## PhotoMath: review of a free smartphone app

As some of you might know, I taught mathematics for decades and am the author of *The Words of Mathematics*, a dictionary that the Mathematical Association of America has kept in print since that organization published it in 1994. Because of my math background, a friend of mine recently told me about a *free* smartphone app called PhotoMath and suggested that I write a review of it. As *Portraits of Wildflowers* is my main presence on the Internet, I’m posting the review here, with the understanding that readers interested in this blog’s normal subjects of nature photography and native plants may be surprised by the topic. On the other hand, it may be just what you’ve always wanted.

According to MicroBlink, the Croatian maker of PhotoMath, the app “uses a mobile phone camera to recognize mathematical expressions. It instantly solves a recognized expression, and displays step-by-step solution.” The program can currently handle arithmetical expressions that include fractions, decimals, powers and roots, and it can solve first-degree equations. It also recognizes some trigonometric, logarithmic, and exponential functions.

Once launched, the program brings up your phone’s camera screen and imposes a conspicuous red-cornered frame in the center of it. You maneuver your phone over the page of a book or worksheet (sorry, no handwritten expressions) and use the frame to isolate a problem. You can swipe horizontally or vertically with a finger to change the dimensions of the frame to make it better fit over the shape of the printed problem. For a long expression or equation, holding your phone in landscape orientation may be better than keeping it in portrait orientation. Most problems in schoolbooks and worksheets are numbered, so you have to be careful not to include the problem number in the frame. As soon as PhotoMath recognizes a framed problem, it emits a loud click and displays the answer in a small red cartouche centered at the base of the red frame. The answer can appear so quickly it seems like magic, and that’s certainly part of the program’s appeal.

But wait, as cheesy television commercials are fond of saying, that’s not all: the lower portion of the screen displays a larger red cartouche inviting you to press to see the steps leading to the answer.

For the expression shown here, PhotoMath takes three steps to simplify the fraction:

To maintain continuity between steps, each new screen begins with what came last on the previous screen.

Notice that as you advance from step to step the number of black dots at the lower left increases to show you which step you’ve reached, while the number of black dots at the lower right decreases with the number of steps remaining. The dots reinforce the information more largely conveyed by “step 1/3,” “step 2/3,” and step “3/3.”

The page for the last step in the working of a problem introduces a dotted line, below which the result appears. This is akin to your math teacher telling you to circle or box your answer at the end of a problem.

Speaking of students and teachers, it struck me that the latter might worry about the former using PhotoMath to do homework assignments. Here’s what the website says about that: “Let’s be honest: many kids cheat anyway, and an app which solves math problems automatically won’t make this problem worse. However, PhotoMath can be really helpful to many children when they are stuck with their homework and there is no one around to help them to figure it out. If we can eliminate kids’ frustration at the point when they can’t do anything else but helplessly stare at the book, we’ll feel awesome. It’s as simple as that.” Well, perhaps not quite that simple, but you can decide for yourself.

PhotoMath did well on problems that aren’t unusual in some way. For example, with the equation

3x + 2 = 5x – 8

it gave these steps:

3x – 5x = -8 – 2

3x – 5x = -10

-2x = -10

x = 5

So far so good, but I wondered if the app would “break” when confronted with special cases. For instance, with

4x – 2 = 4x – 2

it correctly converted the equation to

0 = 0

but left the user to interpret that truism to mean that any value of *x* will solve the equation. In the case of the equation

4x – 2 = 4x – 1

the app just sat there and did nothing. Not all users will understand that the lack of activity came from the fact that the equation is a contradiction (how can something be 1 more than itself?) and therefore has no solution.

The program also did nothing when confronted with 9/0, which is undefined because no real number times the 0 in the denominator would make the 9 that’s in the numerator. The app likewise had no response to 0/0, which is undefined because any number times the 0 in the denominator would make the 0 that’s in the numerator.

For the algebraic expressionPhotoMath gave

which is true but not particularly helpful. On the other hand, when I tried

the program multiplied out the factors and correctly gave

The PhotoMath website says the app handles basic trigonometric functions, so I tried cos (30°) and was baffled by a result of .540302 rather than the correct value of approximately .866. When I tried cos 30° without parentheses the program returned an “answer” of cos 1. Then I realized that the app had treated the degree symbol as the exponent zero: 30 to the power 0 is 1, and sure enough, the cosine of 1 *radian* is .540302. Apparently PhotoMath evaluates trigonometric functions only for arguments that are expressed in radians.

Switching to logarithms, I was pleased to see Photomath correctly give log(2) as .30103 and ln(2) as .693147. When I thought about inverse functions of logarithms, though, and tried a natural exponential expression, the program stared at

and did nothing. After I rewrote that as exp(3) the app correctly gave me 20.085537. Curiously, when I looked at the one and only step the program had taken to get that result, here’s what I saw:

So PhotoMath can display *e* cubed but can’t recognize it via the camera. Strange. It’s also quite a limitation, because math textbooks almost always use an exponential form like *e* cubed rather than exp(3), which is more at home in the world of computer programming.

Just as important in mathematics as *e* is *π*, but PhotoMath apparently doesn’t recognize that special constant either, because when I aimed the camera at the expression π over 2, the app interpreted it as 71 over 2 and therefore mistakenly returned a value of 35 and a half.

The PhotoMath website says the app does roots, but when I tried the cube root of 7 the program misread it as 3 times the square root of 7. When I tried the cube root of 1.331, the program misinterpreted the expression in the same way and gave an incorrect value of 3.461069; in one instance (I tried this expression several times), it even threw away the decimal point in 1.331 and came up with a false result ten times as large.

PhotoMath’s success when I stuck to arithmetic expressions was pretty good. The compound fraction given as a sample on the website,

offered no trouble when I tried it. The app also did a good job with first-degree equations, even a disguised one like

for which it returned the correct value of x = nine fourths. Systems of linear equations aren’t supported, however.

When PhotoMath accesses your phone’s camera, in addition to the red frame at the center of the screen it shows four icons across the bottom, which you can see in this view from the program’s help section:

The History button takes you to a list of recently read problems. I don’t know how many items the list can retain, but after trying out the program for a day I found that my list had more than 50 items in it. Tapping on an item brings back all the solving steps, so you can review them later. The Steps button brings up the steps in a problem that has just been solved.

I assume the Light button is intended to turn on the phone’s light in case the page you’re aiming the camera at isn’t bright enough, but I never could get the light to come on with my iPhone 5 running the latest version of iOS 8. The Help button offers some very basic information about using the program.

Given PhotoMath’s hit-and-miss record when I put it through its paces, I’m tempted to say *caveat emptor*, let the buyer beware, but this is a free app, so no money is at stake. Still, caution is in order, and users should examine results for plausibility: the cube root of 1.331 couldn’t possibly be the 3.46 that PhotoMath claimed, or any number more than a bit larger than 1 (in fact the cube root of 1.331 is exactly 1.1). A good strategy might be to look at all the steps PhotoMath offers as its justifications for an answer, because then any misinterpretation is likely to be obvious (like π being misread as 71).

On October 23, 2014, the blog on the PhotoMath website glowingly announced that “the PhotoMath video on Vimeo has very quickly reached 2 million views. Our web page has over 9000 page views each minute, and the iOS app alone was downloaded more than 1.6M times in less than three days, becoming the top free app in most countries around the world.” That’s pretty impressive, and I encourage you to head over to PhotoMath and increase those numbers by trying out the program for yourself.

PhotoMath is currently at version 1.1.1 and is available for Apple (it requires iOS 7.0 or later; is compatible with iPhone, iPad, and iPod touch; is optimized for iPhone 5 and iPhone 6) and for Microsoft (Windows Phone 8 or 8.1). The makers of PhotoMath say that an eagerly awaited Android version should launch in early 2015.

From one math lover to another, that is really cool!!!

photosfromtheloonybinNovember 26, 2014 at 11:22 AM

It’s good to see you add the first comment; your rational reaction positively equals mine.

Steve SchwartzmanNovember 26, 2014 at 4:19 PM

🙂

photosfromtheloonybinNovember 26, 2014 at 6:07 PM

That’s a prime example of an emoticon.

Steve SchwartzmanNovember 26, 2014 at 10:50 PM

So, do we have prime emoticons now like prime numbers? LOL 🙂

photosfromtheloonybinNovember 27, 2014 at 5:48 AM

I see you caught my drift. On a day like today (Thanksgiving in the U.S.), people are also primed to think a lot about π, even if they spell it pie. And that reminds me of an old joke about someone who mentioned the formula for the area of a circle, π r squared. Another person heard that and said: “No, pie are round.”

Steve SchwartzmanNovember 27, 2014 at 5:55 AM

LOL!!! You must have been a totally awesome math teacher :). I mean, how many people can make math fun like this and actually know how to type the symbol for π on a computer. Have a fantastic Thanksgiving today and send me some leftover π, whether squared or round :).

photosfromtheloonybinNovember 27, 2014 at 6:01 AM

Thanks, Cindy. Sometimes I do miss teaching, but I have a lot more time for photography, and this blog is at least partially a form of teaching.

I don’t know how much credit I can take for knowing how to type π on my computer because the Macintosh makes that easy: option-p.

Let’s hope that by the time we get to Internet 3.0 we’ll be able to send pies back and forth as quickly and easily as we send e-mails now.

You’ve already had your Canadian Thanksgiving but we Americans give you leave to have a second one today. Bon appétit.

Steve SchwartzmanNovember 27, 2014 at 6:30 AM

Ooh, does that mean I can take the day off? 🙂

photosfromtheloonybinNovember 27, 2014 at 7:05 AM

Yes, just tell your boss I said it was okay.

Steve SchwartzmanNovember 27, 2014 at 7:12 AM

He didn’t go for it :(. He says I have to work – how rude!

photosfromtheloonybinNovember 27, 2014 at 9:29 AM

I think he’s doing a number on you, and an irrationally negative one at that. I’d give his opinion zero credence.

Steve SchwartzmanNovember 27, 2014 at 9:44 AM

I usually do LOL!!

photosfromtheloonybinNovember 27, 2014 at 12:38 PM

I really enjoyed this, Steve. As a grandma helping grandson with his 5th grade equations, this can be helpful when I can’t reach daughter #2 the math teacher to help us out.

hmmmm…would this be the google translate of equations?

georgettesullinsNovember 26, 2014 at 11:33 AM

That’s a good way to put it, Georgette: the Google Translate of equations. If you try out the app with your grandson, let us know how well it worked for him and whether you think it helped him.

Steve SchwartzmanNovember 26, 2014 at 4:21 PM

Unusual blogging territory for you…but I am aware of your mathematical background. Nicely done.

lensandpensbysallyNovember 26, 2014 at 1:59 PM

Thanks, Sally. I seldom venture far from this blog’s theme, but once in a rare while I go off on a tangent.

Steve SchwartzmanNovember 26, 2014 at 4:23 PM

That is a very interesting app. Amazing what can be done with the camera in the phone. I like the idea. Of course, one needs to use with some cautions as you point out.

Does it not get into calculus?

Jim in IANovember 26, 2014 at 10:24 PM

No, no calculus at all, but yes, the app is still impressive for how much it can accomplish through the phone’s camera. I expect future versions will add functionality.

If you have either of the two currently supported operating systems I hope you’ll give the app a try soon (or early next year if you require an Android version) and let us know what you experience.

Steve SchwartzmanNovember 26, 2014 at 10:49 PM

I wrote myself a note to check it out after the holiday. I will pass along your review to a former math coordinator for our district to get her reaction.

Jim in IANovember 27, 2014 at 7:34 AM

Here is her response…

I have mixed emotions about the app. Lazy students would use it to cheat, as he points out, if all they care about is getting the correct answer. On the other hand, students who may need a hint about procedure will find it useful. It could also be a great tool for parents who have forgotten their algebra. It is only useful for algorithmic exercises; won’t help on real problem solving requiring actual thought and logic. I’d rather see students use graphing calculators where they can graph solutions to have a visual image as well as solve them algebraically. But as tools and apps go, it’s actually pretty amazing.

Jim in IANovember 28, 2014 at 9:34 PM

Thanks for passing along her reaction. She makes a good point that the app is limited to algorithmic exercises, yet manipulations of expressions and linear equations do form a large part of a standard first-year algebra course. My years as a math teacher left me wishing that students were more adept at carrying out those manipulations and that students learned which manipulations the rules of mathematics allow and which ones they don’t allow.

Steve SchwartzmanNovember 28, 2014 at 10:55 PM

I’m afraid that I am the dim candle. The calculator in my phone is more than I can figure out beyond the basic math actions…add, subtract, multiply, divide and oops.

Steve GingoldNovember 27, 2014 at 7:13 AM

That last one is a great mathematical operation. Many of my students were especially fond of it.

Steve SchwartzmanNovember 27, 2014 at 7:18 AM

I’d fit right in then. Once I run out of fingers and toes I am in trouble.

Steve GingoldNovember 27, 2014 at 7:27 AM

Your facility with fingers and toes should put you right at home in the digital world.

Steve SchwartzmanNovember 27, 2014 at 8:00 AM

Oops is a technical term in the accounting world too, although I don’t think it actually appears on the CPA exam. . . yet.

Nancy

dogear6November 27, 2014 at 3:59 PM

We could say there’s no accounting for that absence, Nancy.

Steve SchwartzmanNovember 27, 2014 at 10:32 PM

That is a really interesting app. I just downloaded it and will have to check it out. Math was always my best subject in school so this should be fun to play with!

Michael GloverNovember 28, 2014 at 6:49 PM

Go for it, Michael. After you do, if anything particularly stands out, you’re welcome to tell us.

Steve SchwartzmanNovember 28, 2014 at 7:10 PM

I don’t even understand math enough to understand your post enough to understand what the app does. !!! Glad *you* got a kick out of fiddling with it!

kathryningridDecember 1, 2014 at 7:15 PM

I understand.

Steve SchwartzmanDecember 1, 2014 at 8:22 PM