# Binary Numbers The Computer Number System

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1 Binary Numbers The Computer Number System Number systems are simply ways to count things. Ours is the base-0 or radix-0 system. Note that there is no symbol for 0 or for the base of any system. We count,2,3,4,5,6,7,8,9, and then put a 0 in the first column and add a new left column, starting at again. Then we count -9 in the first column again. Each column in our system stands for a power of 0 starting at 0 0. Example: People use the base-0 system because we have 0 fingers! ,357,896 = x one million + 3 x one hundred thousand + 5 x ten thousand + 7 x one thousand + 8 x one hundred + 9 x ten + 6 x one.

2 Positional Notation A History Heritage of western culture: The (difficult) Roman representation of numbers: MCMXCVI = 996, but MM = 2000! (M = 000, C = 00, X = 0, V = 5, I = ) VII = 7 (5++), but XC = 90 (00 0), and (worst yet!) XLVII = 47 ( ). Even worse: X C = M, L/V=X. Ouch! A better idea -- positional notation: Each digit in a column represents a multiplier of the power of the base (0) represented by that column. The first column on the right is the zeroth power of 0. Succeeding columns to the left represent higher powers of 0. M? I? Examples of positional notation: = x x x x = 2x0 3 2

3 The Computer Number System All computers use the binary system : Binary number system: Base = 2. Thus there are 2 numbers: 0 and. A single binary number is called a Binary digit, or bit. Computers perform operations on binary number groups called words. Today, most computers use 32- or 64- bit words: Words are subdivided into 8-bit groups called bytes. One-half a byte is sometimes referred to as a nibble (a term not often used anymore). Computer numbers are and 0! A simple electronic switch can represent both binary computer numbers = = 0 3

4 Binary Numeric Representation 4 A 32-bit binary number: We will see ways to make this number more comprehensible below. As mentioned earlier, in the decimal system, each column represents a higher power of ten, starting at the right end with 0 0, e.g.: = x x x x 0 0, and 2002 = 2 x x x x 0 0. Likewise, in the binary number system, which is also positional, each position represents a larger power of two, starting with 2 0 on the right end of the whole number. Consider the binary number = 2 : = x x x x x x x 2 + x 2 0 = 2 8. Or, 23 0 = 0 x x x x x x x 2 + x 2 0 = 0 0 = 0

5 Reading Binary Numbers 5 In a decimal number, a non-0 digit in a column is treated as the multiplier of the power of 0 represented by that column (0 s clearly add no value). We read binary numbers the same way; 0 s count nothing and a in any column means that the power of 2 represented by that column is part of the magnitude of the number. That is:

6 Binary Number Examples = x x 2 = = x x 2 + x 2 0 = 4 + = = x x 2 0 = 8 + = = x x 2 2 = = = x x x x 2 0 = = = X 2 = ½ = = X 2 + X X 2 3 = = = X 2 + X 2 5 = = = x x x x 2 2 = = = x 2 + x X 2 3 = = 3.25 X = x 2 + X X 2 4 = =

7 Exercise # Convert the binary numbers to decimal:

8 Easier Ways to Express Binary Numbers Unfortunately, we were not born with 4 (or 8!) fingers per hand. The reason is that it is relatively difficult to convert binary numbers to decimal, and vice-versa. However, converting hexadecimal (base-6) numbers back and forth to binary is very easy (the octal, or base-8, number system was also used at one time). Since 6 = 2 4, it is very easy to convert a binary number of any length into hexadecimal form, and vice-versa: 0 6 = 0 0 = = 4 0 = = 8 0 = C 6 = 2 0 = = 0 = = 5 0 = = 9 0 = 00 2 D 6 = 3 0 = = 2 0 = = 6 0 = 00 2 A 6 = 0 0 = 00 2 E 6 = 4 0 = = 3 0 = = 7 0 = 0 2 B 6 = 0 = 0 2 F 6 = 5 0 = 2 The letters that stand for hexadecimal numbers above 9 can be upper or lower case both are used. Note that one nibble = one hex digit. 8

9 Binary-Hexadecimal Since 2 4 = 6, each hex digit effectively represents the same numeric count as four binary digits. Another way to say this is that one column in a hex number is the same as four columns of a binary number = 0x 95B.7A* *Note: The 0x prefix before a number signifies hexadecimal. 9

10 Hexadecimal-Binary Conversion Most computers process 32 or 64 bits at a time. In a 32-bit computer such as we will study, each data element in the computer memory (or word ) is 32 bits. Example: Separate into 4-bit groups, starting at the right: Converting: A A 6 F 6 B 6 E 6 } } } } } } } } Or, = 0x 78A5AFBE Another example: Grouping: = = (00) = 2 5 C. F 3 = 0x 25C.F3 0

11 Binary-Hex and Hex-Binary Examples More binary-hex conversions*: = = 0x BA = (000) = 0x 6E = (00) = 0x 3FD.87. To convert hex-binary, just go the other direction! 0x 2375 = (00) = x CD.89 = = x 37AC.6 = (00) (0) = x 3.DCAB = (00) = * Note that leading zeroes are added or removed as appropriate in the conversion processes.

12 Some Tricky Conversions Converting hex numbers to binary where leading or trailing zeros result is simple: Just drop the extra zeros! Examples: 0x 7A.F8 = (0)(00).()(000) = 00. 0x 29.3C = (000)(00).(00)(00) = The binary-hex conversion is a little trickier: Starting at the binary point, create group of 4 bits, then convert to hex (Go for fractions, for integers). Add 0 s to either end of the number to complete a group of four if necessary = (000).(00)(0) = 0x 8.3D ( No 0 s needed) = ([0])(00).(00)([00]) = 0x 7A.5C = ([00]0)(0).(00)([000]) = 0x 2E.58 While leading zeroes are only moderately important (you could probably figure out the hex number without completing the group of 4), trailing zeroes are imperative. If a fractional binary number does not have 4 bits in its last group, zeroes must be added to complete the group or the hex number will not be correct! Trailing zeroes are especially important to the value of the fraction. 2

13 More Conversions With Fractions Hex/binary conversion rule: On either side of the hexadecimal point, convert each hex digit to an equivalent 4-bit binary number, and drop leading integral and trailing fractional zeroes. Examples: 0x 79.EA = (0)(00).(0)(00) = x 2D.58 = (000)(0).(00)(000) = Binary/hex conversion rules: () Group binary digits in sets of 4, starting at the binary point (go left for integers, go right for fractions), (2) Add leading zeroes to get a full group of four bits on the left, add trailing zeroes to get a last group on the right, if necessary (The zeroes on the right are very important!), (3) Convert: (4-bit binary number) (hex digit). Examples: = ([0]0)(0).(00)(000) = 0x 67.A = ([00]0)(0).(0)(0[0]) = 0x 2B.7A 3

14 Exercise #2 Convert the following numbers: to hex: 0x 23d.b to binary: 00. to hex: 0x e5.8c to binary: 4

15 Integer Binary/Decimal Conversions Sadly, we live in a decimal world, so we need to convert from binary to decimal and vice-versa. As we saw in earlier slides, for binary decimal, brute-force is easiest: Ignore zeroes; for any in a column i, add the decimal number represented by 2 to the i th power. Thus: Number to be converted: Power of 2 of columns: Number represented: Thus the decimal number is: = 73. Another example: Number to be converted: Power of 2: Number represented: Thus the decimal number is: =

16 Fractional Binary/Decimal Conversions For numbers right of the binary point, we use the same approach, remembering that the powers of 2 are negative. Number to be converted: 0. 0 Power of 2: Number represented: Thus the decimal number is: = A second example: Number to be converted: Power of 2: Number represented: The decimal number is: =

17 Converting Mixed Binary Numbers Binary numbers with binary point fractions are handled by combining the two techniques shown above: Number to be converted: 0. 0 Power of 2: Number represented: Thus the decimal number is: = Other examples: 0.0 = = = = = = = =

18 Decimal Binary Integer Conversions The easiest way to convert decimal-to-binary integers is the method of successive division. In this method, we simply divide the decimal number by 2: The quotient becomes the new number to divide again. The remainder, which will always be or 0, becomes one bit of the binary number, least significant digit first. The last quotient will always be 0, the last remainder always. Division continues until the quotient is 0 (with a last remainder of ). Thus, converting 26 to binary: 26 2 = 3, remainder = 6, remainder = 3, remainder = =, remainder 2 = 0, remainder

19 Another Decimal-to-Binary Example Convert 7 0 to binary: 7 2=58, remainder 58 2=29, remainder =4, remainder 4 2=7, remainder 0 7 2=3, remainder 3 2=, remainder 2=0, remainder Read this way Starting from the bottom (MSB), 7 0 = Check: 00 2 = =

20 Converting Decimal Fractions to Binary Fractions The method of successive multiplication: Decimal to binary fractions. Example Convert 0.25 to binary: 0.25 x 2 = 0.5 ; note 0 to left of binary point. 0.5 x 2 =.0 ; note to left of binary point. There is no part of the decimal fraction left, thus we are done. Read the binary fraction as the numbers to left of binary point in the two results, the remainder from the first multiplication first. Thus, = Decimal-fraction-to-binary-fraction conversion rules: Multiply decimal fraction by 2. Record the number left of the binary point. This will always be a or a 0. Multiply the remaining fraction by 2, repeating above action. Continue until the fraction is eliminated. 20

21 Converting Decimal Binary Fractions (2) Additional examples: 0.375: x 2 = x 2 = x 2 =.0 Thus = : x 2 = x 2 = x 2 =.46 Then, x 2 = (Finite decimal fractions can x 2 =.664 result in repeating binary x 2 =.328 fractions!*) x 2 = x 2 =.32 2 * Thus we must change the last rule on the previous page to read: Continue until the decimal fraction is eliminated or the binary fraction has at least twice as many places as the decimal fraction. This is because each column position in a binary fraction is much more significant than each column position in a decimal fraction.

22 Exercise #3 Convert as directed: 237 to binary: to binary: to binary:

23 Converting Mixed Decimal Numbers Converting mixed decimal numbers means that we must perform two operations. For the integer part of the number, we do the method of successive division. For the decimal part, we do the method of successive multiplication. We recognize that for decimal numbers with a fraction part, we may not be able to convert the number exactly, since we could get a repeating fraction. In that case, we simply do successive multiplication enough times to get the accuracy of the binary fraction that we desire, at least twice the number of decimal places. Note that if the last of the 2x places is a 0, continue until you get a ). 23

24 Mixed Decimal Conversions -- Examples 36.25: 36/2=8, 0 rem; 8/2=9, 0 rem; 9/2=4, rem; 4/2=2, 0 rem; 2/2=, 0 rem, /2=0, rem. Read for integers,.25x2=0.25;.25x2=0.5;.5x2=.0 Read for fractions. Then = : 9/2=9, rem; 9/2=4, rem; 4/2=2, 0 rem; 2/2=, 0 rem; /2=0, rem..375x2=0.75;.75x2=.5;.5x2=.0. Then = : 7/2=3, rem; 3/2=, rem; /2=0, rem..33x2=0.66;.66x2=.32;.32x2=0.64;.64x2=.28 Then : 0/2=5, 0 rem; 5/2=2, rem; 2/2=, 0 rem; /2=0. rem..7x2=0.34;.34x2=0.68;.68x2=.36;.36x2=0.72;.72x2=.44 Then * *In the last example, we had to go to five places, since the fourth place was a 0. 24

25 Doing Binary Math Adding in binary is identical to the operations in our own decimal system. You simply have to remember the Addition table. Thus: Subtraction works the same way but we don t have to worry about subtraction. More on that next lecture. We also don t have to worry about adding more than two numbers. Why? Because a computer can add only two numbers! However, we know that in adding two numbers together, we may actually have to add three numbers there could be a carry from the previous column. 25

26 Assume we are adding two binary numbers together, each with several digits. As in decimal math, we add on a columnar basis. Thus, in any column i, we add the ith digit of number a, the ith digit of number b, and any carry from the (i ) column. Or in any column, the sum is a i plus b i plus c i- ; Sum = a i + b i + c i-. Our new add table is: Adding With A Carry c i- +a i +b i T

29 Homework 29 Anybody in here ever see the movie 50 First Dates? Hopefully, you have listened carefully and perhaps even made a few notes today. Before bedtime tonight, consult your notes (if any), re-read the lecture on-line and make two lists: () things that you thought were important today, and (2) things you did not completely understand. Making list () reinforces your learning; list (2) gives you things to ask about when visiting me during office hours! Remember: those office hours are for YOUR benefit.

30 Hexadecimal-to Decimal Conversion Since hex numbers are used in computer displays, it is useful to convert decimal hex and back. For hex decimal, we use the same brute-force method, as for binary-to-decimal conversion. Consider 3FB7 6 : Number in hex: 3 F B 7 Position as a power of 6: Decimal value of 6 n : F B The decimal number is then 3(4096)+5(256)+(6)+7() = 6,3. 30

31 Hexadecimal-to Decimal Conversion (2) For fractions, the conversion is the same, remembering that hex digits to the right of the hexadecimal point are multipliers of negative powers of 6: 0x 0.2A6 = (2/6)+(0/[6] 2 )+(6/[6] 3 ) = Mixed numbers are treated similarly: 0x B7.CE = (6) + 7() + 2/6 + 4/256 =

32 Integer Decimal-to-Hex Conversion For converting decimal to hexadecimal integers, we use the method of successive division, as for decimal/binary conversions: Convert to hex: We perform successive divisions by =23, remainder 4 (= E) 23 6=, remainder 7 Read in reverse order as before. 6=0, remainder In reverse order, the hexadecimal number is 7E, or = 7E 6. Similarly, converting 65 0 : 65 6=40, remainder (=B) Note that as in binary conversion, 40 6=2, remainder 8 the last quotient will always be =0, remainder 2 However, the last remainder may Thus, 65 0 =28B 6. be anything from to F. 32

33 Integer Decimal-to-Hex Conversion (2) Additional examples of integer conversion ( read for the answer ): 00 =? 00/6=6, 4 rem; 6/6=0, 6 rem; thus 00 = 0x =? 4096/6=256, 0 rem; 256/6=6, 0 rem; 6/6=, 0 rem; /6=0, rem; thus 4096 = 0x =? 335/6=20, 5 (= 0x F) rem; 20/6=, 4 rem; /6=0, rem; thus 335 = 0x 4F =? 23795/6=487, 3 rem; 487/6=92, 5 (= 0x F) rem; 92/6=5, 2 (= 0x C) rem; 5/6=0, 5 rem; thus = 0x 5CF =? 024/6=64, 0 rem; 64/6=4, 0 rem; 4/6=0, 4 rem; thus 024 = 0x 400. Note that remainders that are > 9 must be converted to the hex digits A-F to get the correct hexadecimal number. 33

34 34 The University of Texas at Dallas Fraction Decimal-to-Hex Conversion For decimal to hexadecimal fraction conversions, we use the method of successive multiplication, as we did for decimal to binary conversions: (we also read for the answer) Convert 0.25 to base 6: 0.25x6=2.0; thus 0.25 = 0x 0.2. Convert 0.335: 0.335x6=5.36;.36x6=5.76;.76x6=2.6, thus x 0.55C. As in the binary case, exact decimal fractions can give hexadecimal repeating fractions. However, since the base 6 is > 0, we do not have to carry out the fraction to so many places in this case the same number of places as the decimal fraction is sufficient. Final example: 0.95 =?.95x6=5.2;.2x6=3.2; thus x 0.F3 (note that 0x 0.F3 = ). Once again, fractional numbers shown as > 9 must be converted to the hex digits A-F (or a-f) to get the correct hexadecimal fraction.

35 Mixed Number Decimal/Hex Conversion For decimal numbers with integers and fractions, we use both techniques just as for decimal-binary conversion: Convert to hexadecimal: 254/6=5, 4 remainder (= E); 5/6=0, 5 remainder (= F)..76x6=2.6;.6x6=2.56 (note that 2 = C). Thus, x FE.C2. Convert to hexadecimal: 58/6=3, 0 remainder (= A); 3/6=0, 3 remainder..642x6=0.272;.272x6=4.352;.352x6=5.632 (again, 0 = A). Thus, x 3A.A45. 35

36 Exercise Do the following conversions and check answers with those shown. Convert to decimal : 0x 2.c x 3e x bcd.4f (approx.) Convert to hex: x x 2b5.e 36

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