LESSONS IN ARITHMETIC.—II.

THE ROMAN METHOD OF NOTATION.

The symbols by which the Romans expressed all numbers were:—

I  denoting  one

V  "  five

X  "  ten

L  "  fifty

C  denoting  a hundred

D or IƆ "  five hundred

M or CIƆ "  a thousand

By combining these symbols according to the following rules all numbers can be represented:— When two symbols are placed together, if the one denoting the less value is on the left of the other, then the less number is to be subtracted from the greater; if on the right hand, it is to be added to it. Thus IX denotes ten with one subtracted, or nine; XI denotes eleven; XL denotes forty; LX, sixty. If the symbols are of equal value, then they are simply to be added. Thus XX denotes twenty; CC, two hundred, etc. The value represented by IƆ is increased tenfold by every additional Ɔ placed on the right. Thus 5,000 is denoted by IƆQOƆ, and 50,000 by IƆƆƆ. The value of the symbol CIQ becomes increased tenfold by the addition of C and Ɔ, one on each side of the line I. Thus 100,000 is denoted by CCCIƆƆƆ, 1,000,000 by CCCCIƆƆƆƆ, and so on. A straight line placed over any one of these symbols increases its value a thousand-fold. Thus I̅ denotes 1,000; V̅, 5,000; L̅, 50,000; C̅, 100,000. 2,000 was usually denoted by CIƆCIƆ, but sometimes by IICIƆ, or IIM, or MM. Similarly, 4,000 was denoted by IVCIƆ, etc.

The above remarks will sufficiently explain the following Table of Roman Numerals:

I denotes one

II " two

III " three

IV " four

V " five

VI " six

VII " seven

VIII " eight

IX " nine

X " ten

XI denotes eleven

XII " twelve

XIII " thirteen

XIV " fourteen

XV " fifteen

XVI " sixteen

XVII " seventeen

XVIII " eighteen

XIX " nineteen

XX " twenty

XXI denotes twenty-one

XXII " twenty-two etc.

XXX " thirty

XL " forty

L " fifty

LX " sixty

LXX " seventy

LXXX " eighty

XC " ninety

C denotes one hundred

CI " one hundred and one

CX " one hundred and ten

CC " two hundred

CCC " three hundred

CCCC " four hundred

D (see also above) five hundred

DC " six hundred

DCC denotes seven hundred

DCCC " eight hundred

DCCCC " nine hundred

M or CIƆ " one thousand

MM (or see also above) two thousand

MDCCCLXVII one thousand eight hundred and sixty-seven etc., as above.

Exercise 3.

1. Write out the names of all the numbers from one to a hundred, and express them in figures.

2. Write out the names of the numbers which immediately follow:—

1. One hundred.

2. One hundred and ninety-nine.

3. Four hundred and ninety-nine.

4. Nine thousand nine hundred and ninety-nine.

5. One million.

3. Express, in figures, the numbers named in the preceding example, and those which immediately follow them.

4. Write the names of the numbers which are next to the following numbers, and express both sets in figures:—

1. One million and ninety-nine.

2. One million five thousand nine hundred and ninety-nine.

3. Nine millions nine hundred and ninety-nine thousand nine hundred and ninety-nine.

5. Read or express the following numbers in words:—

1. 202

2. 1001

3. 15608

4. 306042

5. 5678914

6. 26312478

7. 20030208

8. 1010101

9. 9999999

10. 347125783

11. 202021010

12. 9090909090

13. 100010001000

14. 3000000000000

15. 777666555444

16. 123456789123

17. 48484848484848

18. 10210230430400

6. Write or express the following numbers in figures:—

1. Four hundred and four.

2. Three thousand and thirty-two.

3. Twenty-four thousand and eighty-six.

4. Six hundred and five thousand and nineteen.

5. Eleven thousand eleven hundred and eleven.

6. Three hundred and forty-one thousand seven hundred and eighty-two.

7. One million.

8. Nine thousand nine hundred and ninety-nine millions, nine hundred and ninety-nine thousand nine hundred and ninety-nine.

9. Write the number which follows this last one in order.

10. One trillion and three.

11. Eighty millions two hundred and three thousand and two.

12. Two hundred and two millions twenty thousand two hundred and two.

13. Twenty thousand millions.

14. Two hundred thousand and twenty millions two thousand.

15. The next number to thirty thousand billions nine hundred and ninety-nine thousand.

ADDITION.

1. The process of uniting two or more numbers together, so as to form a single number, is called Addition. The number thus formed is called the sum of the separate numbers.

2. The sign + placed between two numbers indicates that they are to be added together. This symbol is called plus. The sign = placed between two numbers denotes that they are equal. Thus 2 + 3 = 5, expresses that 2 and 3 added together are equal to 5.

3. Suppose that it be required to add the two numbers 3452 and 4327 together.

These are respectively—

3 thousands, 4 hundreds, 5 tens, and 2 units,

4 thousands, 3 hundreds, 2 tens, and 7 units,

which, added together, are equal to—

7 thousands, 7 hundreds, 7 tens, and 9 units.

The sum, therefore, of 3452 and 4327 is 7 thousands, 7 hundreds, 7 tens, and 9 units, which, according to our system of notation, will be written 7779.

This is got by putting down the two numbers one under the other, the units under the units, the tens under the tens, and so on; and then adding up the lower to the upper figure in each place, thus:—

4. In the example we have taken, the sum of the numbers of the thousands amounts only to a number expressed by one figure, namely, 7; and similarly for the hundreds, the tens, and units.

Suppose, however, that we have a case in which this is not so; for instance, to add

8976 and 4368.

These are respectively equal to

8 thousands, 9 hundreds, 7 tens, and 6 units,

4 thousands, 3 hundreds, 6 tens, and 8 units;

or, added together, to

12 thousands, 12 hundreds, 13 tons, and 14 units.

This, however, is not at present in a form which can be at once written down according to our system of notation. We must, therefore, alter its form.

Now 14 units are the same as 1 ten and 4 units; therefore 13 tens and 14 units are the same as 14 tens and 4 units.

But 14 tens are the same as 1 hundred and 4 tens ; therefore 12 hundreds and 14 tens are the same as 13 hundreds and 4 tens.

But 13 hundreds are the same as 1 thousand and 3 hundreds; therefore 12 thousands and 13 hundreds are the same as 13 thousands and 3 hundreds.

Hence we see that 12 thousands, 12 hundreds, 13 tens, and 14 units, are the same as 13 thousands, 3 hundreds, 4 tens, and 4 units, which, by our notation, is written 13344.

5. The preceding process will sufficiently explain the following Rule for Addition:—

Write down the numbers under each other, so that units may stand under units, tens under tens, etc., and draw a line beneath them. Then, beginning with the units, add the columns separately. Whenever the sum of the figures in a column is a number expressed by more than one figure, write down the right-hand figure of such number under the column, and add the other figure or figures into the next column. Proceed in this way