5.1 Introduction to computer organization  (Page 2/2)

Humans use base 10, commonly assumed to be due to us having ten fingers. Digital computers use the base 2 or binary number representation, each digit of which is known as a bit ( b inary dig it ).

Number representations on computers

While this system represents integers well, how about numbers having nonzero digits to the right of the decimal point?In other words, how are numbers that have fractional parts represented? For such numbers, the binary representation system is used, butwith a little more complexity. The floating-point system uses a number of bytes - typically 4 or 8 - to represent the number, but withone byte (sometimes two bytes) reserved to represent the exponent $e$ of a power-of-two multiplier for the number - the mantissa $m$ - expressed by the remaining bytes.

A computer's representation of integers is either perfect or onlyapproximate, the latter situation occurring when the integer exceeds the range of numbers that a limited set of bytes canrepresent. Floating point representations have similar representation problems: if the number $x$ can be multiplied/divided by enough powers of two to yield a fraction lying between 1/2 and1 that has a finite binary-fraction representation, the number is represented exactly in floatingpoint. Otherwise, we can only represent the number approximately, not catastrophically in error as with integers.For example, the number 2.5 equals $0.6252^2$ , the fractional part of which has an exact binary representation. See if you can find this representation. However, the number $2.6$ does not have an exact binary representation, and only be represented approximately in floating point. In single precision floating point numbers, which require 32 bits (one byte for the exponent and theremaining 24 bits for the mantissa), the number 2.6 will be represented as $2.600000079...$ . Note that this approximation has a much longer decimal expansion. This levelof accuracy may not suffice in numerical calculations. Double precision floating point numbers consume 8 bytes, and quadruple precision 16 bytes. The more bits used in the mantissa, the greater the accuracy. This increasing accuracy means that more numberscan be represented exactly, but there are always some that cannot. Such inexact numbers have an infinite binaryrepresentation. Note that there will always be numbers that have an infinite representation in any chosen positional system. The choice ofbase defines which do and which don't. If you were thinking that base 10 numbers would solve this inaccuracy,note that $1/3=0.333333....$ has an infinite representation in decimal (and binary for that matter), but has finite representation in base 3. Realizing that real numbers can be only represented approximately is quite important, and underlies the entirefield of numerical analysis , which seeks to predict the numerical accuracy of any computation.

So long as the integers aren't too large, they can be represented exactly in a computer using the binary positionalnotation. Electronic circuits that make up the physical computer can add and subtract integers without error.(This statement isn't quite true; when does addition cause problems?)

Computer arithmetic and logic

The binary addition and multiplication tables are

Note that if carries are ignored, A carry means that a computation performed at a given position affects other positions as well. Here, $1+1=10$ is an example of a computation that involves a carry. subtraction of two single-digit binary numbers yields the samebit as addition. Computers use high and low voltage values to express a bit, and an array of such voltages express numbersakin to positional notation. Logic circuits perform arithmetic operations.

The variables of logic indicate truth or falsehood. $A\cap B$ , the AND of $A$ and $B$ , represents a statement that both $A$ and $B$ must be true for the statement to be true. You use this kind of statement to tell searchengines that you want to restrict hits to cases where both of the events $A$ and $B$ occur. $A\cup B$ , the OR of $A$ and $B$ , yields a value of truth if either is true. Note that if we represent truth by a "1" andfalsehood by a "0," binary multiplication corresponds to AND and addition (ignoring carries) toXOR . XOR, the exclusive or operator, equals the union of $A\cup B$ and $A\cap B$ . The Irish mathematician George Boolediscovered this equivalence in the mid-nineteenth century. It laid the foundation for what we now call Boolean algebra,which expresses as equations logical statements. More importantly, any computer using base-2 representations andarithmetic can also easily evaluate logical statements. This fact makes an integer-based computational device much morepowerful than might be apparent.