Sem 2 - Conversions

$N_{(b)}=N_{(k)}^{\prime }$

1. Rapid conversions: $b,h\in \{2,2^2,\dots \}$
2. Substitution method
   • $b<h$
   • operations in base $h$
3. Successive divisions and multiplications
   • $b>h$
   • operations in base $b$ Exercise 1. $73162,451_{(8)}={?}_{(16)}$

&=0111\ 0110\ 0111 \ 0010,1001\ 0100\ 1000_{(2)}=7672,948_{(16)}\end{align}$$ Exercise 2. $7672,948_{(16)}=?_{(8)}$ $$\begin{align} 7672,948_{(16)}= 13121302,211010_{(8)} \end{align}$$ 1 hexa digit is transformed into 2 octal digits $7_{(16)}=1\times4^1+3\times4^0$ Exercise 3. $$\begin{gathered}\underbrace{10\ldots0}_{n}=2^n\\ \underbrace{11\ldots1}_{n}=2^{n+1}-1\\ 0,\underbrace{0\ldots1}_n=2^{-(n+1)}\\ 0,\underbrace{1\ldots1}_n=1 \end{gathered}$$ Exercise 4. $1342,23_{(5)}=?_{(8)}$ \ $$\begin{align} 1342,23_{(5)}&=1_{(5)}\times 5^3+3_{(5)}\times 5^2+4_{(5)}\times5+2_{(5)}\times5^0+2_{(5)}\times 5^{-1}+3_{(5)}\times5^{(-2)}\\ &=1_{(8)}\times 5_{(8)}^3+3_{(8)}\times 5^2+4_{(8)}\times5_{(8)}+2_{(8)}\times5_{(8)}^0+2_{(8)}\times 5_{(8)}^{-1}+3_{(8)}\times5^{(-2)}\\ \end{align} $$\begin{array}{rl}\stackrel{3}{0},\stackrel{3}{7}7& {}_{(16)}\times \\ 7& \\ & \\ 3,41& {}_{(16)}\end{array}$$

Exercise 7.

$$\begin{array}{r}765,23_{(8)}={?}_{(6)}\text{with 2 digits}\end{array}$$