Probably the most famous case of synesthesia was the one written up over a period of thirty years from the 1920s by the Russian psychologist A. R. Luria of a journalist called Shereshevsky with a prodigious memory. "S," as Luria called him in his notes for the book The Mind of a Mnemonist, had a highly visual memory which allowed him to "see" words and numbers as different shapes and colors. "S" was able to remember a matrix of 50 digits after studying it for three minutes, both immediately afterwards and many years later. Luria credited Shereshevsky's synesthetic experiences as the basis for his remarkable short- and long-term memory.
Using my own synesthetic experiences since early childhood, I have grown up with the ability to handle and calculate huge numbers in my head without any conscious effort, just like the Raymond Babbitt character. In fact, this is a talent common to several other real-life savants (sometimes referred to as "lightning calculators"). Dr. Darold Treffert, a Wisconsin physician and the leading researcher in the study of savant syndrome, gives one example, of a blind man with "a faculty of calculating to a degree little short of marvelous" in his book Extraordinary People:
When he was asked how many grains of corn there would be in any one of 64 boxes, with 1 in the first, 2 in the second, 4 in the third, 8 in the fourth, and so on, he gave answers for the fourteenth (8,192), for the eighteenth (131,072) and the twenty-fourth (8,388,608) instantaneously, and he gave the figures for the forty-eighth box (140,737,488,355,328) in six seconds. He also gave the total in all 64 boxes correctly (18,446,744,073,709,551, 616) in forty-five seconds.
My favorite kind of calculation is power multiplication, which means multiplying a number by itself a specified number of times. Multiplying a number by itself is called squaring; for example, the square of 72 is 72 x 72 = 5,184. Squares are always symmetrical shapes in my mind, which makes them especially beautiful to me. Multiplying the same number three times over is called cubing or "raising" to the third power. The cube, or third power, of 51 is equivalent to 51 x 51 x 51 = 132,651. I see each result of a power multiplication as a distinctive visual pattern in my head.
As the sums and their results grow, so the mental shapes and colors I experience become increasingly more complex. I see 37's fifth power -- 37 x 37 x 37 x 37 x 37 = 69,343,957 -- as a large circle composed of smaller circles running clockwise from the top around.
When I divide one number by another, in my head I see a spiral rotating downwards in larger and larger loops, which seem to warp and curve. Different divisions produce different sizes of spirals with varying curves. From my mental imagery I'm able to calculate a sum like 13 ÷ 97 (0.1340206...) to almost a hundred decimal places.
I never write anything down when I'm calculating, because I've always been able to do the sums in my head, and it's much easier for me to visualize the answer using my synesthetic shapes than to try to follow the "carry the one" techniques taught in the textbooks we are given at school. When multiplying, I see the two numbers as distinct shapes. The image changes and a third shape emerges -- the correct answer. The process takes a matter of seconds and happens spontaneously. It's like doing math without having to think.