1. EGRE 254 Number Systems and Codes 1/12/09
Digital Logic Design
EGRE 254
Number Systems and Codes
1/12/09
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2. Positional Number Systems
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Positional Number Systems
Numbers are commonly represented in the base 10 positional number system.
For example
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3. In general (see page 26)
A number D of radix or base r with p digits to the left of the radix point and n digits to the right of the radix point can be expressed as:
di – in base 10
di – in base r
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4. 11/20/2018
Examples
= (1x x x x10-1 )8
= (1x82 + 5x81 + 6x80 + 7x8-1 )10
= ( )10 =
559 = (5x x100)9 = (5x91 + 5x90)10
= (45 + 5)10 = 5010
1212 = (1x x120)10 = (12 + 2)10 = 1410 =
= = 44.75
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5. Examples Does r10 = 10r ? Prove it! Does rb = br ? Prove it!
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Examples
Does r10 = 10r ? Prove it!
Does rb = br ? Prove it!
111b = Find b.
(41/3)b = 13b Find b.
(33/3)b = 11b Find b.
1. 10r = (1 x r1 + 0 x r0)10 = r10
2. Consider 23 = 210 = 102 != 112
3. 111b = 1xb2 + 1xb1 + 1 = b2 + b + 1 = 133
b2 + b = 0
(b-11)(b+12) = 0
b = 11, b = -12
(-12)2 + (-12) + 1 = 144 –
Note: (12)-12 = 1x(-12)1 + 2x(-12)0 = = -10.
Q. What is the advantage of a number system with a negative base?
We don’t need the “-” sign.
4. 4b + 1 = 3(b+3) = 3b + 9
b= 9-1 = 8
5. 3b + 3 = 3(b + 1) = 3b + b
? |b| > 3
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6. Decimal (10) Binary (2) Octal (8) Hexadecimal (16) 1 2 10 3 11 4 100 51 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F
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7. Example (5D4.A2)16 = (5x102+Dx101+4+Ax10-1+2x10-2)16
= (5x162+13x x16-1+2x16-2)10
= ( )10
= ( )10
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8. How do we convert from base 10 to base r?
We could use previous techniques, but we would have to do arithmetic in base r.
Not desirable.
Consider two cases.
Integer number.
Fractional number.
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9. Base 10 to base r (integer case)
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10. Example – Integer to Hex
1492 = (d3 d2 d1 d0)16 = d3x163+ d2x162+ d1x16+ d0
1492/16 = 93+4/16 = d3x162+ d2x16+ d1+ d0/16
d0 = 4
93 = d3x162+ d2x16+ d1
93/16 = 5 +13/16 = d3x16+ d2+ d1/16
d1 = 1310 = D16
5 = d3x16+ d2
5/16 = 0 + 5/16 = d3 + d2/16, d3 = 0, d2 = 5
= 5D416
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11. In practice You may get mixed up on direction to read the answer.
5D416 or 4D516?
Note that 5D4 = 0…05D4
4D5 not same as 4D50…0
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12. Example fraction to hex
( )10 = (.d-1 d-2 d-3…)16
= (d-1x16-1+ d-2x16-2+ d-3x16-3+…)10
How do we find d-1? Multiply by 16.
16 x =
= d-1+ d-2x16-1+ d-3x16-2+…
d-1 = 10 and .125 = d-2x16-1+ d-3x16-2+…
Multiply by 16. Then,.125 x 16 = 2.00 = d-2+ d-3x16-1+…
d-2 = 2 and d-n = 0 for n > 2.
Therefore, ( )10 = (.A2)16
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13. Example
For a number containing both an integer and a fractional part. Compute the two parts separately and combine.
= 5D4.A216
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14. Special Case converting between base r and rn
Convert to binary.
Could convert to decimal and then binary.
Easier way.
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15. Examples Convert (11010.11)2 to octal, and to hexadecimal.
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Examples
Convert ( )2 to octal, and to hexadecimal.
( )2 = (11, )2 = 32.68
( )2 = (1, )2 =1A.C16
Convert ( )3 to base 9 = 32.
Convert ( )8 to hexadecimal.
Hint 8 = 23 and 16 = 24.
( )8 = (001,010,011, ,110,111)2
= ( )2 = (29C.BB8)16
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