Ahmad Ali Ahmad Othman - Section 1 - Sheet 2

1. Convert these binary numbers to decimal.

A. $0001{0110}_2$

$$\begin{array}{c}{10110}_2=1\ast 2^4+0\ast 2^3+1\ast 2^2+1\ast 2^1+0\ast 2^0=\\ 16+4+2={22}_{10}\end{array}$$

B. $1001{0101}_2$

$$\begin{array}{c}{10010101}_2=1\ast 2^7+0\ast 2^6+0\ast 2^5+1\ast 2^4+0\ast 2^3+1\ast 2^2+0\ast 2^1+1\ast 2^0=\\ \\ 128+16+4+1={149}_{10}\end{array}$$

C. $10010000{1001}_2$

$$\begin{array}{c}{100100001001}_2=1\ast 2^{11}+0\ast 2^{10}+0\ast 2^9+1\ast 2^8+0\ast 2^7+0\ast 2^6+0\ast 2^5+\\ 0\ast 2^4+1\ast 2^3+0\ast 2^2+0\ast 2^1+1\ast 2^0=\\ \\ 2048+256+8+1={2313}_{10}\end{array}$$

D. $0110{1011}_2$

$$\begin{array}{c}{1101011}_2=1\ast 2^6+1\ast 2^5+0\ast 2^4+1\ast 2^3+0\ast 2^2+1\ast 2^1+1\ast 2^0=\\ \\ 64+32+8+2+1={107}_{10}\end{array}$$

E. $1111{1111}_2$

$$\begin{array}{c}{11111111}_2=1\ast 2^7+1\ast 2^6+1\ast 2^5+1\ast 2^4+1\ast 2^3+1\ast 2^2+1\ast 2^1+1\ast 2^0=\\ \\ 128+64+32+16+8+4+2+1={255}_{10}\end{array}$$

F. $0110{1111}_2$

$$\begin{array}{c}{1101111}_2=1\ast 2^6+1\ast 2^5+0\ast 2^4+1\ast 2^3+1\ast 2^2+1\ast 2^1+1\ast 2^0=\\ \\ 64+32+8+4+2+1={111}_{10}\end{array}$$

G. $00111101{0111}_2$

$$\begin{array}{c}{1111010111}_2=1\ast 2^9+1\ast 2^8+1\ast 2^7+1\ast 2^6+0\ast 2^5+\\ 1\ast 2^4+0\ast 2^3+1\ast 2^2+1\ast 2^1+1\ast 2^0=\\ \\ 512+256+128+64+16+4+2+1={983}_{10}\end{array}$$

H. $1101{1111}_2$

$$\begin{array}{c}{11011111}_2=1\ast 2^7+1\ast 2^6+0\ast 2^5+1\ast 2^4+1\ast 2^3+1\ast 2^2+1\ast 2^1+1\ast 2^0=\\ \\ 128+64+16+8+4+2+1={223}_{10}\end{array}$$

2. Convert the following decimal values to binary.

n	$2^n$	n	$2^n$
0	1	6	64
1	2	7	128
2	4	8	256
3	8	9	512
4	16	10	1024
5	32	11	2048

A. $37$

$$\begin{array}{c}{37}_{10}=32+4+1\\ =2^5+0+0+2^2+2^1+0=0010{0110}_2\end{array}$$

B. $13$

$$\begin{array}{c}{13}_{10}=8+4+1\\ =2^3+2^2+0+2^0={1101}_2\end{array}$$

C. $189$

$$\begin{array}{c}{189}_{10}=128+32+16+8+4+1\\ =2^7+0+2^5+2^4+2^3+2^2+0+2^0=1011{1101}_2\end{array}$$

D. $1000$

$$\begin{array}{c}{1000}_{10}=512+256+128+64+32+8\\ =2^9+2^8+2^7+2^6+2^5+0+2^3+0+0+0=00111110{1000}_2\end{array}$$

E. $77$

$$\begin{array}{c}{77}_{10}=64+8+4+1\\ =2^6+0+0+2^3+2^2+0+2^0\\ =0100{1101}_2\end{array}$$

F. $390$

$$\begin{array}{c}{390}_{10}=256+128+4+2\\ =2^8+2^7+0+0+0+0+2^2+2^1+0\\ =00011000{0110}_2\end{array}$$

G. $205$

$$\begin{array}{c}{205}_{10}=128+64+8+4+1\\ =2^7+2^6+0+0+2^3+2^2+0+2^0\\ =1100{1101}_2\end{array}$$

H. $2133$

$$\begin{array}{c}{2133}_{10}=2048+64+16+4+1\\ =2^{11}+0+0+0+0+2^6+0+2^4+0+2^2+0+2^0\\ =01000010{0101}_2\end{array}$$

I. $511$

$$\begin{array}{c}{511}_{10}=256+128+64+32+16+8+4+2+1\\ =2^8+2^7+2^6+2^5+2^4+2^3+2^2+2^1+2^0\\ =00011111{1111}_2\end{array}$$

3.What is the largest decimal value that can be represented by:

A. 8-bit binary number?

$2^7+2^6+2^5+2^4+2^5+2^4+2^3+2^2+2^1+2^0={255}_{10}$ or simply $2^8-1={255}_{10}$

B. 16-bit number?

$2^{16}-1=65,{536}_{10}$

4. Convert each hex number to its decimal equivalent.

A. ${743}_{16}$

$$\begin{array}{c}{743}_{16}=7\ast {16}^2+4\ast {16}^1+3\ast {16}^0=\\ 1792+64+3={1859}_{10}\end{array}$$

B. ${36}_{16}$

$$\begin{array}{c}{36}_{16}=3\ast {16}^1+6\ast {16}^0=\\ 48+6={54}_{10}\end{array}$$

C. $37F{D}_{16}$

$$\begin{array}{c}37F{D}_{16}=3\ast {16}^3+7\ast {16}^2+15\ast {16}^1+13\ast {16}^0=\\ 12288+1792+240+13={14077}_{10}\end{array}$$

D. ${2000}_{16}$

$$\begin{array}{c}{2000}_{16}=2\ast {16}^3+0\ast {16}^2+0\ast {16}^1+0\ast {16}^0=\\ 8192+0+0+0={8192}_{10}\end{array}$$

E. ${165}_{16}$

$$\begin{array}{c}{165}_{16}=1\ast {16}^2+6\ast {16}^1+5\ast {16}^0=\\ 256+96+5={357}_{10}\end{array}$$

F. $ABCD{0}_{16}$

$$\begin{array}{c}ABCD{0}_{16}=10\ast {16}^4+11\ast {16}^3+12\ast {16}^2+13\ast {16}^1+0\ast {16}^0=\\ 655360+45056+3072+208+0={703696}_{10}\end{array}$$

G. $7F{F}_{16}$

$$\begin{array}{c}7F{F}_{16}=7\ast {16}^2+15\ast {16}^1+15\ast {16}^0=\\ 1792+240+15={2047}_{10}\end{array}$$

H. ${1204}_{16}$

$$\begin{array}{c}{1204}_{16}=1\ast {16}^3+2\ast {16}^2+0\ast {16}^1+4\ast {16}^0=\\ 4096+512+0+4={4612}_{10}\end{array}$$

5. Convert each of the following decimal numbers to hex.

A. ${59}_{10}$

$$\begin{array}{c}59/16=3.6875\Rightarrow 11\\ 3/16=0\Rightarrow 3\\ {59}_{10}=3B_{16}\end{array}$$

B. ${372}_{10}$

$$\begin{array}{c}372/16=23.25\Rightarrow 4\\ 23/16=1.4375\Rightarrow 7\\ 1/16=0\Rightarrow 1\\ {372}_{10}={174}_{16}\end{array}$$

C. ${919}_{10}$

$$\begin{array}{c}919/16=57.4375\Rightarrow 7\\ 57/16=3.5625\Rightarrow 9\\ 3/16=0\Rightarrow 3\\ {919}_{10}={397}_{16}\end{array}$$

D. ${1024}_{10}$

$$\begin{array}{c}1024/16=64\Rightarrow 0\\ 64/16=4\Rightarrow 0\\ 4/16=0.25\Rightarrow 4\\ {1024}_{10}={400}_{16}\end{array}$$

E. ${771}_{10}$

$$\begin{array}{c}771/16=48.1875\Rightarrow 3\\ 48/16=3\Rightarrow 0\\ 3/16=0.1875\Rightarrow 3\\ {771}_{10}={303}_{16}\end{array}$$

F. ${2313}_{10}$

$$\begin{array}{c}2313/16=144.5625\Rightarrow 9\\ 144/16=9\Rightarrow 0\\ 9/16=0.5625\Rightarrow 9\\ {2313}_{10}={909}_{16}\end{array}$$

G. $65,{536}_{10}$

$$\begin{array}{c}65536/16=4096\Rightarrow 0\\ 4096/16=256\Rightarrow 0\\ 256/16=16\Rightarrow 0\\ 16/16=1\Rightarrow 0\\ 1/16=0.0625\Rightarrow 1\\ {65536}_{10}={10000}_{16}\end{array}$$

H. ${255}_{10}$

$$\begin{array}{c}255/16=15.9375\Rightarrow 15(F)\\ 15/16=0.9375\Rightarrow 15(F)\\ {255}_{10}=F{F}_{16}\end{array}$$

6. Convert each of the hex values from Problem 4 to binary.

Hex	Binary	Hex	Binary
0	0000	8	1000
1	0001	9	1001
2	0010	A	1010
3	0011	B	1011
4	0100	C	1100
5	0101	D	1101
6	0110	E	1110
7	0111	F	1111

A. ${743}_{16}$

$$\begin{array}{c}{7}_{16}\Rightarrow {0111}_2\\ 4_{16}\Rightarrow {0100}_2\\ 3_{16}\Rightarrow {0011}_2\\ {743}_{16}=01110100{0011}_2\end{array}$$

B. ${36}_{16}$

$$\begin{array}{c}{3}_{16}\Rightarrow {0011}_2\\ 6_{16}\Rightarrow {0110}_2\\ {36}_{16}=0011{0110}_2\end{array}$$

C. $37F{D}_{16}$

$$\begin{array}{c}{3}_{16}\Rightarrow {0011}_2\\ 7_{16}\Rightarrow {0111}_2\\ F_{16}\Rightarrow {1111}_2\\ D_{16}\Rightarrow {1101}_2\\ 37F{D}_{16}=001101111111{1101}_2\end{array}$$

D. ${2000}_{16}$

$$\begin{array}{c}{2}_{16}\Rightarrow {0010}_2\\ 0_{16}\Rightarrow {0000}_2\\ 0_{16}\Rightarrow {0000}_2\\ 0_{16}\Rightarrow {0000}_2\\ {2000}_{16}=001000000000{0000}_2\end{array}$$

E. ${165}_{16}$

$$\begin{array}{c}{1}_{16}\Rightarrow {0001}_2\\ 6_{16}\Rightarrow {0110}_2\\ 5_{16}\Rightarrow {0101}_2\\ {165}_{16}=00010110{0101}_2\end{array}$$

F. $ABCD{0}_{16}$

$$\begin{array}{c}{A}_{16}\Rightarrow {1010}_2\\ B_{16}\Rightarrow {1011}_2\\ C_{16}\Rightarrow {1100}_2\\ D_{16}\Rightarrow {1101}_2\\ 0_{16}\Rightarrow {0000}_2\\ ABCD{0}_{16}=1010101111001101{0000}_2\end{array}$$

G. $7F{F}_{16}$

$$\begin{array}{c}{7}_{16}\Rightarrow {0111}_2\\ F_{16}\Rightarrow {1111}_2\\ F_{16}\Rightarrow {1111}_2\\ 7F{F}_{16}=01111111{1111}_2\end{array}$$

H. ${1204}_{16}$

$$\begin{array}{c}{1}_{16}\Rightarrow {0001}_2\\ 2_{16}\Rightarrow {0010}_2\\ 0_{16}\Rightarrow {0000}_2\\ 4_{16}\Rightarrow {0100}_2\\ {1204}_{16}=000100100000{0100}_2\end{array}$$

7. Convert the binary numbers in Problem 1 to hex.

A. $0001{0110}_2$

$$\begin{array}{c}{0001}_2\Rightarrow 1_{16}{0110}_2\Rightarrow 6_{16}0001{0110}_2={16}_{16}\end{array}$$

B. $1001{0101}_2$

$$\begin{array}{c}{1001}_2\Rightarrow 9_{16}{0101}_2\Rightarrow 5_{16}1001{0101}_2={95}_{16}\end{array}$$

C. $10010000{1001}_2$

$$\begin{array}{c}00010000{1001}_2{0001}_2\Rightarrow 1_{16}{0000}_2\Rightarrow 0_{16}{1001}_2\Rightarrow 9_{16}00010000{1001}_2={109}_{16}\end{array}$$

D. $0110{1011}_2$

$$\begin{array}{c}{0110}_2\Rightarrow 6_{16}{1011}_2\Rightarrow B_{16}0110{1011}_2=6B_{16}\end{array}$$

E. $1111{1111}_2$

$$\begin{array}{c}{1111}_2\Rightarrow F_{16}{1111}_2\Rightarrow F_{16}1111{1111}_2=F{F}_{16}\end{array}$$

F. $0110{1111}_2$

$$\begin{array}{c}{0110}_2\Rightarrow 6_{16}{1111}_2\Rightarrow F_{16}0110{1111}_2=6F_{16}\end{array}$$

G. $000111101{0111}_2$

$$\begin{array}{c}00011110{1011}_2{0001}_2\Rightarrow 1_{16}{1110}_2\Rightarrow E_{16}{1011}_2\Rightarrow B_{16}00011110{1011}_2=1E{B}_{16}\end{array}$$

H. $1101{1111}_2$

$$\begin{array}{c}{1101}_2\Rightarrow D_{16}{1111}_2\Rightarrow F_{16}1101{1111}_2=D{F}_{16}\end{array}$$

9. When a large decimal number is to be converted to binary, it is sometimes easier to convert it first to hex, and then from hex to binary. Try this procedure for ${2133}_{10}$ and compare it with the procedure used in Problem 2 (h).

First: decimal to hex

$$\begin{array}{c}2133/16=133.3125\Rightarrow 5\\ 133/16=8.3125\Rightarrow 5\\ 8/16=0.5\Rightarrow 8\\ {2133}_{10}={855}_{16}\end{array}$$

Second: hex to binary

$$\begin{array}{c}{8}_{16}\Rightarrow {1000}_2\\ 5_{16}\Rightarrow {0101}_2\\ 5_{16}\Rightarrow {0101}_2\\ {855}_{16}=10000101{0101}_2\end{array}$$

10. How many hex digits are required to represent decimal numbers up to 20,000?

General rule: digits equals the $lo{g}_{base}$ of maximum (ceiled) => $\lceil lo{g}_{base}(maximum)\rceil$

Answer: $\lceil lo{g}_{16}(20000)\rceil =4$ => We need 4 digits.

11. Convert these hex values to decimal.

A. ${92}_{16}$

$$\begin{array}{c}{92}_{16}=9\ast {16}^1+2\ast {16}^0=\\ 144+2={146}_{10}\end{array}$$

B. $1A{6}_{16}$

$$\begin{array}{c}1A{6}_{16}=1\ast {16}^2+10\ast {16}^1+6\ast {16}^0=\\ 256+160+6={422}_{10}\end{array}$$

C. $37F{D}_{16}$

$$\begin{array}{c}37F{D}_{16}=3\ast {16}^3+7\ast {16}^2+15\ast {16}^1+13\ast {16}^0=\\ 12288+1792+240+13={14333}_{10}\end{array}$$

D. $ABC{D}_{16}$

$$\begin{array}{c}ABC{D}_{16}=10\ast {16}^3+11\ast {16}^2+12\ast {16}^1+13\ast {16}^0=\\ 40960+2816+192+13={43981}_{10}\end{array}$$

E. $000F_{16}$

$$\begin{array}{c}000F_{16}=0\ast {16}^3+0\ast {16}^2+0\ast {16}^1+15\ast {16}^0=\\ 0+0+0+15={15}_{10}\end{array}$$

F. ${55}_{16}$

$$\begin{array}{c}{55}_{16}=5\ast {16}^1+5\ast {16}^0=\\ 80+5={85}_{10}\end{array}$$

G. $2C{0}_{16}$

$$\begin{array}{c}2C{0}_{16}=2\ast {16}^2+12\ast {16}^1+0\ast {16}^0=\\ 512+192+0={704}_{10}\end{array}$$

H. $7F{F}_{16}$

$$\begin{array}{c}7F{F}_{16}=7\ast {16}^2+15\ast {16}^1+15\ast {16}^0=\\ 1792+240+15={2047}_{10}\end{array}$$

12. Convert these decimal values to hex.

A. ${75}_{10}$

$$\begin{array}{c}75/16=4.6875\Rightarrow 11\\ 4/16=0.25\Rightarrow 4\\ {75}_{10}=4B\end{array}$$

B. ${314}_{10}$

$$\begin{array}{c}314/16=19.625\Rightarrow 10\\ 19/16=1.1875\Rightarrow 3\\ 1/16=0.0625\Rightarrow 1\\ {314}_{10}=13A_{16}\end{array}$$

C. ${2048}_{10}$

$$\begin{array}{c}2048/16=128\Rightarrow 0\\ 128/16=8\Rightarrow 0\\ 8/16=0.5\Rightarrow 8\\ {2048}_{10}={800}_{16}\end{array}$$

D. ${24}_{10}$

$$\begin{array}{c}24/16=1.5\Rightarrow 8\\ 1/16=0.0625\Rightarrow 1\\ {24}_{10}={18}_{16}\end{array}$$

E. ${7245}_{10}$

$$\begin{array}{c}7245/16=452.8125\Rightarrow 13\\ 452/16=28.25\Rightarrow 12\\ 28/16=1.75\Rightarrow 12\\ 1/16=0.0625\Rightarrow 1\\ {7245}_{10}=1CC{D}_{16}\end{array}$$

F. ${498}_{10}$

$$\begin{array}{c}498/16=31.125\Rightarrow 2\\ 31/16=1.9375\Rightarrow 15\\ 1/16=0.0625\Rightarrow 1\\ {498}_{10}=1F{2}_{16}\end{array}$$

G. $25,{619}_{10}$

$$\begin{array}{c}25619/16=1601.1875\Rightarrow 3\\ 1601/16=100.0625\Rightarrow 1\\ 100/16=6.25\Rightarrow 4\\ 6/16=0.375\Rightarrow 6\\ {25619}_{10}={6413}_{16}\end{array}$$

H. ${4095}_{10}$

$$\begin{array}{c}4095/16=255.9375\Rightarrow 15\\ 255/16=15.9375\Rightarrow 15\\ 15/16=0.9375\Rightarrow 15\\ {4095}_{10}=FF{F}_{16}\end{array}$$