#set page(margin: (x: .5in, y: .5in)) #let solvein(solution) = { let outset = 3pt h(outset) box( outset: outset, stroke: blue + .3pt, fill: rgb(0, 149, 255, 15%), radius: 4pt, )[#solution] } #let solve(content) = [ #align( center, block( inset: 5pt, stroke: blue + .3pt, fill: rgb(0, 149, 255, 15%), radius: 4pt, )[#align(left)[#content]], ) ] #let notein(content) = { let outset = 3pt h(outset) box( outset: outset, stroke: luma(20%) + .3pt, fill: luma(95%), radius: 4pt, )[#content] } #let note(content) = [ #align( center, block( inset: 5pt, stroke: luma(20%) + .3pt, fill: luma(95%), radius: 4pt, )[#align(left)[#content]], ) ] #align(center)[ = CS 3333 Mathematical Foundations Homework 3 (100 points)\ #underline[Price Hiller] *|* #underline[zfp106] ] #line(length: 100%, stroke: .25pt) = Submission: Same as HW1. = Questions Convert a number in a number system to another one. *Write down the intermediate steps of your calculations.* + Decimal numbers to binary numbers. (10 pts) #enum( numbering: "a.", [ $157$ #notein[ $ (157)_10 &= 2 ⋅ 78 &+ #text(red)[1]\ 78 &= 2 ⋅ 39 &+ #text(red)[0]\ 39 &= 2 ⋅ 19 &+ #text(red)[1]\ 19 &= 2 ⋅ 9 &+ #text(red)[1]\ 9 &= 2 ⋅ 4 &+ #text(red)[1]\ 4 &= 2 ⋅ 2 &+ #text(red)[0]\ 2 &= 2 ⋅ 1 &+ #text(red)[0]\ 1 &= 2 ⋅ 0 &+ #text(red)[1] $ #solve[$(157)_10 = (10011101)_2$] ] ], [ $39.25$ #notein[ #align(center)[#underline[Whole Number]] $ (39)_10 &= 2 ⋅ 19 &+ #text(red)[1]\ 19 &= 2 ⋅ 9 &+ #text(red)[1]\ 9 &= 2 ⋅ 4 &+ #text(red)[1]\ 4 &= 2 ⋅ 2 &+ #text(red)[0]\ 2 &= 2 ⋅ 1 &+ #text(red)[0]\ 1 &= 2 ⋅ 0 &+ #text(red)[1] $ #align(center)[#underline[Fraction]] $ .50 &= 0.25 ⋅ 2 #note[#text(red)[0]]\ 1.00 &= 0.50 ⋅ 2 #note[#text(red)[1]] $ #solve[$(39.25)_10 = (100111.01)_2$] ] ], ) + Binary numbers to decimal numbers. (10 pts) #enum( numbering: "a.", [ $(10111010)_2$ #notein[ $ (1 ⋅ 2^7) + (0 ⋅ 2^6) + (1 ⋅ 2^5) + (1 ⋅ 2^4) + (1 ⋅ 2^3) + (0 ⋅ 2^2) + (1 ⋅ 2^1) + (0 ⋅ 2^0) = #solve[$(186)_10$] $ ] ], [ $(1101.011)_2$ #notein[ $ (1 ⋅ 2^3) + (1 ⋅ 2^2) + (0 ⋅ 2^1) + (1 ⋅ 2^0) + (0 ⋅ 2^-1) + ( 1 ⋅ 2^-2) + (1 ⋅ 2^-3) = #solve[$(13.375)_10$] $ ] ], ) + Octal integers to binary numbers. (10 pts) #enum( numbering: "a.", [ $(527)_8$ #notein[ #table( columns: (auto, auto, auto, auto), [*Octal*], [$5$], [$2$], [$7$], [*Binary*], [$101$], [$010$], [$111$], ) #solve[$(527)_8 = (101010111)_2$] ] ], [ $(4361)_8$ #notein[ #table( columns: (auto, auto, auto, auto, auto), [*Octal*], [$4$], [$3$], [$6$], [$1$], [*Binary*], [$100$], [$011$], [$110$], [$001$], ) #solve[$(4361)_8 = (100011110001)_2$] ] ], ) + Octal integers to decimal numbers. (10 pts) #enum( numbering: "a.", [ $(527)_8$ #notein[ $ (5 ⋅ 8^2) + (2 ⋅ 8^1) + (7 ⋅ 8^0) = #solve[$(343)_10$] $ ] ], [ $(4361)_8$ #notein[ $ (4 ⋅ 8^3) + (3 ⋅ 8^2) + (6 ⋅ 8^1) + (1 ⋅ 8^0) = #solve[$(2289)_10$] $ ] ], ) + Binary numbers to octal numbers. (10 pts) #enum( numbering: "a.", [ $(10 110 110)_2$ #notein[ #table( columns: (auto, auto, auto, auto), [*Binary*], [$#text(red)[0]10$], [$110$], [$110$], [*Octal*], [$2$], [$6$], [$6$], ) #solve[$(10 110 110)_2 = (266)_8$] ] ], [ $(11 110.011)_2$ #notein[ #table( columns: (auto, auto, auto, auto), [*Binary*], [$#text(red)[0]11$], [$110$], [$011$], [*Octal*], [$3$], [$6$], [$3$], ) #solve[$(11 110.011)_2 = (36.3)_8$] ] ], ) + Hexadecimal integers to binary numbers. (10 pts) #enum( numbering: "a.", [ (F$6$A$)_16$ #notein[ #table( columns: (auto, auto, auto, auto), [*Hexadecimal*], [F], [$6$], [A], [*Binary*], [$1111$], [$0110$], [$1010$], ) #solve[(F6A$)_16 = (1111 0110 1010)_2$] ] ], [ (D$0$EB$)_16$ #notein[ #table( columns: (auto, auto, auto, auto, auto), [*Hexadecimal*], [D], [$0$], [E], [B], [*Binary*], [$1101$], [$0000$], [$1110$], [$1011$], ) #solve[(D$0$EB$)_16 = (1101 0000 1110 1011)_2$] ] ], ) + Hexadecimal integers to decimal numbers. (10 pts) #enum( numbering: "a.", [ (F6A$)_16$ #notein[ $ (15 ⋅ 16^2) + (6 ⋅ 16^1) + (10 ⋅ 16^0) = #solve[$(3946)_10$] $ ] ], [ (D0EB$)_16$ #notein[ $ (13 ⋅ 16^3) + (0 ⋅ 16^2) + (14 ⋅ 16^1) + (11 ⋅ 16^0) = #solve[$(53483)_10$] $ ] ], ) + Binary numbers to hexadecimal numbers. (10 pts) #enum( numbering: "a.", [ $(10 1101. 011)_2$ #notein[ #table( columns: (auto, auto, auto, auto), [*Binary*], [$#text(red)[00]10$], [$1101$], [$011#text(red)[0]$], [*Hexadecimal*], [$2$], [D], [$6$], ) #solve[$(10 1101. 011)_2 = (2$D$.6)_16$] ] ], [ $(1 1011 1101)_2$ #notein[ #table( columns: (auto, auto, auto, auto), [*Binary*], [$#text(red)[000]1$], [$1011$], [$1101$], [*Hexadecimal*], [$1$], [B], [D], ) #solve[$(1 1011 1101)_2 = (1$BD$)_16$] ] ], ) + Octal numbers to hexadecimal numbers. (10 pts) #enum( numbering: "a.", [ $(605.35)_8$ #notein[ 1. #underline[Octal to Binary] #table( columns: (auto, auto, auto, auto, auto, auto, auto), [*Octal*], [$6$], [$0$], [$5$], [$.$], [$3$], [$5$], [*Binary*], [$110$], [$000$], [$101$], [$.$], [$011$], [$101$], ) #note[$=11 000 0101 .011 101$] //$=1 1000 0101.0111 01$ 2. #underline[Binary to Hexadecimal] #table( columns: (auto, auto, auto, auto, auto, auto, auto), [*Binary*], [$#text(red)[000]1$], [$1000$], [$0101$], [$.$], [$0111$], [$01#text(red)[00]$], [*Hexadecimal*], [$1$], [$8$], [$5$], [$.$], [$7$], [$4$], ) #solve[$(605.35)_8 = (185.74)_16$] ] ], ) + Hexadecimal numbers to octal numbers. (10 pts) #enum( numbering: "a.", [ (C$9$A.$3$B$)_16$ #notein[ 1. #underline[Hexadecimal to Binary] #table( columns: (auto, auto, auto, auto, auto, auto, auto), [*Hexadecimal*], [C], [$9$], [A], [$.$], [$3$], [B], [*Binary*], [$1100$], [$1001$], [$1010$], [$.$], [$0011$], [$1011$], ) #note[$=1100 1001 1010 . 0011 1011$] //$=110 010 011 010.001 110 11 2. #underline[Binary to Octal] #table( columns: (auto, auto, auto, auto, auto, auto, auto, auto, auto), [*Binary*], [$110$], [$010$], [$011$], [$010$], [$.$], [$001$], [$110$], [$11#text(red)[0]$], [*Octal*], [$6$], [$2$], [$3$], [$2$], [$.$], [$1$], [$6$], [$6$], ) #solve[(C$9$A.$3$B$)_16 = (6232.166)_8$] ] ], )