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…