A Short Introduction to Binary Numbers

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1 A Short Introduction to Binary Numbers Brian J. Shelburne Department of Mathematics and Computer Science Wittenberg University 0. Introduction The development of the computer was driven by the need to mechanize arithmetic; that is, to construct a machine that would perform arithmetic quickly and accurately and thereby speed the solution of calculation intensive problems. But if arithmetic was to be performed by "mechanical (or electronic) devices" like computers, efficient ways had to be found to represent numbers on these machines. What follows is a short introduction to the standard methods used to represent numbers on a computer. 1. Unsigned Binary Integers In our everyday decimal notation, we use the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to represent the first ten whole number quantities starting at zero. (Note: There is no digit for the quantity we call ten.) To represent whole numbers greater than nine, we use a "string" of digits where the position of a digit in the string determines the weight given to that digit. The weights used are powers of ten. Thus the number 12 represents one ten and two ones. The number 257 represents 2 hundreds (ten to the second power) plus 5 tens (ten to the first power) plus 7 ones (ten to the zero power) or 257 = 2x x x10 0 This is called "positional notation" and any whole number quantity can be represented this way. Now computers are built out of electronic components which are in one of two discrete states, OFF or ON. We assign the symbol 0 to the OFF state and the symbol 1 to the ON state. Using only two binary digits or bits (bit is a contraction of binary digit), integers can be represented using base 2 or binary notation. Like decimal notation, the position of a binary digit in binary notation determines the weight given it. However, the weights are powers of two, not ten. For example the binary number 1101 represents 1 "eight" (two to the third power) plus 1 "four" (two to the second power) plus 0 "twos" (two to the first power) plus 1 "one" (two to the zero power) or 1101 = 1x x x x2 0 = = 13 This rewriting of a binary number as a "sum of powers of two" is also a simple technique to convert it to its corresponding decimal representation. Note: It is useful to know the powers of two from 2 0 up to 2 10 (or even 2 16 ). The following "table" is easily generated by starting at 2 0 = 1 and doubling each value until 2 16 is obtained. 2 0 = = = = = = = = = = = = = = = = =

2 2. Converting Binary to Decimal Representation As mentioned above, one method to convert binary representation to decimal representation is to write the binary representation as a "sum of powers of two". Exercise: Convert the following binary representations to their decimal equivalents by expanding each as a sum of powers of two. a b A more "elegant" technique for converting binary to decimal is called Horner's Method or Synthetic Division. It is a technique that does not require computing the decimal values for each power of two (thus it minimizes the amount of multiplication). The set-up for Horner's method (see below) uses three lines placing the binary representation on the first line, a single 0 in the first (left-most) column of the second line and leaves the third line empty except for a "2 " (the 2 is the multiplier) off to the left. A bar separates the second and third lines. Horner's Method Set-Up <- line line line 3 Horner's method works from left to right. Starting with the left-most column, sum the two digits on lines 1 and 2 (1 + 0) and place the sum below the bar in line 3. Then multiply this sum by 2 transferring the product to the next column of line 2 (see below). Repeat this "sum, multiply by 2, transfer to next column" process for each digit in line 1. The final sum in the right-most column of line 3 is the corresponding decimal representation. Initial Set-Up After 1 Iteration Final Result ^ decimal result Exercise: Use Horner's Method to convert the binary representations in the above exercise to decimal. Note: Horner s Method for binary to decimal conversion can be easily (?) implement in Python. 2

3 3. Converting Decimal to Binary Representation An easily remembered method for converting decimal representation to binary is to repeatedly subtract out powers of two from the decimal representation. Begin by determining the largest power of two less than or equal to the decimal value. (For example 2 4 = 16 is the largest power of two less than or equal to 31 while 2 5 = 32 is the largest power of two less than or equal to 32.) Subtract the largest power of two from the decimal value and repeat by finding the largest power of two less than or equal to the remainder. Subtract again. Repeat until you reach zero all the while keeping track of the powers of two subtracted out. The powers of two subtracted out make up the binary representation. Example : Convert 13 (decimal) to binary 13-8 = 5 where 8 = = 1 where 4 = = 0 where 1 = 2 0 Therefore 13 = 1x x x x2 0 = 1101 A useful trick to use with the "Subtraction of Powers of Two" method is to first list the powers of two from right to left leaving a blank line below each power Scanning from left to right, compare each power of two with the number to be converted. If the power of two is strictly larger, put a 0 in the blank below it; if it's less than or equal to the number, put a 1 in the blank below it, subtract the power of two from the number, and continue with the remainder.. Subtracting out powers of two to convert decimal to binary is the inverse of expanding by powers of two to convert binary to decimal. Example: Convert 104 (decimal) to binary Therefore 104 decimal equals binary This method often gives leading zeros which, as shall be seen below, is necessary when representing binary integers on the computer. A more elegant method, called the Integer Division Algorithm, involves repeated integer division of the decimal value by 2 while keeping track of the remainders. The remainders produce the binary representation in reverse order (i.e. the first remainder is the right-most or least significant bit). In the example worked out below, 13 is first divided by 2 to yield quotient 6 remainder 1. Continue by dividing the quotient 6 by 2 to obtain a new quotient 3 and a remainder 0. When the quotient eventually reaches 0, stop. The reminders (in reverse order) give you the binary representation. 3

4 As noted below, "toppling" over the remainders will yield the correct binary representation. Example : Convert 13 (decimal) to binary r 1 Note : If you "topple over" --- the "remainder tower" 2 3 r 0 you will have the --- binary representation. 2 1 r r 1 => Note the use of the "upside-down" division sign. Exercise: Using both methods described above, Subtracting Powers of Two and Integer Division, (use each to check the results of the other) find the binary representation for a. 47 b. 212 Note: Like Horner s Method for Binary ti Decimal conversion, the Integer Division method is also easy (?) to implement in Python 4. Representing Unsigned Binary Integers on the Computer. Computers use a fixed number of bits to store a binary integer. The number of bits used is a function of the way the memory of the computer is configured. Generally the individual bits of memory are not individually accessible (i.e. "addressable"). Instead a fixed number of bits is grouped into "cells" and only these larger "cells" can be accessed. The number of bits in an addressable cell varies from computer architecture to computer architecture. Today s computers are byte addressable meaning the smallest addressable cell is an 8-bit byte. Since the largest unsigned integers that can be stored in 8 bit is 255 (can you figure this out? see below) today s computers use 32 bits (4 bytes) or 64 bits (8 bytes) to store an integer. For example, 13 decimal is (note leading zeros). The largest unsigned integer (in decimal) is = 4,294,967,295 which is sufficiently large for most purposes. Exercises 1. What is the largest unsigned binary integer representable by the following memory cell sizes and what are their corresponding decimal values? a. a 4 bit "nybble" (Yes, a 4 bit "half" byte is sometimes referred to as a "nybble") b. an 8 bit byte c. a 12 bit "word" d. a 16 bit "word" 2. Suppose you added 1 to the largest integer represented by 16 bits. That is = The latter value is a 1 followed by 16 zeros. What is the value of this binary representation? (Careful - it's NOT 2 17.) If the largest binary integer representable by 16 bits is one less than the above answer, use this method to find the decimal value of the largest integer representable by a 32 bit double word. 4

5 5. Binary Addition and Subtraction Binary Addition of two digits (with no carry in) is trivial; there are only four "rules" Note that the last sum has a "carry out"; that is 1 plus 1 is 0 "carry" 1. More interesting and useful is binary addition with a "carry in" which is the "carry out" from the previous column <- carry in Using the above eight "rules", any two binary integers can be added. Example: Binary Subtraction is more complicated because of the need to "borrow" from the next column. This is indicated by placing a -1 above the column to the left to indicate that 1 must be borrowed from the next column over borrow -> The only time a borrow is needed is when subtracting 1 from 0 (as in the second subtraction above). Essentially, you borrow 10b from the next column over so you subtract 1 from 10b leaving a difference of 1. The effect of a borrow from a previous column on the right can get a bit complicated. If the current minuend (upper digit) is a 1, it will be used by the borrow from the previous column and so it effectively becomes a zero (see subtractions 3 and 4 below). If the minuend is a 0 a borrow "in" from the right must be propagated to the next column on the left which has the ultimate effect of making the 0 minuend into a 1. (When you borrow 10b from the column on the left, since 10b = 1 + 1, one of the 1's is used by the borrow-in" from the right while the remaining 1 becomes the new minuend). See subtractions 1 and 2 below. It helps to distinguish between a "borrow-in" (a borrow made by the column on the right) and a lend or "borrowout" (a borrow from the next column on the left). borrow borrow borrow borrow out in out in in out in

6 If you think about it, subtraction with binary integers is just like subtraction with decimal integers. However, as we will see below, there is easier way to do binary subtraction. Example : Exercises: Evaluate the following a b c d e Multiplication and Division of Binary Integers There are only four "rules" for multiplication of binary integers x 0 x 0 x 1 x Note that multiplication by 1 repeats the multiplicand (i.e = 101 ) while multiplication by 0 zeros things out (i.e = 0 ).So computing partial products is trivial; what makes multiplication difficult is the need to add the partial products obtained by multiplying the multiplicand or the "upper" integer by each bit of the multiplier or the "lower" integer <- multiplicand (13d) 101 <- multiplier (5d) <- partial products 0000 < < <- final product (65d) Division of binary integers is just plain difficult (except for the fact that binary integers are easy to deal with). Assuming you are fairly "fluent" in binary subtraction for unsigned integers where the subtrahend is less than or equal to the minuend (for instance subtract 101 from 1000), binary division can be done like decimal division. Example: Calculate (17d) 101 (5d) 11 <- quotient divisor -> 101 / <- dividend <- remainder This standard "pencil and paper" technique is the basis for the "shift, test and restore" algorithm used to divide binary numbers on the computer. 6

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