Answer to Question #250006 in Discrete Mathematics for Danny

Question

] Carry out the following calculations and conversions by hand (showing all your working). (a) Compute the sum (1011010)2 + (11011)2 (without changing the base). (b) Compute the sum (135)8 + (2357)8 (without changing the base). (c) Convert (197)10 to octal. (d) Convert (20A5.76)12 to decimal.

Answer 1

a)

$$\def\arraystretch{1.5} \begin{array}{c} & & 1 & 1 & & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0_2 \\ & & 1 & 1 & 0 & 1 & 1_2\\ \hline 1 & 1 & 1 & 0 & 1 & 0 & 1_2\\ \end{array}$$

b)

$$\def\arraystretch{1.5} \begin{array}{c} & 1 & 1 & \\ & 1 & 3 & 5_8 \\ 2 & 3 & 5 & 7_8\\ \hline 2 & 5 & 1 & 4_8\\ \end{array}$$

c)

$$197\div8=24 \ R5$$

$$24\div8=3\ R0$$

$$(197)_{10}=(305)_8$$

d)

$$(20A5.76)_{12}$$

Integer part:

$$2\cdot12^3+0\cdot12^2+10\cdot12^1+5\cdot12^0$$

$$=3456+0+120+5=3581$$

Fraction part:

$$7\cdot\dfrac{1}{12}+6\cdot\dfrac{1}{12^2}=\dfrac{15}{24}=0.625$$

$$(20A5.76)_{12}=(3581.625)_{10}$$