Bits and Beyond

Claude Shannon founded the field of information theory. A core fact in information theory is that there is a basic unit of information, called a “bit¹” or a “bit of entropy.” Roughly speaking, a “bit” is an amount of information that is about as surprising as the result of a single coin flip. In the sentence “please pass the salt” the word “salt” has less than a bit of entropy; most of the time someone says “please pass the” the next word they say is “salt,” so adding that word provides very little more information. On the other hand, the word that comes after “is” in “Hello, my name is” has many bits of entropy; no matter what the word is, it was quite surprising.

Digital Information

How much information can we transmit over a wire? If we put voltage on one end, because wire conducts well we’ll very soon see the same voltage at the other end. Presumably, the more precise our measurement of that voltage can be, the more information we can collect. If we can distinguish between 1000 voltage levels, for example, we’ll get \(\log_2(1000) = 10\) bits of information per measurement; whereas a less sensitive voltmeter that can only distinguish between two voltage levels gets only \(\log_2(2) = 1\) bit per reading. This seems to suggest that the more precise I can make the voltage generator and the more sensitive the voltmeter, the more information I can transmit per reading.

The problem with this assumption is that it takes longer to transmit higher-resolution data. This is partly a consequence of fundamental mathematical and physical laws, like the Heisenberg uncertainty principle, but it is also more tellingly a consequence of living in a noisy world. To tell the difference between 8.35 and 8.34 volts, you need to ensure that the impact of wire quality and the environment through which it passes contributes significantly less than 0.01 voltage error to the measurement; generally this requires watching the line of a while to see what is the noisy variation and what is the true signal. By contrast, telling the difference between 10 volts and 0 volts is much simpler and much more robust against noise. It is quite possible to make several dozen good 1-bit measurements in the time it’d take to make one 10-bit measurement.

This observation led to advances in digital signals: signals composed of a large number of simple “digits²” instead of one fine-grained “analog³” signal. We can communicate much more information with much less error by transmitting a large number of single-bit signals than we can by transmitting a smaller number of signals with more information in each.

Saying Anything with Bits

Information theory asserts that any information can be presented in any a format that contains enough bits. There are many interesting theorems and algorithms in that field about how this can best be done, but for most of computing we only need to consider a handful of simple encodings, described in the subsections below.

Place-value numbers

Base-10, “decimal”

When you were still a small child you were taught a place-value based numbering system. Likely at that age you never considered why we called it “place-value,” but the name is intended to suggest that the value of each digit depends not only on the digit itself, but also on its placement within the string of digits. Thus in the number string 314109, the first 1’s value is a hundred times larger than the second 1’s value. If we write out the full place values

we can see that the number’s total value is \(3 × 10^{5} + 10^{4} + 4 × 10^{3} + 10^{2} + 9 × 10^{0}\) or three-hundred fourteen thousand one hundred nine.

Base-2, “binary”

There is no particular magic to the 10s in the above example. Because it is easier to distinguish two things than ten, we might reasonably try to use 2s instead. Thus, instead of the digits 0 through 10 − 1 = 9 we use the digits 0 through 2 − 1 = 1. Thus in the base-2 number string 110101 the first 1’s value is thirty-two times larger than the last 1’s value. If we write out the full place values

we can see that the number’s total value is \(2^{5} + 2^{4} + 2^{2} + 2^{0}\) or fifty-three.

Base-2 numbers of this sort are widely used to convert sequences of single-bit data into a larger many-bit datum. Indeed, we call a binary digit a “bit,” the same word information theory uses to describe the fundamental unit of information, and often refer to any single-bit signal as being a 1 or 0 even if it is more accurately a high or low voltage, the presence or absence of light in a fiber, a smooth or pitted region on an optical disc, or any of the wide range of other physical signals.

Math works the same in base-2 as it does in base-10:

            1      11      11    1 11    1 11
  1011    1011    1011    1011    1011    1011
+ 1001  + 1001  + 1001  + 1001  + 1001  + 1001
------  ------  ------  ------  ------  ------
             0      00     100    0100   10100

Binary is useful for hardware because it takes less space and power to distinguish between two voltage states (high and low) than between three or more. However, for most uses we find it more useful to cluster groups of bits together, typically either in 4-bit clusters called “nibbles” or “nybbles” or in 8-bit clusters called “bytes” or “octets.”

Hexadecimal

Base-16, or hexadecimal, is useful because humans can read it without getting as easily lost in the long sequence of 0s and 1s, but it has a trivial mapping to the binary that hardware actually stores.

Hexadecimal digits are taken from the set of nibbles: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f}. Place-value notation is used with base 16. Thus the number d02\(_{16}\) (also written 0xd02) represents the number

Hexadecimal is particularly noteworthy because it is easy to convert to and from binary: each nibble is simply 4 bits. Thus

Hexidecimal is commonly used by humans who want to communicate in binary without getting lost in the middle of a long string of 0s and 1s.

Math works the same in base-16 as it does in base-10:

            1       1       1    1  1    1  1
  d0b2    d0b2    d0b2    d0b2    d0b2    d0b2
+ 300f  + 300f  + 300f  + 300f  + 300f  + 300f
------  ------  ------  ------  ------  ------
             1      c1     0c1    00c1   100c1

Bytes or octets

Base-256 uses octets or bytes, each being 2 nibbles or 8 bits. Octets are typically written using pairs of nibbles directly; thus the digits are taken from the set of bytes: {00, 01, 02, … fd, fe, ff}. Humans almost never do math directly in base-256.

Octets are noteworthy because, while processor mathematical circuits use base-2, most of the rest of computing is based on octets instead. Memory, disk, network, datatypes, assembly—all used bytes, not bits, as the smallest unit⁴ of communication.

Base-2 logs and exponents

Power-of-two numbers show up a lot in hardware and hardware-interfacing software. It is worth learning the vocabulary used to express them.

In all cases above, the i is usually dropped to just say (e.g.) G instead of Gi. The i clarifies that we mean the base-2 power, not the base-10 power. G could mean either \(2^{30}\) or \(10^{9}\), numbers the differ by about 7%; which one is meant can only be guessed from context unless the clarifying i is present (which it usually is not).

Only the first three rows in the table above (K, M, and G) will show up often in this course. The only entry in the base-10 column we expect you to learn is the first, \(2^{10}\) = 1024.

\(2^{27}\) = \(2^{7}\) \(2^{20}\) = 128M. This pattern works for any power of 2: the 1’s digit of the exponent becomes a number, the 10’s digit becomes a letter. Thus

If you memorize the values of \(2^{0}\) through \(2^{9}\), you can then do these exponentiations in your head. That is worth memorizing; these things show up too often to be worth wasting thought on all the time; memorize them now and you’ll have less cognitive load in the future.

Logarithms with base 2 (written \(\log_2(n)\) or \(\lg(n)\) or sometimes just \(log(n)\)) do the inverse of this: \(\lg(64G) = \lg(2^{6} 2^{30}) = lg(2^{36}) = 36\). Again, these show up so often that you should be able to do them automatically.

Negative numbers

We are accustomed to having a special symbol that tells us if a number is negative. When communicating in bits with just two symbols, that is not an option, so several other approaches to representing negative numbers are used. Three such approaches are often taught in courses like this, and three are actually used, but they are not the same three. We’ll explore the three that are common in hardware each in terms of decimal numbers first, then see how they map to binary.

All of these solutions depend on access to a finite number of bits/digits/nibbles/bytes to store numbers in. Our examples will assume we have four digits.

Two’s Complement

This version picks a number (typically half of the maximum number we can write, rounded up) and decides that that number and all numbers bigger than it are negative. How negative? To understand that, we need to observe a ring-like behavior of numbers.

What is 0000 - 0001 ? The answer is based on place-value subtraction’s notion of borrowing: 0 - 1 is 9 borrow 1. We end up borrowing from all spots to get 0000 - 0001 = 9999 with a stray borrow “overflowing” and getting lost due to finite precision. Since 0000 - 0001 is obviously negative 1, we set 9999 to be negative 1. One less than negative 1 is negative 2 = 9998, and so on.

Two’s complement is nice because the three most common mathematical operators (addition, subtraction, and multiplication) work the same for signed and unsigned values. Division is messier, but division is always messy.

In hardware, two’s complement is typically used for signed integers.

Visualization of two's complement numbers in binary. Zero is all zero bits. Adding and subtracting 1 makes numbers one larger or smaller, except at the transition between 01...1 and 10...0. There is one more negative number than there are positive numbers. Two's complement is commonly used for integral values. Swapping sign is equivalent to flipping all bits and adding one to the result.

Biased

This version picks a magical fixed number (typically half of the maximum number we can write, rounded down) and calls it “the bias.” For our numbers (0000 through 9999) that bias would be 4999. We then define the meaning of number x to be the value of x − bias. Thus 5006 means 5006 − 4999 = seven; 4992 means 4992 − 4999 = negative seven.

Biased numbers are nice in that addition and subtraction and comparison operators (< and > and their friends) work exactly like they do for unsigned numbers. However, multiplication and division are messier and humans generally find them confusing to read.

Biased numbers are used by hardware to represent the exponent of floating-point numbers stored in binary scientific notation.

Visualization of biased numbers in binary. Zero is 01...1. Adding and subtracting 1 makes numbers one larger or smaller, except at the transition between 1...1 and 0...0. There is one more positive number than there are negative numbers. Biased numbers are commonly used for the exponent of floating-point numbers, but not for many other purposes.

Sign bit

This is the version we all learned in grade school. Negative seven is written exactly like 7, but with a special symbol in the most-significant digits place: -7. With our fixed-width constraint, we’d write that as -007 or +007.

The main downside to sign-bit values is that they have two zeros: -000 and +000. These two are written differently but have the same numerical value, so should -000 == +000 be true or false? We can make any decision we want, of course, but ambiguity is not popular.

In binary, the negative sign is generally a 1 in the first bit, while a 0 there is the equivalent of a positive sign instead. Sign bits are used in floating-point numbers.

Visualization of sign-bit numbers in binary. There are two zeros and two discontinuities. The negative numbers move backwards, increasing in value when the unsigned representation decreases. Sign-bit integers, as displayed here, are not used in any common hardware; sign bits are more common for floating-point numbers instead.

One’s Complement

There is also a representation called “Ones’ complement” that is often taught in courses like this but that is not used by common hardware today.

On non-integral numbers

At one time there were many alternative representations of non-integral numbers, but IEEE standards 754 and 854 define a particular representation that has received very widespread implementation. Numbers in this and related formats are called IEEE-style floating-point numbers or simply “floating-point numbers” of “floats”.

A floating-point number consists of three parts:

1. A sign bit; 0 for positive, 1 for negative. Always present, always in the highest-order bit’s place, always a single bit.

2. An exponent, represented as a biased integer. Always appears between the sign bit and the fraction.

   If the exponent is either all 1 bits or all 0 bits, it means the number being represented is unusual and the normal rules for floating-point numbers do not apply.

3. A fraction, represented as a sequence of bits. Always occupies the low-order bits of the number. If the exponent suggested this was not the normal-case number, may have special meaning.

There are four cases for a floating-point number:

Normalized

The exponent bits are neither all 0 nor all 1.

The number represented by s eeee ffff is ± 1.ffff × \(2^{eeee − bias}\). The value 1.ffff is called “the mantissa”.

Denormalized

The exponent bits are all 0.

The number represented by s 0000 ffff is ± 0.ffff × \(2^{1 − bias}\). The value 0.ffff is called “the mantissa”. Note that the exponent used is “1 − bias” not “0 − bias”.

Infinity

The exponent bits are all 1 and the fraction bits are all 0.

The meaning of s 1111 0000 is ±∞.

Not a Number

The exponent bits are all 1 and the fraction bits are not all 0.

The value s 1111 ffff is Not a Number, or NaN. There is meaning to the fraction bits, but we will not explore it in this course.