Approximately
A calculation does not always have to be exact, so mental calculation doesn't either. To go fast without getting lost in unnecessary precision, one can therefore develop approximate mental calculation techniques. These are not limited to the occasional replacement of a number with a simpler number. From logarithms to square roots, once the path of approximation is chosen, a whole set of new techniques can unfold to allow one to evaluate the result of certain problems mentally.
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Grains on the chessboard and other high powers
Are billions, trillions, quadrillions and the like beyond our grasp? Not necessarily. With approximate calculations, many immense numbers become perfectly intelligible.
Heron's method for approximating square roots
What is the square root of 50? Since 50 is not a perfect square, the best we can give is an approximation. Put away our pencils and smartphones, and let's try to work it out in our heads.
The rule of 72
Interest-rate and monthly-payment problems are hard for our brains to grasp, because we struggle to picture an exponential phenomenon. During the Renaissance, however, trade expanded, bringing with it a growing need for calculations of this kind. This led to the development of new shortcuts.
A mnemonic for the number π
Whether for doing mental arithmetic or impressing the crowd, certain numerical values need to be learned by heart. The digits of π are an important example, but a particularly tricky one to memorize, since they follow no simple pattern. To overcome the difficulty, nothing beats alexandrine verse.
Logarithms to the rescue
Mental arithmetic usually brings to mind the four basic operations, or perhaps a few square-root calculations. Yet a little additional theoretical groundwork can take us much further, allowing us to tackle in our heads problems that might otherwise seem impossible without a calculator.
Fermi's wacky problems
They contain no data, yet a little ingenuity and plenty of common sense are all you need to work them out in your head: welcome to the world of the wacky problems invented by physicist Enrico Fermi.