W' oops fram alpha?

I like Wolfram|Alpha. I use it in my classroom and even recommend it to my students as a great way to play around with mathematical ideas at home... but sometimes I get surprised at the strange things it won't do.

For instance, this morning in class.

We are working with inverse relations with my kids; trying to show that every mapping has an inverse mapping, but that doesn't mean either one of them is a function... and I showed them a complicated implicit function. I started withx^2 + 3xy + 2xy^2 = 7 just picked something off the top of my head...



It was interesting, then we talked about what it would look like if we flipped it around by interchanging the x's and y's.

y^2+3yx+2yx^2=7 but???





I couldn't imagine what was amiss, but retyped it as y^2 + 3xy + 3x^2 y=7




Curious, I experimented...was it the order yx verses xy that confused it? tried a rectangular hyperbola

xy=1 no problem.... yx=1 this does not compute???


Ok, so what happens if I type in x=y^2... I get the typical parabola orientation you would expect with y=x^2, centered at the origin and opening up along the y? axis. Neither axis was labeled in this image, but the graph is labeled "y from -1 to 1"




y^2 = x shows the right opening parabola you would expect, with both axes labeled.

One of the many idiosyncrasies of Wolfram alpha.

Mathematical Induction, A Brief History of the Term

I have had an interest in the history and etymology of mathematical terms for many years, as witnessed by my MathWords web page. Recently I came across a couple of old journal articles, (1915-1918) related to the history of mathematical induction, and to the term itself. Most of this comes from an article by Cajori, and the very early dates above. Certainly those who know more about the current status of the usage could help by sharing their information.

The logical and scientific process called induction dates back as far as Cicero's translation of Aristotle. Cicero used the latin term "inductio", for the Greek "epagoge", which translates as "leading to."  Levi ben Gerson wrote Art of Calculation (or Art of the Computer) in 1321. It deals with arithmetical operations, including extraction of square roots and cube roots. In this work he also gives formulas for the sum of squares and the sum of cubes of natural numbers as well as studying the binomial coefficients. In proofs, he uses induction making this one of the earliest texts to use this important technique. 

Induction has always existed in mathematics, but the formal concept of mathematical induction did not appear until it was developed by Maurolycus in 1575 to prove that the sum of the first n odd numbers is n². While the roots of formal mathematical induction are nested in works from Fermat all the way back, one might say, to Euclid's proof of the infinity of the primes, the work of Maurolycus was unique in the formal use of attaching one term to the next in a general way.

The method of Maurolycus was repeated and extended in the works of Pascal to be a much more clear illustration of the present method but none of them used a particular name for their logical process. Then in his Arithmetica infinitorum in 1656 Wallis decided to name the term. On page 15 he creates the term "per modum inductionis" to prove that the limit of the ratio of the sum of the first n squares to n³ + n² was 1/3. His inductive method followed very much the unnamed method of Maurolycus.

Later Bernoulli gives an improvement to Wallis' method by showing the argument from n to n+1 as a general proof; this was the real foundation of modern mathematical induction. Bernoulli gives no specific name to his process, but uses his method as an improvement on the "incomplete induction" earlier used.

For the next 150 years, mathematicians used induction in both senses, to refer to the process of observing a relationship from a pattern , and in the method of Bernoulli to prove such an induced relationship by arguing from n to n+1. Then early in the 19th century, George Peacock uses the term "demonstrative induction" in his 1830 Treatise on Alebra. Then several years later, Augustus De Morgan proposes the name "successive induction" but then at the end of the article he talks about the method as "mathematical induction."

Isaac Todhunter used both names in his chapter on the method, but he used only Mathematical Induction in the chapter heading. When he defined and introduced the term "mathematical induction" (1838), he gave the process a rigorous basis and clarity that it had previously lacked.   Several popular textbook authors, Jevons and Ficklin, for example, used both terms. But among several others, Chrystal, Hall and Knight, used only the term mathematical induction. The same name seems to have been common in the early part of the 20th century in America and Europe, with Germany seemingly clinging to a single term for both "complete" and "incomplete" induction. Cajori, in 1918, says the Germans most commonly use the term, "vollstandige Induktion". I do not know if there is currently a more appropriate notation for the true mathematical induction of Bernoulli in Germany. If a reader is familiar with the current situation in German mathematics, please update me.

Student Confusion about Order of Operations

In America they say "PEMDAS" and the British often use "BODMAS" and some decry any mnemonics at all, but everyone teaches essentially the same rules of evaluation.

Consider the poor Algebra I or II student, who has seen and heard the following in his first few weeks of school:

"Multiplication and Division are handled as equals, from left to right."

"When we say 6a we mean 6 times a, where a just represents any number that we might choose to be use in place of it"

And then later we write on the board 6a² / 3a = ?
and so the student says, "I get it, that means 6 times a² then I divide by 3 and multiply by a. It must be 3a³."
Then, wanting to follow instructions, he types the expression into his Ti-84 + silver edition calculator after storing a value for x, and sure enough, he gets the value of 3x ³.


But of course, you explain, that's wrong. when we divide like this we mean that the 3a is intended to be a monomial term, taken as a single unit.
Your student is compliant, so he/she just nods and murmurs "Ok" and as you walk back to the front of the room, turns to their neighbor and says "You understand that?" and gets a shrug and a side to side head-shake... all just part of the mystery of mathematics.

If you realize what is confusing them, you may become very conscientious about writing problems with the fractions written out with horizontal fraction bars $\frac{6x^{2}}{2x}$ and talk about "implied parentheses". Unfortunatly many of your students have only heard the term "implied" used in prejudicial situations;"He implied I had stolen it!" or similar, and really have no idea what you mean. So maybe you get REALLY conscientious and write every fractional expression using actual parentheses $\frac{(6x^{2})}{(2x)}$.... and then that night on the homework, the book does NOT use them.

The repercussions of this order of operations confusion leads to students unsure of whether to write the linear term in an equation as $\frac{5x}{2}$, or 5$\frac{x}{2}$ or $\frac{5}{2}$ x.. but sometimes that even leads to $\frac{5}{2x}$ .

Embrace the World with Both Arms

I've mentioned before my admiration for micro-credit (see the big Kiva link in my sidebar... go on, kick in $25 and make the world a little better place).

Here is a wonderful talk from Jacqueline Novogratz(say that three times as fast as you can!) about what she has termed as patient capitalism.. A middle ground between profit at all risk capitalism and charity that too often forces the poor into a worse position than they started from. It's not enough to talk about world peace, you gotta make it happen...
Hope you enjoy it