Powers: International Mantissa of Mystery

A kind viewer of my Youtube lecture on Data has written that I used the term mantissa in a rather old-fashioned way on Slide 8. And he or she is correct. There is also an error on that slide which is annoying. So let's sort out both problems at once.

This is my old Slide 8.

And here is the new one:

What changed? Well firstly I corrected the typo. Of course the number 3.14159 is equal to 314159* 10-5.

Secondly I relabelled the mantissa to something called the "significand". The problem with the word mantissa is that it means two slightly different things. When we are talking about logarithms the word mantissa means the fractional part of the logarithm (usually base-10). Now what is puzzling in the Wikipedia article on "significand" is the quote from Donald Knuth. Knuth is one of the icons of Computer Science - when Knuth says something we all listen - and he quoted as saying

"Other names are occasionally used for this purpose, notably 'characteristic' and 'mantissa'; but it is an abuse of terminology to call the fraction part a mantissa, since that term has quite a different meaning in connection with logarithms. Furthermore the English word mantissa means 'a worthless addition."

So let's deal with those two points. Does it really mean "worthless addition"? No. The Oxford English Dictionary does indeed list it with that meaning, but it is clearly marked as obsolete. Their last citation of the word in that meaning is 1781. If we are going to preclude words because they used to mean something else in 1781 then I'm afraid we had better stop using the word algorithm. Furthermore the obsolete meaning is "something of comparatively small importance". Well compared to the exponent, the mantissa is of comparatively small importance. So, sorry, that argument does not cut the mustard.

But is a logarithmic "mantissa" "quite different" to a floating point mantissa? No. Not if we are talking about logarithms to base 10 and floating point numbers with base 10 and particular choices of representing the significand and exponent. But, yes, if we are talking about logarithms and floating point numbers to different bases.

So, let's say this quietly, since it is equivalent to saying that Mother Theresa wasn't a very nice person, but, in this case, Knuth's arguments do not stack-up.

Nevertheless,, the IEEE floating point standard does use the word significand (probably because they were scared to not obey strictures of Knuth) and so let's let it rest - it is, after all, a very small supplement, or should I say mantissa, to human knowledge.