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Representing numbers in different bases

Representing numbers in different bases. D = a n-1 a n-2 … a 0 . a -1 a -2 …. In base r. N = a n-1 * r n-1 + a n-2 * r n-2 + … + a 0 + a -1 *r -1 + a -2 *r -2 + …. In base 10:. Representing numbers in different bases. Convert:. (0.41) 10 to () 4.

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Representing numbers in different bases

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  1. Representing numbers in different bases D = an-1 an-2 … a0 . a-1 a-2 … In base r N = an-1 * rn-1 + an-2 * rn-2 + … + a0 + a-1*r-1 + a-2*r-2 + … In base 10:

  2. Representing numbers in different bases Convert: (0.41)10 to ()4

  3. Representing numbers in different bases 0.41 = an-1 * 4n-1 + an-2 * 4n-2 + … + a0 + a-1*4-1 + a-2 * 4-2 + … =0 0.41 = a-1*4-1 + a-2* 4-2 + …

  4. Representing numbers in different bases 0.41 = a-1*4-1 + a-2 * 4-2 + … 4 1.64 = a-1 + a-2 * 4-1 + a-3 * 4-2 <1 a-1 = 1

  5. Complement to Base r Definition: n digits m digits r-complement Number xxxxxxxx . yyyyyy D rn - D n=4 24 2-complement (1101)2 10000-1101 =0011 102 10-complement n=2 100-12 = 88 (12)10

  6. Complement-1 to Base r Definition: n digits m digits (r-1) complement Number xxxxxxxx . yyyyyy D rn-r-m - D n=3 m=2 1-complement (1101.11)2 1111.11-1101.11 =0010.00 9-complement n=1 99-12 = 87 (12)10

  7. Another representation of 2 complement BCD Weight: 2n-1 an-1 an-2 … a0 . a-1 a-2 … 2-complement Weight: -2n-1 BCD Coding Two complement -23 + 22 + 1 1101 = -3 - 0011

  8. Calculating the r complement r complement (r-1) complement rn - D rn-r-m - D +r-m Number (base 2): 1101 +1 1-complement: 0010 0011

  9. 0 in complement to 1 1-complement Number 00000 11111 Two representations to 0!

  10. Complement to 1 vs. 2 We usually use 2-complement

  11. Subtraction using 1-complement M – N  M + 2n-N-1 = 2n+(M-N-1) M>N-1 M<N-1 Carry <0 >0 2n+(M-N-1) 2n+(M-N-1) No Carry Take the complement and put (-) Carry exists Add it to the result ] -[ 2n – (2n+(M-N-1)) -1 -(N-M) (M-N)

  12. Example I 0011 +1010 3 -5 101 No Carry -010 = -2

  13. Example II 011 +101 3 -2 Carry 000 1 1 001

  14. Changing number of bits Given a number in 2 complement with n bits What is the representation with m>n bits ?

  15. Changing number of bits 0011 1011 11 1011 00 0011

  16. Binary Multiplication 1101 X 0011 13 X 03 1101 1101 100111 39

  17. 2-Complement multiplication -3 5 1101 0101 X X 111101 00000 1101 Carry 110001 1

  18. 2-Complement multiplication -3 -5 1101 1011 X X ?????

  19. 2-Complement multiplication -3 -5 1101 1011 X X 1111101 111101 00000 Remember: Last digit has negative weight 0011 0001111 =15

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