Expert: Josh - 4/20/2007

Question
Hello, I was wondering if you could help me w/ these two division problems at that bottom...I'm sure you'll find them quite easy, but I've been having some problems figuring them out...I would find it most helpful if you could answer the two division problems in algorithmic form; in other words long division...thank you

             ______                     _____
        1984/55.00                23958/11979

Answer
Hello Dan,

Doing long division using the keyboard is not easy in this medium. Let D(n) be the dividend, S be the divisor and Q(n) be the quotient in step n of the algorithm.
The truncation operator <x/y> means dividing x by y, then rounding answer down to nearest integer.

e.g.1, <7/3>=round_down(2.3333...)=2,
e.g.2, <2/3>=round_down(0.6666...)=0,
e.g.3, <2/30>=round_down(0.0666...)=0,

Initially, we identify D(0)=55.00, S=1984.

When n=0: Let b(0) = <D(0)/S>. We find that b(0) = round_down(55/1984) yields 0. Q(0)=1*b(0)=0.xxx
Update: D(1)=550.0

When n=1: Let b(1) = <D(1)/S>. We find that b(1) = round_down(550/1984) yields 0. Q(1)=Q(0)+0.1*b(1)=0.0xxx
Update: D(2)=5500

When n=2: Let b(2) = <D(2)/S>. We find that b(2) = round_down(5500/1984) yields 2. Q(2)=Q(1)+0.01*b(2)=0.02xxx
Update: D(3)=10*(D(2)-b(2)*S) = 10(5500-2*1984) = 15320

When n=3: Let b(3) = <D(3)/S>. We find that b(3) = round_down(15320/1984) yields 7. Q(3)=Q(2)+0.001*b(3)=0.027xxx
Update: D(4)=10*(D(3)-b(3)*S) = 10(15320-7*1984) = 14320

When n=4: Let b(4) = <D(4)/S>. We find that b(4) = round_down(14320/1984) yields 7. Q(4)=Q(3)+0.0001*b(4)=0.0277xxx
Update: D(5)=10*(D(4)-b(4)*S) = 10(14320-7*1984) = 4320

When n=5: Let b(5) = <D(5)/S>. We find that b(5) = round_down(4320/1984) yields 2. Q(5)=Q(4)+0.00001*b(5)=0.02772xxx
Update: D(6)=10*(D(5)-b(5)*S) = 10(4320-2*1984) = 3520

When n=6: Let b(6) = <D(6)/S>. We find that b(6) = round_down(3520/1984) yields 1. Q(6)=Q(5)+0.000001*b(6)=0.027721xxx
Update: D(7)=10*(D(6)-b(6)*S) = 10(3520-1*1984) = 15360

When n=7: Let b(7) = <D(7)/S>. We find that b(7) = round_down(15360/1984) yields 7. Q(7)=Q(6)+0.0000001*b(7)=0.0277219xxx
Update: D(8)=10*(D(7)-b(7)*S) = 10(15360-7*1984) = 14720

and so forth. If D(0) is an irrational number, the algorithm must be terminated after achieving a predetermined level of precision.

Long division is a hundred times harder to illustrate in this forum than it is on paper.

I'll let you deal with the other one.