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Intro to CS – Honors I Representing Numbers

Intro to CS – Honors I Representing Numbers. Georgios Portokalidis gportoka@stevens.edu. Today’s Lecture. Numerical systems How do computers represent numbers Operations on integers Signed and unsigned numbers. Numerical Systems. We all know well the decimal system Base 10 system

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Intro to CS – Honors I Representing Numbers

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  1. Intro to CS – Honors IRepresenting Numbers Georgios Portokalidis gportoka@stevens.edu

  2. Today’s Lecture • Numerical systems • How do computers represent numbers • Operations on integers • Signed and unsigned numbers

  3. Numerical Systems • We all know well the decimal system • Base 10 system • The number can be actually expressed as: 4x102 + 6x101 + 6x100 • So a number N base 10 is denoted as (N)10 • uses digits 0..9 • And a digit d in position i has the value dix10i • How about base 2, (N)2? • Can you also do base 16 or 8? Tens Ones Hundreds 4 6 6 102 100 101

  4. Decimal to Binary • A simple algorithm • Let D= the number we wish to convert from decimal to binary • Find P, such that 2P is the largest power of two smaller or equal to D. • Repeat until P<0 • If 2P<=D then • Put 1 into column P • Subtract 2P from D • Else • Put 0 into column P • Subtract 1 from P • (356)10 (??)2

  5. Decimal to Binary (cont’d) • All binary numbers N can be represented as • B = dix2i+ di-1x2i-1+...+d1x21+ d0x20 , • where dibinary digit in position i • Odd numbers have d0=1 and even d0=0 • This reveals the rightmost bit of N • We need to shift the number to the right by 1 bit to again calculate its last bit • (B - d0x20)/2  dix2i-1+ di-1x2i-2+...+d1x20 • An alternative algorithm • Let D= the number we wish to convert from decimal to binary • Repeat until D equals 0 • Divide D with 2 (D/2) • Prepend the remainder of the division to the left of the binary number • Let D be the quotient of the division • (257)10 (??)2

  6. Hexadecimal 4 binary digits correspond to one hexadecimal digit • Base 16 number system • Digits are [0..9] [A..F] • Letters are case insensitive • Converting from binary to hexadecimal • There are some benefits 16 being a power of 2 • What is 1010 0010 in hex? • Answer: 0xC2 • Note the ‘0x’ prefix! • Converting from decimal to hex • Use the same algorithm as for converting decimal to binary • (76)10 (??)16

  7. Number Representation • Computer systems use the binary numerical system • Numbers need to be stored in fixed-size elements, such as the registers, hence they cannot be arbitrarily long • The smallest addressable piece of memory is a byte - 8 bits • CPU registers are 32-bit or 64-bit long • Most Significant Bit (MSB) = Leftmost bit in number, the bit with the highest value • Least Significant Bit (LSB) = Rightmost bit in number, the bit with the smallest value

  8. Binary Addition • 1010 • +1111 • -------- • Cheat sheet: • 0+0=0 • 1+0=1 • 1+1=10 • Column 20: 0+1=1.Record the 1. Temporary Result: 1; Carry: 0 • Column 21: 1+1=10. Record the 0, carry the 1.Temporary Result: 01; Carry: 1 • Column 22: 1+0=1 Add 1 from carry: 1+1=10. Record the 0, carry the 1.Temporary Result: 001; Carry: 1 • Column 23: 1+1=10. Add 1 from carry: 10+1=11.Record the 11. Final result: 11001

  9. Binary Multiplication • 1010 • x 11 • --------- • Cheat sheet • 0x0=0 • 1x0=0 • 0x1=0 • 1x1=1 • Multiplying by 2 is easy, just shift in (append) a 0 from the right • How about multiplying with 4 or 8?

  10. Overflows • Integer overflows occur when the result of an operation requires more bits than the length of the number involved Example with unsigned bytes 1001 1100 +0110 0111 ---------------- 1 0000 0011 The overflowing bit is lost

  11. Binary Division • 111011 / 11 • Same rules as in decimal division • When dividing integers the remainder is ignored

  12. Bitwise NOT or One’s Complement • Bitwise operations are logical operations on the individual bits of one or two numbers • Performs logical negation of each bit in the number • Just flip each bit’s value • Example: • NOT 0100 0110 = • 1011 1001

  13. Bitwise AND • Takes two numbers and performs the logical AND operation on each pair of corresponding bits • Example: • 0100 0110 • AND 0110 1100 = • 0100 0100

  14. Bitwise OR • Takes two numbers and performs the logical OR operation on each pair of corresponding bits • Example: • 0100 0110 • OR 0110 1100 = • 0110 1110

  15. Bitwise XOR • Takes two numbers and performs the logical XOR (eXclusiveOR) operation on each pair of corresponding bits • Example: • 0100 0110 • XOR 0110 1100 = • 0010 1010

  16. Signed Numbers • Sign-magnitude notation • Use the leftmost bit of a number as the equivalent of a sign: “0” is “+”, “1” is “-” • Example: 12  0000 1100, -12  1000 1100 • One’s complement • Flip all bits. The leftmost bit still indicates signed-ness • Example: 12  0000 1100, -12  1111 0011 • Two’s complement • Flip all bits, and add 1. The leftmost bit still indicates signed-ness • Example: 12  0000 1100, -12  1111 0100 • Excess 2(m-1) • m is the length of the number in bits. Every number is represented by adding it to 2(m-1) • Example: 12  27+12=140  1000 1100, -12  27-12=116  0111 0100 • Numbers are the same as two’s complement with sign bit flipped The largest signed number is smaller, than the largest unsigned number

  17. Why So Many Ways? • Let’s perform the following binary additions (-5+12), (-12+-5), and (12+-12) • Two’s complement makes the addition of both signed and unsigned integers • Different programming languages may use different ways of representing numbers • Java uses two’s complement for signed numbers

  18. Overflows and Signed Integers • Integer overflows occur when the result of an operation requires more bits than the length of the number involved Example with unsigned bytes 1001 1100 +0110 0111 ---------------- 1 0000 0011 Example with signed bytes 0001 1100 +0110 0111 ---------------- 1000 0011 The overflow happens into the sign bit The overflowing bit is lost

  19. Shifting Bits • The bits of a number can be moved or shifted to the left or right • Numbers are stored in fixed-size registers in the CPU • Moving bits can cause bits to shift-out and others to shift-in • Logical shift • Integers are treated as bit-strings • Appropriate for unsigned integers Example: 1001 1000 shift left 3 = 1100 0000 Example: 1001 1000 shift right 2 = 0010 0110 Zeroes are shifted in Shifted-out bits are lost

  20. Shifting Bits • The bits of a number can be moved or shifted to the left or right • Numbers are stored in fixed-size registers in the CPU • Moving bits can cause bits to shift-out and others to shift-in • Logical shift • Integers are treated as bit-strings • Appropriate for unsigned integers • Arithmetic shift • Integers are treated as numbers • Appropriate for signed integers Example: 1001 1000 shift left 3 = 1100 0000 Example: 1001 1000 shift right 2 = 1110 0110 Zeroes are shifted in Shifted-out bits are lost The leftmost bit is used when shifting-in bits from the left

  21. Summary • Numbers can be represented in different numerical systems • Computer use the binary system • Arithmetic operations like addition, multiplication, and division are straightforward • Bitwise operations perform actions on individuals bits of numbers • There are multiple ways to represent signed numbers • Two’s complement is the one used in most computers due to its simplicity • Numbers in the CPU have fixed length, so beware of overflows • Shift operations allow you to move bits in a number • The bits shifted-in from the left depend on whether we are performing an arithmetic or logical shift

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