1 / 16

Digital Logic Design

Digital Logic Design. EGRE 254 Number Systems and Codes 1/12/09. Positional Number Systems. Numbers are commonly represented in the base 10 positional number system. For example. In general (see page 26).

hsmall
Download Presentation

Digital Logic Design

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Digital Logic Design EGRE 254 Number Systems and Codes 1/12/09

  2. Positional Number Systems • Numbers are commonly represented in the base 10 positional number system. • For example

  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

  4. Examples • 156.78 = (1x102 + 5x101 + 6x100 + 7x10-1 )8 = (1x82 + 5x81 + 6x80 + 7x8-1 )10 = (64 + 40 + 6 + .875)10 = 110.87510 • 559 = (5x101 + 5x100)9 = (5x91 + 5x90)10 = (45 + 5)10 = 5010 • 1212 = (1x121 + 2x120)10 = (12 + 2)10 = 1410 • 101100.112 =

  5. Examples • Does r10 = 10r? Prove it! • Does rb = br? Prove it! • 111b = 13310 Find b. • (41/3)b = 13b Find b. • (33/3)b = 11b Find b.

  6. Example • (5D4.A2)16 • = (5x102+Dx101+4+Ax10-1+2x10-2)16 = (5x162+13x161+4+10x16-1+2x16-2)10 = (1280+208+4+.625+.0078125)10 = (1492.6328125)10

  7. 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.

  8. Base 10 to base r (integer case)

  9. 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 149210 = 5D416

  10. 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

  11. Example fraction to hex (.6328125)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 .6328125 = 10.125 = 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, (.6328125)10 = (.A2)16

  12. Example • For a number containing both an integer and a fractional part. Compute the two parts separately and combine. • 1492.632812510 = 5D4.A216

  13. Special Case converting between base r and rn • Convert 132.68 to binary. • Could convert to decimal and then binary. • Easier way.

  14. Examples • Convert (11010.11)2 to octal, and to hexadecimal. (11010.11)2 = (11,010.110)2 = 32.68 (11010.11)2 = (1,1010.1100)2 =1A.C16 • Convert (121.121)3 to base 9 = 32. • Convert (1234.567)8 to hexadecimal. • Hint 8 = 23 and 16 = 24.

More Related