## Posts tagged ‘1’

### A 1-Derful Post for 1/1

My sons have refrigerator magnets with digits and binary operators, which they use to create expressions, equations, and dates. Recently, they created the following equation:

1 + 1 ÷ 1 – 1 × 1 = 1

They asked if it was correct. *Oh, no, that’s not how things work in this house.* “You tell me,” I said.

Eli said, “One plus one is two, divided by one is two, minus one is one, times one is one. It’s true.”

Anyone who teaches middle school has seen students make this type of order of operations error. The equation is true if operations are performed left-to-right but not if the conventional order of operations is applied.

On a calculator, I entered

1 + 2 ÷ 2

and asked, “What is the value of this expression?” Sure enough, they thought it would be three-halves, and they were surprised to see two displayed when the ENTER key was pressed.

Eli looked puzzled, and Alex looked cross. “Oh, right,” said Alex. “We have to do multiplication and division first.”

They then concluded that their equation was indeed true, and this time for the right reasons.

But it made me wonder:

What is the probability that the following equation will be true, if the four binary operators are randomly placed in the blanks with each operator used only once?

1 __ 1 __ 1 __ 1 __ 1 = 1

I’ll tell you that (a) I was surprised by the results and (b) I didn’t have to check every possible equation to arrive at the answer; in fact, I didn’t even have to check a quarter of them.

Surgeon: I have so many patients to see today! Who should I do surgery on first?

Nurse: Follow the order of operations.How many calculus teachers does it take to screw in a light bulb?

0.99999999…

### Prime Time

One of the great joys of my current job is that I get to visit math classes. This is awesome, and I am incredibly grateful to the teachers who invite me to their classrooms. I’ve thought about returning to the classroom myself, but visiting is much better — I get to see magic and interact with kids, but I don’t have to worry about correcting misbehaviors, creating or grading tests, or filling out report cards.

I recently witnessed several great classes at Tincher Prep, a K-8 school in California. The students were the most collectively polite group of kids I’ve ever met, and the faculty was filled to capacity with intelligent, dedicated professionals. Students in first grade measured things with paper cut-outs of their foot, to get an appreciation for why we have standard measures. Kindergarten kids happily sang number songs and then counted by 5’s to figure out that it was the 75th day of the school year. Students in a middle school class were jumping out of their seats with excitement when playing a review game. In every class I visited, students were excited to be learning. What an awesome environment!

In one classroom, students were given the following assignment:

Complete this list of the first 10 prime numbers:

1, 2, ___, ___, ___, ___, ___, ___, ___, ___

John Derbyshire claims that Henri Lebesque was the last mathematician who considered 1 to be a prime number. The primary reason it should not be considered a prime number is that the Fundamental Theorem of Arithmetic — which states that every integer greater than 1 can be represented as the product of a unique set of prime numbers — will not hold. It also causes a problem with Euler’s Totient Function: for prime numbers, φ(*n*) = *n* – 1, but this rule is violated if 1 is considered a prime number.

The teacher who posed this problem to students, however, shouldn’t feel bad for including 1 as a prime number. Lots of professionals have trouble figuring out which numbers are prime…

- Mathematician: 3 is prime, 5 is prime, 7 is prime, and by induction, every odd integer greater than 2 is prime.
- Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is an experimental error,

11 is prime, … - Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, …
- Programmer: 3 is prime, 5 is prime, 7 is prime, 7 is prime, 7 is prime, …
- Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 — we’ll do the best we can, …
- Software Salesperson: 3 is prime, 5 is prime, 7 is prime, 9 will be prime in the next release, …
- Biologist: 3 is prime, 5 is prime, 7 is prime, results have not yet arrived for 9, …
- Lawyer: 3 is prime, 5 is prime, 7 is prime, there is not enough evidence to prove that 9 is not prime, …
- Accountant: 3 is prime, 5 is prime, 7 is prime, 9 is prime if 2/3 is deducted for taxes, …
- Statistician: Let’s try several randomly chosen numbers: 17 is prime, 23 is prime, 11 is prime, …
- Professor: 3 is prime, 5 is prime, 7 is prime, and the rest are left as an exercise for the student.
- Computational Linguist: 3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a
*very*odd prime, … - Psychologist: 3 is prime, 5 is prime, 7 is prime, 9 is prime but tries to

suppress it, … - Casino Card Counters: 3 is prime, 5 is prime, and 7 is prime, but I’ll take 21 over any of them.