Computers and Floating-Point Numbers: In Layman’s Terms

There exist WiKiPedia pages that do explain how single- and double-precision floating-point numbers are formatted – both used by computers – but which are so heavily bogged down with tedious details, that the reader would need to be a Computer Scientist already, to be able to understand them. And in that case, those articles can act as a quick reference. But they would do little, to explain the subject to laypeople. The mere fact that single- and double-precision numbers are explained on the WiKi in two separate articles, could already act as a deterrent for most people to try understanding the basic concepts.

I will try to explain this subject in basic terms.

While computers store data in bits that are organized into words – those words either being 32-bit or 64-bit words on most popular architectures – even by the CPU, those words are interpreted as representing numbers in different ways. One way is either as signed or as unsigned ‘integers’, which is another way of saying ‘whole numbers’. And another is either as 32-bit or as 64-bit floating-point numbers. Obviously, the floating-point numbers are used to express fractions, as well as very large or very small values, as well as fractions which are accurate to a high number of digits ‘after the decimal point’.

A CPU must be given an exact opcode, to perform Math on the different representations of numbers, where what type of number they are, is already reflected at compile-time, by which opcode has been encoded, to use the numbers. So obviously, some non-trivial Math goes into defining, how these different number-formats work. I’m to focus on the two most-popular floating-point formats.

Understanding how floating-point numbers work on computers, first requires understanding how Scientists use Scientific Notation. In the Engineering world, as well as in the Household, what most people are used to, is that the number of digits a number has to the left of the decimal-point, be grouped in threes, and that the magnitude of the number is expressed with prefixes such as kilo- , mega- , giga- , tera- , peta- , or, going in the other direction, with milli- , micro- , nano- , pico- , femto- or atto- .

In Science, this notation is so encumbering that the Scientists try to avoid it. What Scientists will do, is state a field of decimal digits, which will always begin with a single (non-zero) digit, followed by the decimal point, followed by an arbitrary number of fractional digits, followed by a multiplication-symbol, followed by the base of 10 raised to either a positive or a negative power. This power also states, how many places further right, or how many places further left, the reader should visualize the decimal point. For example, Avogadro’s number is expressed as

6.022 × 1023

IF we are told to limit our precision to 3 places after the decimal point. If we were told to give 6 places behind the decimal point, we would give it as

6.022141 × 1023

What this means, is that relative to where it is written, the decimal point would need to be shifted to the right 23 places, to arrive at a number, that has the correct order of magnitude.

When I went to High-School, we were drilled to use this notation ad nauseum, so that even if it seemed ridiculous, we would answer in our sleep that to express how much ‘a dozen’ was, using Scientific Notation, yielded

1.2 × 10+1

More importantly, Scientists feel comfortable using the format, because they can express such ideas as ‘how many atoms of regular matter are thought to exist in the known universe’, as long as they were not ashamed to write a ridiculous power of ten:

1 × 1080

Or, how many stars are thought to exist in our galaxy:

( 2 × 1011 … 4 × 1011 )

The latter of which should read, ‘from 200 billion to 400 billion’.

When Computing started, its Scientists had the idea to adapt Scientific Notation to the Binary Number System. What they did was to break down the available word-sizes, essentially, into three fields:

  1. A so-called “Significand”, which would correspond to the Mantissa,
  2. An Exponent,
  3. A Sign-bit for the entire number.

The main difference to Scientific Notation however was, that floating-point numbers on computers, would do everything in powers of two, rather than in powers of ten.

A standard, 32-bit floating-point number reserves 23 bits for the fraction, and 8 bits for the exponent of 2, while a standard, 64-bit floating-point number reserves 52 bits for the fraction, and 11 bits for the exponent of 2. This assignment is arbitrary, but sometimes necessary to know, for implementing certain types of subroutines or hardware.

But one thing that works as well in binary as it does in decimal, is that bits could occur after a point, as easily as they could occur before a point.

Hence, this would be the number 3 in binary:


While this would be the fraction 3/4 in binary:


(Updated 11/13/2017 : )

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Finding the Multiplicative Inverse within a Modulus

The general concept in RSA Cryptography is, there exists a product of two prime numbers, call them (p) and (q), such that

T ^ E ^ D mod (p)(q) = T


T is an original plaintext document,

E is an encryption key,

D is the corresponding decryption key,

E and D are successively-applied exponents, of T,

(p)(q) is the original modulus.

A famous Mathematician named Euler found, that in order for exponent operations on T to be consistent in the modulus (p)(q), the exponents’ product itself must be consistent in the modulus (p-1)(q-1). This latter value is also known as Euler’s Totient Product of (p)(q).

There is a little trick to this form of encryption, which I do not see explained often, but which is important. Since E is also the public key, a relatively small prime number is used in practice, that being (2^16 + 1), or 65537. D could be a 2048 or a 4096-bit exponent. The public key consists of the exponent E and the modulus (p)(q) packaged together.

Thus, when the key-pair is created, (p) and (q) must be known separately, so that D can be computed, and then (p-1)(q-1), which hopefully never touched the hard-drive, can be discarded, after D is saved, along with (p)(q) again, this time forming the private key.

Therefore, D must be the multiplicative inverse of E, in the modulus of (p-1)(q-1). But how does one compute that?

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