Decimal Number System □ Most computers count in binary, which we can easily understand from the decimal so ingrained in us 35462
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Computer Fundamentals:
Number Systems
Dr Robert Harle
Today's Topics
yThe significance of the bit and powers of 2 yData quantities (B, kB, MB, GB, etc) yNumber systems (decimal, binary, octal , hexadecimal) yRepresenting negative numbers (sign-magnitude, 1's complement, 2's complement) yBinary addition (carries, overflows) yBinary subtraction So... yWhat is a bit?The Significance of the Bit
yA bit (Binary digIT) is merely 0 or 1 yIt is a unit of information since you cannot communicate with anything less than two states yThe use of binary encoding dates back to the 1600s with Jacquard's loom, which created textiles using card templates with holes that allowed needles throughBits and Computers
yThe nice thing about a bit is that, with only two states, it is easy to embody in physical machinery yEach bit is simply a switch and computers moved from vacuum tubes to transistors for this e-Decimal Number System
yMost computers count in binary, which we can easily understand from the decimal so ingrained in us 35462Decimal Number System
yMost computers count in binary, which we can easily understand from the decimal so ingrained in us354622x100
Decimal Number System
yMost computers count in binary, which we can easily understand from the decimal so ingrained in us354622x1006x101+
Decimal Number System
yMost computers count in binary, which we can easily understand from the decimal so ingrained in us354622x1006x101+4x102+
Decimal Number System
yMost computers count in binary, which we can easily understand from the decimal so ingrained in us354622x1006x101+4x102+5x103+
Decimal Number System
yMost computers count in binary, which we can easily understand from the decimal so ingrained in us354622x1006x101+4x102+5x103+3x104+
Binary
yBinary is exactly the same, only instead of ten digits/states (0 to 9) we have just two, so the base becomes 2:101100x201x21+1x22+0x23+1x24+
Binary
yBinary is exactly the same, only instead of ten digits/states (0 to 9) we have just two, so the base becomes 2:10110b = 22d0x201x21+1x22+0x23+1x24+
Binary
yBinary is exactly the same, only instead of ten digits/states (0 to 9) we have just two, so the base becomes 2:10110b = 22d0x201x21+1x22+0x23+1x24+
Most Signiificant
Bit (MSB)Least Signiificant
Bit (LSB)
Works for Fractional Numbers too...
35.4622x10-36x10-2+4x10-1+5x100+3x101+
Works for Fractional Numbers too...
10.110b = 2.75d0x2-31x2-2+1x2-1+0x20+1x21+35.4622x10-36x10-2+4x10-1+5x100+3x101+
Check11.011b
Check11.011b = 3.375d1x2-31x2-2+0x2-1+1x20+1x21+
Representable Numbers
yWith d decimal digits, we can represent 10d different values, usually the numbers 0 to (10d-1) inclusive yIn binary with n bits this becomes 2n values, usually the range 0 to (2n-1) yComputers usually assign a set number of bits (physical switches) to an instance of a type. yAn integer is often 32 bits, so can represent positive integers from 0 to 4,294,967,295 incl. yOr a range of negative and positive integers...Other Common Bases
yHigher bases make for shorter numbers that are easier for humans to manipulate. e.g.6654733d=11001011000101100001101b
yWe traditionally choose powers-of-2 bases because this corresponds to whole chunks of binaryOther Common Bases
yHigher bases make for shorter numbers that are easier for humans to manipulate. e.g.6654733d=11001011000101100001101b
yWe traditionally choose powers-of-2 bases because this corresponds to whole chunks of binary yOctal is base-8 (8=23 digits, which means 3 bits per digit) y6654733d=011-001-011-000-101-100-001-101b= 31305415oOther Common Bases
yHigher bases make for shorter numbers that are easier for humans to manipulate. e.g.6654733d=11001011000101100001101b
yWe traditionally choose powers-of-2 bases because this corresponds to whole chunks of binary yOctal is base-8 (8=23 digits, which means 3 bits per digit) y6654733d=011-001-011-000-101-100-001-101b= 31305415o yHexadecimal is base-16 (16=24 digits so 4 bits per digit) yOur ten decimal digits aren't enough, so we add 6 new ones: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,FOther Common Bases
yHigher bases make for shorter numbers that are easier for humans to manipulate. e.g.6654733d=11001011000101100001101b
yWe traditionally choose powers-of-2 bases because this corresponds to whole chunks of binary yOctal is base-8 (8=23 digits, which means 3 bits per digit) y6654733d=011-001-011-000-101-100-001-101b= 31305415o yHexadecimal is base-16 (16=24 digits so 4 bits per digit) yOur ten decimal digits aren't enough, so we add 6 new ones: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F yBecause we constantly slip between binary and hex, we have a special marker for it yPrefix with '0x' (zero-x). So 0x658B0D=6654733d, 0x123=291d Bytes yA byte was traditionally the number of bits needed to store a character of text yA de-facto standard of 8 bits has now emerged y256 values y0 to 255 incl. yTwo hex digits to describe y0x00=0, 0xFF=255 yCheck: what does 0xBD represent? Bytes yA byte was traditionally the number of bits needed to store a character of text yA de-facto standard of 8 bits has now emerged y256 values y0 to 255 incl. yTwo hex digits to describe y0x00=0, 0xFF=255 yCheck: what does 0xBD represent? yB → 11 or 1011 yD → 13 or 1101 yResult is 11x161+13x160 = 189 or 10111101Larger Units
yStrictly the SI units since 1998 are: yKibibyte (KiB) y1024 bytes (closest power of 2 to 1000) yMebibyte (MiB) y1,048,576 bytes yGibibyte (GiB) y1,073,741,824 bytesLarger Units
yStrictly the SI units since 1998 are: yKibibyte (KiB) y1024 bytes (closest power of 2 to 1000) yMebibyte (MiB) y1,048,576 bytes yGibibyte (GiB) y1,073,741,824 bytes ybut these haven't really caught on so we tend to still use the SI Kilobyte, Megabyte, Gigabyte. This leads to lots of confusion since technically these are multiples of 1,000.The Problem with Ten
Unsigned Integer Addition
yAddition of unsigned integers works the same way as addition of decimal (only simpler!) y0 + 0 = 0 y0 + 1 = 1 y1 + 0 = 1 y1 + 1 = 0, carry 1 yOnly issue is that computers have fixed sized types so we can't go on adding forever... 111+ 001001 + 001
Carry lflag:Carry lflag:
Modulo or Clock Arithmetic
yOverflow takes us across the dotted boundary ySo 7+1=0 (overflow) yWe say this is (7+1) mod 8 000 001 010011100101110111000
001 010 011 100101
110
111
Negative Numbers
yAll of this skipped over the need to represent negatives. yThe naïve choice is to use the MSB to indicate +/- y1 in the MSB → negative y0 in the MSB → positive yThis is the sign-magnitude technique-7 = 1111NegativeNormal positive
representation of 7Difficulties with Sign-Magnitude
yHas a representation of minus zero (10002=-0) so wastes one of our 2n labels yAddition/subtraction circuitry must be designed from scratch 1101+ 0001 1110
Our unsigned addition alg.
Difficulties with Sign-Magnitude
yHas a representation of minus zero (10002=-0) so wastes one of our 2n labels yAddition/subtraction circuitry must be designed from scratch 1101+ 0001 1110
Our unsigned addition alg.+13
+1 +14Unsigned
interpretationDifficulties with Sign-Magnitude
yHas a representation of minus zero (10002=-0) so wastes one of our 2n labels yAddition/subtraction circuitry must be designed from scratch 1101+ 0001
1110-5
+1 -6+13 +1 +14Sign-mag
interpretationUnsigned interpretationOur unsigned addition alg.
Alternatively...
yGives us two discontinuities and a reversal of direction using normal addition circuitry!! 000 001 010011100101110111000
001 010 011 100101
110
111
Ones' Complement
yThe negative is the positive with all the bits flipped y7 → 0111 so -7 → 1000 yStill the MSB is the sign yOne discontinuity but still -0 :-( 000 001 010011100101110111000
001 010 011 100101
110
111
Two's Complement
yThe negative is the positive with all the bits flipped and 1 added (the same procedure for the inverse) y7 → 0111 so -7 → 1000+0001 → 1001 yStill the MSB is the sign yOne discontinuity and proper ordering 000 001 010 011 100101
110
111
000 001 010
011100101110111
Two's Complement
yPositive to negative: Invert all the bits and add 1 yNegative to positive: Same procedure!!1011 (-5) → 0100 → 0101 (+5)0101 (+5) → 1010 → 1011 (-5)
Signed Addition
1101+0001 1110
Our unsigned addition alg.y...it just works with our addition algorithm! +13 +1 +14
Unsigned
Signed Addition
1101+0001
1110-3
+1 -2+13 +1 +142's-compUnsigned
Our unsigned addition alg.y...it just works with our addition algorithm!Signed Addition
1101+0001
1110-3
+1 -2+13 +1 +142's-compUnsigned
Our unsigned addition alg.y...it just works with our addition algorithm! ySo we can use the same circuitry for unsigned and 2s- complement addition :-) yWell, almost.Signed Addition 0100+0100
1000+4
+4 +8UnsignedCarry lflag: 0
ySo we can use the same circuitry for unsigned and 2s- complement addition :-) yWell, almost. yThe problem is our MSB is now signifying the sign and our carry should really be testing the bit to its right :-(Signed Addition 0100+0100
1000+4
+4 -8+4 +4 +82's-compUnsignedCarry lflag: 0
ySo we can use the same circuitry for unsigned and 2s- complement addition :-) yWell, almost. yThe problem is our MSB is now signifying the sign and our carry should really be testing the bit to its right :-( ySo we introduce an overflow flag that indicates this problemSigned Addition 0100+0100
1000+4
+4 -8+4 +4 +82's-compUnsignedCarry lflag: 0
Overlflow: 1
Integer subtraction
yCould implement the "borrowing" algorithm you probably learnt in school yBut why bother? We can just add the2's complement instead.
0100- 0011 0100
+1101