The estimation method is featured in a new book by Rob Eastaway.

Here is the main video on Numberphile:

And here are some extras:

In addition, my resident graph expert Dave Wiley (see previous blog) drew up two graphs to analyse the strength or weaknesses of zequals.

This first one shows what happens when you square a number.

The blue line is the true answer, the red line is a zequal plot.

See a hi-res version of this graph on Flickr.

The second graph is fascinating.

It shows what happens when you multiply two numbers (x and y axis) using the zequals technique.

White areas show where you answer is dead right.

Red areas show where your zequals answer is higher than the true answer... blue areas show where is it is too low!?

Again, a hi-res version is on Flickr.

Nice post/video, a cool way of expressing the idea.

ReplyDeleteIn the video Rob starts with 436 * 68, which becomes 440 * 70 = 28,000 which then becomes 30,000, which is the correct zequals for the original calculation.

Is this always the case, or can the final result be different from not using the intermediate step?

No, counter example:

Delete449*64 zz 400*60 = 24000 zz 20000

449*64 = 28736 zz 30000

Figured as much, thanks.

DeleteSo, am I understanding correctly that the higher the x and y values, the smaller the error? Is that why the colors fade?

ReplyDeleteNo, the graph is self-similar for all orders of magnitude. If you look closely, you can see that the lower left 100x100 square is a smaller version of the full 1000x1000 square. There will be a blue region for numbers between 1000 and 1500, which will break up and fade towards 10000, and so forth.

ReplyDeleteIt is a fractal then. I shows Benford’s Law well. Also, if we could redo the graph for different bases, it would show that higher bases are more accurate under zequals because they are finer grained.

DeleteI think the interesting thing to point out about the chart is that as the x and y values increase (presumably when you would be in greater need of estimation) the % of error decreases (as demonstrated by the chart turning white in the upper right corner).

ReplyDeleteIt would be more informative to me to see the second graph as a 3D surface, where each (x, y) pair maps not to a color, but to height along the z-axis. Would it be possible for you to make another graph in this way?

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