Representing numbers in different bases In base r

Representing numbers in different bases In base r D = an-1 an-2 … a 0. a-1 a-2 … In base 10: N = an-1 * rn-1 + an-2 * rn-2 + … + a 0 + a-1*r-1 + a-2*r-2 + …

Representing numbers in different bases Convert: (0. 41)10 to ()4

Representing numbers in different bases 0. 41 = an-1 * 4 n-1 + an-2 * 4 n-2 + … + a 0 + a-1*4 -1 0. 41 + a-2 * 4 -2 + … = a-1*4 -1 + a-2* 4 -2 + … =0

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 <1 a-1 = 1 * 4 -2

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

Complement-1 to Base r Definition: Number D n=3 m=2 (1101. 11)2 n=1 (12)10 n digits xxxx m digits . yyyyyy 1 -complement 9 -complement (r-1) complement rn-r-m - D 1111. 11 -1101. 11 =0010. 00 99 -12 = 87

Another representation of 2 complement BCD Weight: 2 n-1 2 -complement Weight: -2 n-1 an-2 … a 0. a-1 a-2 … BCD Coding Two complement 1101 = - 0011 -23 + 22 + 1 -3

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

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

Complement to 1 vs. 2 Calculation Zero preserntation 1 -Complement 2 -Complement Easy Harder Dual Singe We usually use 2 -complement

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

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

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

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

Changing number of bits 0011 1011 00 0011 11 1011

Binary Multiplication 1101 X 0011 1101 100111 13 X 03 39

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

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

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