10.1. Unit 10. Signed Representation Systems Binary Arithmetic

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1 0. Unit 0 Signed Representation Systems Binary Arithmetic

2 0.2 BINARY REPRESENTATION SYSTEMS REVIEW

3 0.3 Interpreting Binary Strings Given a string of s and 0 s, you need to know the representation system being used, before you can understand the value of those s and 0 s. Information (value) = s + Context (System) Unsigned Binary system =? BCD System ASCII system 65 0 A ASCII 4 BCD

4 0.4 Binary Representation Systems Integer Systems Unsigned Unsigned (Normal) binary Signed Signed Magnitude 2 s complement Excess-N* s complement* Floating Point For very large and small (fractional) numbers Codes Text ASCII / Unicode Decimal Codes BCD (Binary Coded Decimal) / (842 Code) * = Not fully covered in this class

5 0.5 Review of Number Systems Number systems consist of. A base (radix) r 2. r coefficients [0 to r-] Human System: Decimal (Base 0): 0,,2,3,4,5,6,7,8,9 Computer System: Binary (Base 2): 0, Human systems for working with computer systems (shorthand for human to read/write binary) Octal (Base 8): 0,,2,3,4,5,6,7 Hexadecimal (Base 6): 0-9,A,B,C,D,E,F (A thru F = 0 thru 5)

6 0.6 Binary Examples (00.) 2 = = (0000) 2 = =

7 0.7 Unique Combinations Given n digits of base r, how many unique numbers can be formed? r n What is the range? [0 to r n -] 2-digit, decimal numbers (r=0, n=2) combinations: digit, decimal numbers (r=0, n=3) 000 combinations: bit, binary numbers (r=2, n=4) combinations: bit, binary numbers (r=2, n=6) 64 combinations: Main Point: Given n digits of base r, r n unique numbers can be made with the range [0 - (r n -)]

8 0.8 Approximating Large Powers of 2 Often need to find decimal approximation of a large powers of 2 like 2 6, 2 32, etc. Use following approximations: ( thousand) = Kilo ( million) = Mega ( billion) = Giga ( trillion) = Tera- For other powers of 2, decompose into product of 2 0 or 2 20 or 2 30 and a power of 2 that is less than bit half word: 64K numbers 32-bit word: 4G numbers 64-bit dword: 6 million trillion numbers 2 6 = 2 6 * * 0 3 = 64, = 2 4 * * 0 6 = 6,000, = 2 8 * * 0 6 = 256,000, = 2 2 * * 0 9 = 4,000,000,000

9 0.9 Decimal to Unsigned Binary To convert a decimal number, x, to binary: Only coefficients of or 0. So simply find place values that add up to the desired values, starting with larger place values and proceeding to smaller values and place a in those place values and 0 in all others 25 0 = For 25 0 the place value 32 is too large to include so we include 6. Including 6 means we have to make 9 left over. Include 8 and.

10 0.0 Decimal to Another Base To convert a decimal number, x, to base r: Use the place values of base r (powers of r). Starting with largest place values, fill in coefficients that sum up to desired decimal value without going over = 4 B 0 hex 256 6

11 0. Signed Magnitude 2 s Complement System SIGNED SYSTEMS

12 0.2 Binary Representation Systems Integer Systems Unsigned Unsigned (Normal) binary Signed Signed Magnitude 2 s complement s complement* Excess-N* Floating Point For very large and small (fractional) numbers Codes Text ASCII / Unicode Decimal Codes BCD (Binary Coded Decimal) / (842 Code) * = Not covered in this class

13 0.3 Unsigned and Signed Normal (unsigned) binary can only represent positive numbers All place values are positive To represent negative numbers we must use a modified binary representation that takes into account sign (pos. or neg.) We call these signed representations

14 0.4 Signed Number Representation 2 Primary Systems Signed Magnitude Two s Complement (most widely used for integer representation)

15 0.5 Signed numbers All systems used to represent negative numbers split the possible binary combinations in half (half for positive numbers / half for negative numbers) In both signed magnitude and 2 s complement, positive and negative numbers are separated using the MSB MSB= means negative MSB=0 means positive

16 0.6 Signed Magnitude System Use binary place values but now MSB represents the sign ( if negative, 0 if positive) 4-bit Unsigned to 5 4-bit Signed Magnitude to +7 +/ bit Signed Magnitude to +27 +/

17 0.7 Signed Magnitude Examples 0 = -5 4-bit Signed Magnitude 0 +/- +/ = +3 = -7 Notice that +3 in signed magnitude is the same as in the unsigned system +/ = -9 8-bit Signed Magnitude +/ = +25 +/ Important: Positive numbers have the same representation in signed magnitude as in normal unsigned binary

18 0.8 Signed Magnitude Range Given n bits MSB is sign Other n- bits = normal unsigned place values Range with n- unsigned bits = [0 to 2 n- -] Range with n-bits of Signed Magnitude [ -2 n- to +2 n- ]

19 0.9 Disadvantages of Signed Magnitude. Wastes a combination to represent = 000 = Addition and subtraction algorithms for signed magnitude are different than unsigned binary (we d like them to be the same to use same HW) 4-6 Swap & make res. negative 6-4 -

20 s Complement System Normal binary place values except MSB has negative weight MSB of = -2 n- 4-bit Unsigned to 5 4-bit 2 s complement to bit 2 s complement to

21 0.2 2 s Complement Examples 0 = -5 4-bit 2 s complement = +3 Notice that +3 in 2 s comp. is the same as in the unsigned system = = bit 2 s complement = Important: Positive numbers have the same representation in 2 s complement as in normal unsigned binary

22 s Complement Range Given n bits Max positive value = 0 Includes all n- positive place values Max negative value = Includes only the negative MSB place value Range with n-bits of 2 s complement [ -2 n- to +2 n- ] Side note What decimal value is? - 0

23 0.23 Comparison of Systems Signed Mag s comp

24 0.24 Unsigned and Signed Variables In C, unsigned variables use unsigned binary (normal power-of-2 place values) to represent numbers = In C, signed variables use the 2 s complement system (Neg. MSB weight) to represent numbers =

25 0.25 IMPORTANT NOTE All computer systems use the 2's complement system to represent signed integers! So from now on, if we say an integer is signed, we are actually saying it uses the 2's complement system unless otherwise specified We will not use "signed magnitude" unless explicitly indicated

26 0.26 Zero and Sign Extension Extension is the process of increasing the number of bits used to represent a number without changing its value Unsigned = Zero Extension (Always add leading 0 s): 0 = 000 Increase a 6-bit number to 8-bit number by zero extending 2 s complement = Sign Extension (Replicate sign bit): pos. neg. 000 = = 00 Sign bit is just repeated as many times as necessary

27 0.27 Zero and Sign Truncation Truncation is the process of decreasing the number of bits used to represent a number without changing its value Unsigned = Zero Truncation (Remove leading 0 s): 000 = 0 Decrease an 8-bit number to 6-bit number by truncating 0 s. Can t remove a because value is changed 2 s complement = Sign Truncation (Remove copies of sign bit): pos. neg = = 00 Any copies of the MSB can be removed without changing the numbers value. Be careful not to change the sign by cutting off ALL the sign bits.

28 0.28 Data Representation In C/C++ variables can be of different types and sizes Integer Types (signed and unsigned) C Type Bytes s ATmega328 [unsigned] char 8 byte [unsigned] short [int] 2 6 word [unsigned] long [int] [unsigned] long long [int] Can emulate but has no single-instruction support Floating Point Types C Type Bytes s ATmega328 float 4 32 N/A double 8 64 N/A

29 ARITHMETIC 0.29

30 0.30 Binary Arithmetic Can perform all arithmetic operations (+,-,*, ) on binary numbers Can use same methods as in decimal Still use carries and borrows, etc. Only now we carry when sum is 2 or more rather than 0 or more (decimal) We borrow 2 s not 0 s from other columns Easiest method is to add bits in your head in decimal (+ = 2) then convert the answer to binary (2 0 = 0 2 )

31 0.3 Binary Addition In decimal addition we carry when the sum is 0 or more In binary addition we carry when the sum is 2 or more Add bits in binary to produce a sum bit and a carry bit no need to carry sum bit no need to carry sum bit no need to carry sum bit carry into next column of bits sum bit

32 0.32 Binary Addition & Subtraction 0 (7) (3) 0 0(0) (0) (5) (5)

33 0.33 Binary Addition 0 00(6) (7) 0(3)

34 0.34 Binary Addition (6) (7) 0 0 (3) (6) + 0 (7) + 0 (3) 0 carry bit sum bit carry bit sum bit (6) + 0 (7) + 0 (3) (6) (7) (3) 0 carry bit sum bit carry bit sum bit

35 0.35 Hexadecimal Arithmetic Same style of operations Carry when sum is 6 or more, etc. 4 D 6 + B = 8 0 = = 6 0 =

36 0.36 "Taking the 2's complement" SUBTRACTION THE EASY WAY

37 0.37 Taking the Negative Given a number in signed magnitude or 2 s complement how do we find its negative (i.e. - * X) Signed Magnitude: Flip the sign bit 00 = +6 => 0 = -6 2 s complement: Take the 2 s complement 00 = +6 => -6 = 00 Operation defined as:. Flip/invert/not all the bits ( s complement) 2. Add and drop any carry (i.e. finish with the same # of bits as we start with)

38 0.38 Taking the 2 s Complement Invert (flip) each bit (take the s complement) s become 0 s 0 s become s Add (drop final carry-out, if any) Original number = +9 flip is called the s complement of a number Resulting number = -9 Important: Taking the 2 s complement is equivalent to taking the negative (negating)

39 0.39 Taking the 2 s Complement Original number = -22 Take the 2 s complement yields the negative of a number Resulting number = Original # = 0 Take the 2 s complement 2 s comp. of 0 is Taking the 2 s complement again yields the original number (the operation is symmetric) 000 Back to original = Original # = -8 Take the 2 s complement Negative of -8 is -8 (i.e. no positive equivalent, but this is not a huge problem)

40 s Complement System Facts Normal binary place values but MSB has negative weight MSB determines sign of the number 0 = positive / = negative Special Numbers 0 = All 0 s (00 00) - = All s ( ) Max Positive = 0 followed by all s (0..) Max Negative = followed by all 0 s (00 00) To take the negative of a number (e.g. -7 => +7 or +2 => -2), requires taking the complement 2 s complement of a # is found by flipping bits and adding x = -7 flip ( s comp.) Add -x = -(-7) = +7

41 ADDITION AND SUBTRACTION 0.4

42 s Complement Addition/Subtraction Addition Sign of the numbers do not matter Add column by column Drop any final carry-out Subtraction Any subtraction (A-B) can be converted to addition (A + -B) by taking the 2 s complement of B (A-B) becomes (A + s comp. of B + ) Drop any carry-out

43 s Complement Addition No matter the sign of the operands just add as normal Drop any extra carry out (3) 0(-3) + 000(2) 00(5) (2) (-) Drop final carry-out 0 00 (3) + 0(-2) 000 () 00 0(-3) + 0(-2) 0(-5)

44 0.44 Unsigned and Signed Addition Addition process is the same for both unsigned and signed numbers Add columns right to left Examples: If unsigned If signed (9) (3) (2) (-7) (3) (-4)

45 s Complement Subtraction Take the 2 s complement of the subtrahend and add to the original minuend Drop any extra carry out 00(+3) - 000(+2) 0(-3) - 0(-2) Drop final carry-out _ s comp. of +2 Add _ s comp. of -2 Add

46 0.46 Unsigned and Signed Subtraction Subtraction process is the same for both unsigned and signed numbers Convert A B to A + Comp. of B Drop any final carry out Examples: If unsigned (2) (2) If signed (-4) (2) A s comp. of B Add (0) (-6) If unsigned If signed

47 0.47 Important Note Almost all computers use 2's complement because The same addition and subtraction algorithm can be used on unsigned and 2's complement (signed) numbers Thus we only need one adder circuit (HW component) to perform operations on both unsigned and signed numbers

48 OVERFLOW 0.48

49 0.49 Overflow Overflow occurs when the result of an arithmetic operation is too large to be represented with the given number of bits Conditions and tests to determine overflow depend on sign

50 0.50 Unsigned Overflow Overflow occurs when you cross this discontinuity = Plus With 4-bit unsigned numbers we can only represent 0 5. Thus, we say overflow has occurred

51 0.5 2 s Complement Overflow = = -0 With 4-bit 2 s complement numbers we can only represent -8 to +7. Thus, we say overflow has occurred Overflow occurs when you cross this discontinuity

52 0.52 Overflow in Addition Overflow occurs when the result of the addition cannot be represented with the given number of bits. Tests for overflow: Unsigned: if Cout = Signed: if p + p = n or n + n = p If unsigned (3) (4) (7) If signed (-3) (4) (+) If unsigned (6) (5) () If signed (6) (5) (-5) Overflow Cout = No Overflow n + p No Overflow Cout = 0 Overflow p + p = n

53 0.53 Overflow in Subtraction Overflow occurs when the result of the subtraction cannot be represented with the given number of bits. Tests for overflow: Unsigned: if Cout = 0 Signed: if addition is p + p = n or n + n = p If unsigned (7) (8) (-) If signed (7) (-8) (5) Desired Results 0_ A s comp. of B Add (5) (-) If unsigned Overflow Cout = 0 If signed Overflow p + p = n

54 FLOATING POINT 0.54

55 0.55 Floating Point Used to represent very small numbers (fractions) and very large numbers Avogadro s Number: * 0 23 Planck s Constant: * 0-27 Note: 32 or 64-bit integers can t represent this range Floating Point representation is used in HLL s like C by declaring variables as float or double

56 0.56 Fixed Point Unsigned and 2 s complement fall under a category of representations called Fixed Point The radix point is assumed to be in a fixed location for all numbers [Note: we could represent fractions by implicitly assuming the binary point is at the left A variable just stores bits you can assume the binary point is anywhere you like] Integers: 000. (binary point to right of LSB) For 32-bits, unsigned range is 0 to ~4 billion Fractions:.000 (binary point to left of MSB) storage Fixed point Rep. Range [0 to ) Main point: By fixing the radix point, we limit the range of numbers that can be represented Floating point allows the radix point to be in a different location for each value

57 0.57 Floating Point Representation Similar to scientific notation used with decimal numbers ±D.DDD * 0 ±exp Floating Point representation uses the following form ±b.bbbb * 2 ±exp 3 Fields: sign, exponent, fraction (also called mantissa or significand) Overall Sign of # S Exp. fraction

58 0.58 Normalized FP Numbers Decimal Example *0 5 is not correct scientific notation Must have exactly one significant digit before decimal point: +7.54*0 4 In binary the only significant digit is Thus normalized FP format is: ±.bbbbbb * 2 ±exp FP numbers will always be normalized before being stored in memory or a reg. The. is actually not stored but assumed since we always will store normalized numbers If HW calculates a result of 0.000*2 5 it must normalize to.0000*2 2 before storing

59 0.59 IEEE Floating Point Formats Single Precision (32-bit format) Sign bit (0=pos/=neg) 8 Exponent bits Excess-27 representation More on next slides 23 fraction (significand or mantissa) bits Equiv. Decimal Range: 7 digits x 0 ±38 Double Precision (64-bit format) Sign bit (0=pos/=neg) Exponent bits Excess-023 representation More on next slides 52 fraction (significand or mantissa) bits Equiv. Decimal Range: 6 digits x 0 ± S Exp. Fraction S Exp. Fraction

60 0.60 Floating Point vs. Fixed Point Single Precision (32-bits) Equivalent Decimal Range: 7 significant decimal digits * 0 ±38 Compare that to 32-bit signed integer where we can represent ±2 billion. How does a 32-bit float allow us to represent such a greater range? FP allows for range but sacrifices precision (can t represent all numbers in its range) Double Precision (64-bits) Equivalent Decimal Range: 6 significant decimal digits * 0 ±308

61 0.6 Exponent Representation Exponent needs its own sign (+/-) Rather than using 2 s comp. system we use Excess-N representation Single-Precision uses Excess-27 Double-Precision uses Excess-023 This representation allows FP numbers to be easily compared Let E = stored exponent code and E = true exponent value For single-precision: E = E => E =, E = 28 0 = For double-precision: E = E => E = -2, E = 02 0 = s comp. E' (stored Exp.) Excess Comparison of 2 s comp. & Excess-N Q: Why don t we use Excess-N more to represent negative # s

62 0.62 Single-Precision Examples 2 7 =28 2 = =3 = = * * = = 26 = +.0 *

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