# Floating-point systems: Is the mantissa the whole thing or just the "fraction" part after the decimal?

In the context of floating-point systems, our numerical analysis book defines the terms mantissa and fraction as follows:

I am unable to find any consistent definition of the terms "mantissa" and "fraction" online. Wolfram defines a mantissa to be what the book here calls the fraction:

This definition is more consistent with the Wikihow article on converting decimal floating-point numbers to their binary IEEE representation:

Which is which? I'm finding it very difficult to use outside resources together with my book because I'm not sure who is talking about what.

You're discovering that sometimes the usage of a word in mathematics and related studies is not always as uniform as textbooks often make it out to be.

As the entry in Wiktionary shows, mantissa has at least two very distinct meanings in mathematics. One meaning is the fractional part of a number when that number happens to be used as a base-ten (common) logarithm. Another is an alternative word for significand.

Wolfram MathWorld, as you found, has a definition slightly different from either of these. Their definition of mantissa is the fractional part of a number whether or not that number is being interpreted as a base-ten logarithm.

The definition of mantissa as a synonym for significand is not consistent with the other definitions. If $$x$$ is a number written in scientific notation as $$x = 3.45678 \times 10^2,$$ the significand of $$x$$ is $$3.45678,$$ but since $$x$$ written in ordinary notation is $$x=345.678,$$ the MathWorld definition of mantissa says that the mantissa of $$x$$ is $$0.678.$$

When people start writing about IEEE-754 floating-point representation, things get even more muddled. If you take your example of $$1.010101001_2 \times 2^6,$$ the significand (often called the mantissa in this context) is $$1.010101001_2.$$ But when it comes time to store this number in IEEE-754 representation, the makers of the standard resorted to a clever trick to squeeze one more bit of precision out of the fixed number of bits of any of their numeric formats. They observed that the most significant bit of a number in binary representation is always $$1,$$ so it is not necessary actually to store this bit when storing the value of a floating-point number. So the IEEE-754 format only stores the sign bit, some bits that encode the characteristic, and the fractional part of the significand. This format does not store the most significant bit of the significand.

What the Wikihow article is showing you is the part of the significand (or mantissa) that is actually stored in an IEEE-754 floating-point number or a similar format. But it's not actually very much like the MathWorld definition of mantissa, because the fractional part of the significand of a number $$x$$ (in some floating-point representation) is often quite different from the fractional part of the number $$x$$ itself.

• "They observed that the most significant bit of a number in binary representation is always 1" Why is this true, if anything shouldn't it be 0? Dec 26, 2020 at 8:24
• @aaaakshat A bit is a binary digit. In the context of general base-$n$ notation, either fixed-point or scientific notation, a leading zero is never a significant digit. The most significant digit of a number written in any base-$n$ radix is the leftmost non-zero digit. In base ten this could be any digit from $1$ to $9$. What are the non-zero digits in base two? Now notice what happened to the leftmost $1$ in the number $1.010101001\times2^6$ in the Wikihow article. Dec 26, 2020 at 14:01