As you may recall, a few days ago I posted my

*Peter Poses a Puzzle to Ponder*column. This was instigated by my friend Peter Traneus Anderson (a.k.a. Traneus Rex) who had sent me an email saying:Find ‘a’ and ‘b’ such that (a + b) = (a * b) = 2Shortly thereafter, I received an email from my chum Charles Pfeil who said that since he hadn’t been able to determine the solution himself, he’d tried the free Photomath app on his iPhone. Sadly, that hadn’t worked either, but Charles sent me an image to show me how the app works on a typical problem. As we see, the idea is that the app uses the camera on your iPhone to capture an equation written on a piece of paper as shown on the left-hand side of this image. The app then solves the equation as shown on the right-hand side of the image. I just downloaded this app and played with it a bit to discover that the “= ?” part of the equation is optional. Also, it works if the equation is written on lined paper. In fact, this is just the tip of the iceberg as to what this bodacious beauty can do. You can also get it to show you the steps it used to solve the equation and explain each step. Also, in the case of equations like

*y = mx + c*, you can get it to calculate things like the x-intercept, y-intercept, slope, domain, derivative… the list goes on. Also, it will plot the result and present it as a graph. All I can say about this app is “OMG! This is awesome!” This would have been so useful when I was at high school. It reminds me of the time the guys decided to create an equation solving app on*The Big Bang Theory*. If you pause on the high-level view of the app on the whiteboard, you’ll see it features a “Camera View” area. In turn, this reminded me of the time Dr Leslie Winkle corrects an equation on Sheldon’s board on*The Big Bang Theory*, but we digress… In response to Charles, I pointed him at the MyScript Calculator app, which allows you to write your equations in freehand using your finger or a stylus. Although this isn’t free, it’s well worth the money ($2.99) to see this little rascal in action and gasp in awe. I actually use this calculator quite often to perform on-the-fly calculations. Sad to relate, the MyScript calculator is of no use when it comes to solving Peter’s Puzzle. Happily, the solution becomes obvious when we realize that we can represent*a*and*b*as complex numbers such that:a = (1 + i) b = (1 – i)…where

*i*is an imaginary number; that is, the square root of –1 (in electrical engineering and related fields, the imaginary unit is normally denoted by*j*to avoid confusion with electric current as a function of time, which is conventionally represented by*i(t)*or just*i*). So, let’s start by substituting these complex values for the (*a*+*b*) portion of our problem:(a + b) = (1 + i) + (1 – i) = 1 + i + 1 – i = 2As we see, the +

*i*and –*i*values cancel each other out, leaving us with 1 + 1 = 2. Next, let’s substitute our complex values for the (*a***b*) portion of the problem:(a * b) = (1 + i) * (1 – i) = 1Once again, the +^{2}+ i – i – i^{2}= 2

*i*and –*i*values cancel each other out. Of course, 1^{2}= 1. The only tricky bit is the –*i*^{2}. Since*i*is the square root of –1, then*i*^{2}must be –1, so –*i*^{2}= – –1, and – –1 = + 1, which means we once again end up with 1 + 1 = 2. Tra-la! OK, over to you. What did you think of this puzzle and do you know any similar posers with which you could tease the rest of us? Also, what do you think of the calculator apps presented here and do you know of any other calculator apps with which you could delight the rest of us?