2}$

Part 2: The difference between $\sqrt{t}$ and solving $x^2 = t$

Part 3: Definition of an inverse function and the horizontal line test

Part 4: Why extraneous solutions may occur when solving algebra problems involving a square root

Part 5: Defining $\sqrt{x}$

Part 6: Consequences of the definition of $\sqrt{x}$ : simplifying $\sqrt{x^2}$

Part 7: Defining $\sqrt[n]{x}$ if $n$ is odd or even

Part 8: Rational exponents if the denominator of the exponent is odd or even

Arcsine

Part 9: There are infinitely many solutions to $\sin x = 0.8$

Part 10: Defining arcsine with domain $[-\pi/2,\pi/2]$

Part 11: Pedagogical thoughts on teaching arcsine.

Part 12: Solving SSA triangles: impossible case

Part 13: Solving SSA triangles: one way of getting a unique solution

Part 14: Solving SSA triangles: another way of getting a unique solution

Part 15: Solving SSA triangles: continuation of Part 14

Part 16: Solving SSA triangles: ambiguous case of two solutions

Part 17: Summary of rules for solving SSA triangles

Arccosine

Part 18: Definition for arccosine with domain $[0,\pi]$

Part 19: The Law of Cosines and solving SSS triangles

Part 20: Identifying impossible triangles with the Law of Cosines

Part 21: The Law of Cosines provides an unambiguous angle, unlike the Law of Sines

Part 22: Finding the angle between two vectors

Part 23: A proof for why the formula in Part 22 works

Arctangent

Part 18: Definition for arctangent with domain $(-\pi/2,\pi/2)$

Part 24: Finding the angle between two lines

Part 25: A proof for why the formula in Part 24 works.

Arcsecant

Part 26: Defining arcsecant using $[0,\pi/2) \cup (\pi/2,\pi]$

Part 27: Issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 28: More issues that arise in calculus using the domain $[0,\pi/2) \cup (\pi/2,\pi]$

Part 29: Defining arcsecant using $[0,\pi/2) \cup [pi,3\pi/2)$

Logarithm

Part 30: Logarithms and complex numbers

Different definitions of e: Index

Part 1: Justification for the formula for discrete compound interest

Part 2: Pedagogical thoughts on justifying the discrete compound interest formula for students.

Part 3: Application of the discrete compound interest formula as compounding becomes more frequent.

Part 4: Informal definition of $e$ based on a limit of the compound interest formula.

Part 5: Justification for the formula for continuous compound interest.

Part 6: A second derivation of the formula for continuous compound interest by solving a differential equation.

Part 7: A formal justification of the formula from Part 4 using the definition of a derivative.

Part 8: A formal justification of the formula from Part 4 using L’Hopital’s Rule.

Part 9: A formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 10: A second formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 11: Numerical computation of $e$ using Riemann sums and the Trapezoid Rule to approximate areas under $y = 1/x$ .

Part 12: Numerical computation of $e$ using $\displaystyle \left(1 + \frac{1}{n} \right)^{1/n}$ and also Taylor series.