Section 2.5 Roman Numerals
The Romans used a special method of showing numbers, based on the following symbols: These 7 symbols are the only Roman numerals. We can’t write 0, negative numbers or fractions using Roman numerals.
1 5 10 50 100 500 1000
Numbers greater than 4,000 are formed by placing a dash over the symbol, meaning "times 1,000", but these are not commonly used, and we won’t study them:
5,000 10,000 50,000 100,000 500,000 1,000,000 ______V X L C D M
We never write more than 3 of the same symbol in a row. Notice how we write 4 and 9 and 40 and 90 in the table below. Writing a “larger” number to the left of a smaller number tells you to subtract.
1 2 3 4 5 6 7 8 9
I II III IV V VI VII VIII IX
10 20 30 40 50 60 70 80 90
X XX XXX XL L LX LXX LXXX XC
100 200 300 400 500 600 700 800 900
C CC CCC CD D DC DCC DCCC CM
Now let’s try to convert Roman numerals into common numbers. These common numbers are called Hindu-Arabic numbers.
Example: Convert XIV to Hindu-Arabic numbers.
Step 1: We need to know which number each letter stands for.
I = 1 V = 5 X = 10
Step 2: Locate the largest symbol within the number. This should be the first or second number.
XIV
In our problem the X is the symbol with the largest value.
If there is no symbol before the symbol with the largest value just add the value of this symbol to our total.
In our case there is no symbol preceding the X, so I will add 10 to my total.
Total so far: 10
Step 3: Remove the X from our number and repeat the process.
Here is what is left when you remove the X from our number.
IV
Look for the largest number in our reduced number.
In our problem the V is the symbol with the largest value.
I can’t just add the value of a V which is 5 to our total.
If a number precedes the largest number, I must subtract the value of the symbol before it from its value and add that number to the total.
IV = 5 – 1 = 4
Add this number to our total.
Total so far: 10 + 4
We have use up all of the Roman numerals so I can just finish adding and write my answer.
Answer: XIV = 14 Example: Convert CXCVII to Hindu-Arabic numbers.
I will solve this problem in a table. I will be doing the exact same technique. I think it will be easier to follow in a table. You might not think this seems easier at first. But I think you will after watching me work a few of these on my video.
Steps to fill in the table:
1) Write the number in the upper left corner.
2) Beneath the number write the symbol with largest value in the number. This will be the first or second number. Include both symbols if the largest value is second.
3) Write the value of that symbol beneath the symbol.
4) Remove the symbol or symbols that were used when creating a column and write the remaining symbols in the top row of the next column.
5) Repeat steps 1 – 4 until all symbols have been used.
6) Add the values in the bottom row.
First step: Write the number in the upper left corner. C is the symbol with the largest value. I will write it beneath the number. Write the value of C beneath the C. Remove the C from the number and write what remains in the top of the next column.
Remaining CXCVII XCVII number Largest C symbol value 100
Second step: C is the symbol with the largest value. I will write XC beneath the number. Write the value of XC beneath the XC. Remove the XC from the number and write what remains (VII) in the top of the next column.
Remaining CXCVII XCVII VII number Largest C XC symbol value 100 90
Third step: V is the symbol with the largest value. I will write V beneath the number. Write the value of V beneath the V. Remove the V from the number and write what remains (II) in the top of the next column.
Remaining CXCVII XCVII VII II number Largest C XC V symbol value 100 90 5
Fourth step: I is the symbol with the largest value. I will write I beneath the number. Write the value of I beneath the I. Remove the I from the number and write what remains (I) in the top of the next column.
Remaining CXCVII XCVII VII II I number Largest C XC V I symbol value 100 90 5 1
Fifth step: I is the symbol with the largest value. I will write I beneath the number. Write the value of I beneath the I. Remove the I from the number and write what remains (I) in the top of the next column.
Remaining CXCVII XCVII VII II I Nothing number left Largest C XC V I I symbol value 100 90 5 1 1
Last step: I have used up all of the symbols. I just add the numbers in the bottom row to get my answer.
Answer: 100 + 90 + 5 + 1 + = 197
Example: Write MCMLXIV as a Hindu Arabic numeral.
I will do this as a single table. This may be hard to follow. I won’t show any explanation. I will explain this table in my video.
Remaining MCMLXIV CMLXIV LXIV XIV IV None left number Largest M CM L X IV symbol Value 1000 900 50 10 4
Answer: 1000 + 900 + 50 + 10 + 4 = 1964
Homework: #1-28 Write as a Hindu Arabic numeral.
1) CCXLII 2) DXLVI 3) CDXLVII 4) CDXXXIV
5) LXXXVI 6) LXXVII 7) CXXIII 8) CXXXI
9) XLIX 10) XLIV 11) DCXVI 12) DCCLXVI
13) XXXIV 14) XXXIX 15) XCIX 16) XCIV
17) MCMXXX 18) MCMXCIII 19) MMCMIV 20) MMCMXCV
21) MMXIV 22) MMXIX 23) CMXCVII 24) CMXLVII
25) DCXLVII 26) DCCXLIX 27) CDLVI 28) CDXLVII
Now we need to go “backwards”. These tables will help us go backwards.
“ONES 1 – 9”
Roman I II III IV V VI VII VIII IX numeral Hindu- 1 2 3 4 5 6 7 8 9 Arabic
TENS 10 – 90”
Roman X XX XXX XL L LX LXX LXXX XC numeral Hindu- 10 20 30 40 50 60 70 80 90 Arabic
HUNDREDS 100 – 900”
Roman C CC CCC CD D DC DCC DCCC CM numeral Hindu- 100 200 300 400 500 600 700 800 900 Arabic
THOUSANDS 1000 – 3000 (We won’t study Roman Numerals over 4,000)
Roman M MM MMM numeral Hindu- 1000 2000 3000 Arabic
Example: Write 278 as a Roman numeral
Break the number up into hundreds, tens and ones.
278 = 200 + 70 + 8
Attach a Roman numeral to each. Use the table to get the values.
200 = CC
70 = LXX
8 = VIII
Answer: CCLXXVIII
Example: Write 2419 as a Roman numeral.
Break the number up into thousands, hundreds, tens and ones.
2419 = 2000 + 400 +10 + 9
Attach a Roman numeral to each. Use the table to get the values.
2000 = MM
400 = CD
10 = X
9 = IX
Answer: MMCDXIX
Homework: # 29 – 60 Write as a Roman numeral.
29) 53 30) 57 31) 18 32) 17
33) 125 34) 131 35) 316 36) 722
37) 19 38) 29 39) 34 40) 24
41) 49 42) 44 43) 47 44) 48
45) 95 46) 98 47) 432 48) 446
49) 951 50) 957 51) 948 52) 941
53) 1950 54) 1973 55) 1994 56) 1999
57) 2016 58) 2019 59) 2024 60) 2043
Answers:
1) 242 3) 447 5) 86 7) 123 9) 49 11) 616 13) 34 15) 99 17) 1930 19) 2904 21) 2014 23) 997 25) 647 27) 456 29) LIII 31) XVIII 33) CXXV 35) CCCXVI 37) XIX 39) XXXIV 41) XLIX 43) XLVII 45) XCV 47) CDXXXII 49) CMLI 51) CMXLVIII 53) MCML 55) MCMXCIV 57) MMXVI 59) MMXXIV