hw7920 -- some solutions

23. Convert the following numbers from the base shown to the base ten.
	a. 111 (base 2)
				4 + 2 + 1
	b. 777 (base 8)
				7 * 8^2   +  7 * 8    +  7
	c. FEC (base 16)
				15 * 16^2  +  14 * 16   +  12
	d. 777 (base 16)
				7 * 16^2   +  7 * 16    +  7
	e. 111 (base 8) 
				1*8^2  +  1*8  + 1

31.  Convert the following octal numbers to hexadecimal:
	a. 0777
		000 111 111 111   =   0001 1111 1111
	b. 0605
		000 110 000 101   =   0001 1000 0101
	c. 0443
		000 100 100 011   =   0001 0010 0011 
	d. 0521
		000 101 010 001   =   0001 0101 0001
	e. 1 
			1 = 1

34.  Convert the following decimal numbers to hexadecimal:
	a. 1066	 [Battle of Hastings; Norman conquest]
					[WORK UPWARD!]
		       0  R 1    	zero	(done!)
		   2 ) 1  R 0		odd	0/2 = 0
		   2 ) 2  R 0		even	2/2 = 1
		   2 ) 4  R 0		even	4/2 = 2
		   2 ) 8  R 0		even	8/2 = 4
		  2 ) 16  R 1		even	16/2 = 8
		  2 ) 33  R 0		odd	32/2 = 16
		  2 ) 66  R 1		even	66/2 = 33
		 2 ) 133  R 0		odd	132/2 = 66
		 2 ) 266  R 1		even	266/2 = 133
		 2 ) 533  R 0		odd	532/2 = 266
		2 ) 1066		even	1066/2 = 533

		[READ DOWN]	   10000101010
		    Hex:	 100 0010 1010	0x42A
		    Octal:	10 000 101 010	02052

	b. 1939	 [Invasion of Poland; WW II]

		      _0  R 1
		   2 )_1  R 1
		   2 )_3  R 1
		   2 )_7  R 1
		  2 )_15  R 0
		  2 )_30  R 0
		  2 )_60  R 1
		 2 )_121  R 0
		 2 )_242  R 0
		 2 )_484  R 1
		 2 )_969  R 1
		2 ) 1939

		[READ DOWN]	   11110010011
		    Hex:	 111 1001 0011		0x793
		    Octal:	11 110 010 011		03623

	c. 1
		      _0  R 1
		   2 )_1

	d. 998	 [Otto III retakes Rome.]

		      _0  R 1
		   2 )_1  R 1
		   2 )_3  R 1
		   2 )_7  R 1
		  2 )_15  R 1
		  2 )_31  R 0
		  2 )_62  R 0
		 2 )_124  R 1
		 2 )_249  R 1
		 2 )_499  R 0
		 2 )_998
				   1111100110
				 11 1110 0110		0x3E6
				1 111 100 110		01746

	e. 43		 [One more than the answer.]

		      _0  R 1
		   2 )_1  R 0
		   2 )_2  R 1
		   2 )_5  R 0
		  2 )_10  R 1
		  2 )_21  R 1
		  2 )_43
				 101011
				10 1011		0x2B  [OR not 2B?]
				101 011		053

37.  Perform the following OCTAL additions:
	a.  0770 + 0665
				  ''     (carrys)
				  0770
				+_0665_
				  1655
	b.  0101 + 0707
				  ' '    (carrys)
				  0101
				+_0707_
				  1010
	c. 0202 + 0667 
				  ' '    (carrys)
				  0202
				+_0667_
				  1071

38.  Perform the following HEXADECIMAL additions:
	a. 19ABE + 43
				    ''   (carrys)
				  19ABE
				+____43
				  19B01
	b. AE9 + F
				   '    (carrys)
				  AE9
				+___F
				  AF8
	c. 1066 + ABCD 
				   ''    (carrys)
				  1066
				+_ABCE_
				  BC34

40.  Perform the following HEXADECIMAL subtractions:
	a. ABC - 111
				        (no carrys)
				  ABC
				-_111_
				  9AB
	b. 9988 - AB
				  9988
				-___AB_
				  98DD		  13
		check:		+___AB_		+_11_
				  9988		  24 =0x18
					D+B is 13+11 =24 = 0x18
	c. A9F8 - 1492 	[Columbus]
				  A9F8
				-_1492_
				  9566
		check:		+_1492
				  A9F8_

41.  Why are binary numbers important in computing?