Adding and Subtracting Roman NumeralsDate: 10/07/97 at 15:25:36 From: meghan Subject: Adding and subtracting Roman numerals Dear Dr. Math, I am a student of St. Johns University in Queens, New York. I don't have a specific question, but do you have any suggestions for how to teach adding and subtracting of Roman numerals? It would be great if you could help! Thanks! Meg Date: 10/10/97 at 11:47:54 From: Doctor Luis Subject: Re: Adding and subtracting Roman numerals The basic roman numerals are (i,v,x,L,C,D,M,....) which correspond to groups of the basic numeral i, and, in our decimal notation, to (1,5,10,50,100,500,1000,...) The Roman numerals, defined by the successor property, are the following: i --> i ii --> i+i iii --> ii+i --> i+i+i iiii --> iv --> v-i iiiii --> iv+i --> v-i+i --> v vi --> v+i vii --> vi+i --> v+i+i viii --> vii+i --> ..... --> v+i+i+i viiii --> viii+i--> ix viiiii --> ix+i --> x-i+i --> x xi --> x+i xii --> xi+i . . . Let a be a basic numeral. Then, in general, the juxtaposition or concatenation of a with itself (or another numeral) is taken to signify addition: a = a aa = a+a aaa = a+a+a aaaa = a+a+a+a aaaaa = a+a+a+a+a This last number is generally represented by another Roman numeral b, defined by b = aaaaa and bb is represented by yet another Roman numeral c = bb = aaaaaaaaaa One can clearly see that b is a^5, and that c is a^10 Such usual group definitions are i --> i (identity numeral or basic pure numeral) iiiii --> v iiiiiiiiii --> vv --> x xxxxx --> L xxxxxxxxxx --> LL --> C CCCCC --> D CCCCCCCCCC --> DD --> M Similarly, one can define higher groups of (pure) numerals. Composite numerals are of the form ba, ab, abb, or abbb. (where b<a) and could be defined as ba = a-b (in other words, ba is the number you add to b to get a) ab = a+b abb = a+b+b abbb = a+b+b+b a and b, of course are pure numerals. Usually, groups of the same four pure numerals are written in the composite form bbbb --> bbbbb-b --> a-b --> ba Now, Roman numerals, in general, are written as a concatenation of either pure or composite groups of numerals, and writing the greater groups first: as in 1567 = MDLXVII (1000+500+50+10+5+2) Here I have some examples which you might want to examine. I have included the equivalent operation in decimal numbers to the right (don't you love place-value notation?) (notice that I'm using the definition of the numerals to evaluate the products, sums, and subtractions) ii*iv = ii*(v-i) 2*4 = 2*(5-1) = ii*v - ii*i = 2*5-2*1 = vv - ii = 5+5-2 = v+v-ii = 5+5-2 = v+iiiii-ii = 5+(1+1+1+1+1)-(1+1) = v+iii = 5+(1+1+1) = 5 + 3 = viii = 8 xxi+xxxiv = xxi+xxxiiii 21 + 34 = (20+1)+(30+4) = xxxxxiiiii = (20+30)+(1+4) = LV = 50 + 5 = 55 Civ - xLvii = LLiv - xxxxvii 104 - 47 = (50+50+4) - (40+5+2) = (LL+iv)-(L-x+v+ii) = (50+50+4) - (50-10+5+2) = L+L+iv - L+x-v-ii = 50+50+4 - 50+10-5-2 = (L+L-L)+(x+iv-v-ii) = 50+50-50 + 10+4-5-2 = L + (x+v-i-v-ii) = 50 + 10+5-1-5-2 = L + (x-i-ii) = 50 + 10-1-2 = L + (ix-ii) = 50 + 9-2 = L + (viiii-ii) = 50 + 5+4-2 = L + vii = 50 + 5+2 = Lvii = 57 = LVII If you have any more questions, feel free to ask again :-) -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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