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Representing Numbers Using Bases

Representing Numbers Using Bases

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Representing Numbers Using Bases

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  1. Representing Numbers Using Bases • Numbers in base 10 are called decimal numbers,they are composed of 10 numerals ( ספרות ).9786 = 9*1000 + 7*100 + 8*10 + 6*1 = 9*103 + 7*102 + 8*101 + 6*100 • Numbers in base 2 are called binary numbers, they are composed of the numerals 0 and 1.110101 = 1*32 + 1*16 + 0*8 + 1*4 + 0*2 + 1*1 = 1*25 + 1*24 + 0*23 + 1*22 + 0*21 + 1*20 • 1111 1111 1111 1111 = 32768 + 16384 + 8192 + 4096 + 2048 + 1024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 215 + 214 + … + 21 + 20 = S2n = 216 -1

  2. Converting Between Bases • From base 2 to base 10 is easy: just multiply each numeral by its exponent. 1001b = 1*8 + 1*1 = 9d • From base 10 to base 2 is slightly more complicated: In a loop from 31 to 0 (in the case of a 32 bit word) perform the following: • Divide the decimal number by 2n and write the result in the nth position. • Replace the number with the number minus the result multiplied by 2n • Decrement n by 1.

  3. A Base Conversion Example In C++ it looks like this: for(n=31;n>=0;n--){ res = decimal/(int)pow(2,n); decimal = decimal - res*(int)pow(2,n); cout << res; } 500/29 = 0 ; 500 - 0 = 5004/23 = 0 ; 4 - 0 = 4 500/28 = 1 ; 500 - 256 = 244 4/22 = 1 ; 4 - 4 = 0 244/27 = 1 ; 244 - 128 = 116 0/21 = 0 ; 0 - 0 = 0 116/26 = 1 ; 116 - 64 = 52 0/20 = 0 ; 0 - 0 = 0 52/25 = 1 ; 52 - 32 = 20 20/24 = 1 ; 20 - 16 = 4 500d = 111110100b

  4. Representation (יצוג) of Numbers • int, long, short, char - positive numbers are stored as a binary number where the MSB (Most Significant Bit) is 0A short is represented by 16 bits 100d = 26 + 25 + 22 =0000000001100100An int is represented by 32 bits 65545d = 216 + 23 + 20 = 00000000000000010000000000001001A char is represented by 8 bits‘a’ = 48d = 25 + 24 = 00110000

  5. Signed and Unsigned Numbers • In order to calculate both positive and negative numbers we must be able to distinguish (להבדיל) between them. • The most obvious solution is to add a bit that represents the sign. The name of this scheme is called: sign and magnitude (גודל). • There were several problems with this scheme: • where do we put the sign? • an extra step is needed to calculate the sign bit • there is both a positive and negative zero Thus a new representation was chosen

  6. Two's Complement • In the representation chosen leading 0s means positive and leading 1s means negative. • The positive half of the numbers from 0 to 2,147,483,647d (231 -1) uses the same representation as before. • The smallest negative number is -2,147,483,648 (-231) and is represented by 10000…00000b • It is followed by -2,147,483,647 (1000…0001b) up to -1 (1111…1111b). • There is a negative number (-231) with no positive partner, beware when programming. • The sum of a number and its inverse (נגדי) is -1.

  7. Negative Integers • Negative integers are represented using the two's-complement (משלים לשניים) method. The number is represented in binary, the bits are flipped and 1 is added. Thus the MSB is always 1 • Thus in a char:-100d = -01100100 = 10011011 + 1 = 1001110011111111 = 00000000 + 1 = -1d-128d = -10000000 = 01111111 + 1= 1000000011000101 = 00111010 + 1 = 00111011 = -59d • In a short:-25,000d = -0110000110101000 = 1001111001010111 + 1 = 10011110011000

  8. Unsigned Integers • Are stored as binary numbers. The MSB (sign bit) can be 1 or 0, the number is still positive.Thus in a char:10011100 = 27 + 24 + 23 + 22 = 156 11111111 = 27 + … + 20 = 28 - 1 = 255 • A problem arises when we have to compare numbers. How do we know if the number is an unsigned large number or a negative number. The solution is to add instructions which handle unsigned numbers. sltu $t0,$s0,$s1sltiu $t0,$s0,100

  9. Hexadecimal Numbers • Number in base 16 are called hexadecimalnumbers,they are composed of 16 numerals (0-9,a-f).9786hex = 9*163 + 7*162 + 8*161 + 6*160 = = 9*4096 + 7*256 + 8*16 + 6*1 = 38790d0xabcdef = 10*165 + 11*164 + 12*163 + 13*162 + 14*161 + 15*160 =11259375d • The conversion from binary to hex is very easy, each hex digit is 4 binary digits:0x0 = 0000 0x1 = 0001 0x2 = 0010 0x3 = 00110x4 = 0100 0x5 = 0101 0x6 = 0110 0x7 = 01110x8 = 1000 0x9 = 1001 0xa = 1010 0xb = 10110xc = 1100 0xd = 1101 0xe = 1110 0xf = 1111

  10. Binary to Hexadecimal • Using the previous table it is easy to represent 32 bit binary numbers in a short form:0100 1100 0010 1101 1000 1100 0111 1001 = 4 a 2 d 8 c 7 9 =0x4a2d8c79. • Conversion from decimal to hexadecimal is similar to converting decimal to binary. for(n=7;n>=0;n--){ res = decimal/(int)pow(16,n); decimal = decimal - res*(int)pow(16,n); if(res>9) cout << (char)((res -10) + 'a'); else cout << res; }

  11. Addition and Subtraction • Lets add 6 to 7:00…00111 = 7d00…00110 = 6d00…01101 = 13d • Subtraction uses addition, the operand is simply negated before being added:7 - 6 = 7 + (-6) =00…00111 = 7d11…11010 = -6d00…00001 = 1d • The pseudoinstruction neg $t0,$t1 #$t0=-$t1is implemented by: sub $t0,$zero,$t1

  12. Overflow (גלישה) • Overflow occurs when the result of a operation can't be represented by the hardware. This can happen when we add two numbers of the same sign or subtract two large numbers of opposing signs. • Overflow is detected by the following table:Operation A B Result A+B >=0 >=0 <0 A+B <0 <0 >=0 A- B >=0 <0 <0 A- B <0 >=0 >=0 • How is overflow detected in unsigned numbers? It isn't. The instructions add, addi, sub cause exceptions when overflow occurs. The instructions addu, addiu, subu ignore overflow. Guess what instructions MIPS C compilers use?

  13. Logical Operations • All processors provide logical instructions that operate on the bits of the operands. • and, andi, or, ori, xor and xori perform the AND, OR and XOR (eXlusive OR) operations. • If $t0 contains 10 and $t1 contains 9 then:and $t2,$t0,$t1 # $t2=10&9=8or $t2,$t0,$t1 # $t2=10|9=11xor $t2,$t0,$t1 # $t2=10&9=3 • MIPS provides a NOR (Not OR) instruction as well:nor $t2,$t0,$t1 # $t2=~(10|9)=41100 NOR 1001 = 0100 • The logical NOT is a pseduinstruction in MIPS.

  14. Shift Instructions • In order to access bits or fields of bits the logical operations and the shift operations are used. • They move all bits in a word to the left or right, filling the emptied bits with 0s. The instructions are shift left logical (sll) , and shift right logical(srl).These 2 instructions use the shamt field of the R-type instruction. • sll $t0,$t1,6 # $t0 = $t1<<8;0000 0000 0000 1101 -> 0000 1101 0000 0000srl$t2,$t0,3 # $t2 = $t0>>3;0000 1101 0000 0000 -> 0000 0001 1010 0000 • The instructions sllv and slrv contain the shift amount in a register.sllv $t0,$t1,$t2 # $t0=$t1>>$t2

  15. The Basic Building Blocks

  16. Building an ALU • The arithmetic logic unit or ALU is the device that performs arithmetic and logical operations in the computer. We will build it out of 4 basic building blocks. The AND gate, Or gate, Inverter and Multiplexor. • We have now a 1-bit logical unit for AND and OR:

  17. 1-bit Full Adder • 1-bit addition has 3 inputs (a, b, CarryIn) and 2 outputs (Sum, CarryOut). • The CarryOut is set if at least 2 out of a, b, and CarryIn are 1. • The Sum is set when exactly one input is 1 or all three are 1.

  18. 1-bit ALU • We now have a 1-bit ALU that can perform AND, OR and addition. • By connecting 32 of these 1-bit ALUs we can build a 32-bit ALU. The CarryOut of the N 1-bit ALU is connected to the CarryIn of the N+11-bit ALU. This adder is called a ripple carry or ripple adder. A single carry out in the least significant bit (LSB) can cause a carry out of the MSB.

  19. 32-bit ALU • Subtraction is performed by adding the negative value. Thus we have to add an inverter to our ALU.

  20. Adding the slt instruction • slt produces 1 if rs < rt. Thus slt will set all but the LSB to 0 with the LSB being dependent on the result. We add a new input to the multiplexor called Less, this will be the output of slt. To set the LSB we perform rs-rt. If the result is negative we must set the output. Thus the sign bit is the correct output. But it isn't the output of slt, so a new output is added to the 1-bit ALU of the MSB, Set.

  21. The New ALU • As we can see all the Less inputs are hardwired to zero except the LSB Less input which is wired to the Set output of the MSB 1-bit ALU.

  22. Comparison in the ALU • In order to implement the bne and beq instructions we must compare between two numbers. a is equal to b if a-b=0The output line Zero is set to 1 if all the Results are 0, which happens if a == b.

  23. ALU Summary • Our ALU can perform AND, OR, Addition, Subtraction, Set on less then (slt), and Equality tests. The universal symbol of the ALU is shown here: