ARITHMETIC.

1. ARITHMETIC is that science which treats of number.

2. The theory of Arithmetic treats of the properties of numbers in the most general and abstract manner.

3. Practical Arithmetic applies numbers to the performance of calculations in the various arts and common affairs of life.

DEFINITION OF NUMBERS,
CHARACTERS, &c.

4. A unit is the number one; but is sometimes used to signify any of the nine digits.

5. A whole number consists of one or more units, unbroken, or not divided into parts; as 7, 24, 130, &c.

6. An integer is the whole of any thing; as a pound, a yard, &c. or, 1, 2, 7, 9, &c.

7. A fraction is a part of an integer or whole number; as, , fit, &c.

8. A mixed number is, a whole number with a fraction annexed to it; as, 14, 44, 247, &c.

9. An even number is that which can be divided into two equal whole numbers; as, 8, 12, 64, &c.

10. An odd number is that which cannot be divided into two equal whole numbers; as, 9, 17, 29, &c.

11. A prime number is that which cannot be produced by the multiplication of two others, greater than unity; as 3, 5, 7, 11, 19, &c.

B

12. A composite number is that which can be produced by the multiplication of two or more numbers together; as, 6, 16, 24, 96, &c.

13. A square number is the product of a number multiplied into itself; as 4 is the square of 2; 49 of 7, &c.

14. A cube number is the product of a number multiplied twice into itself; as 8 is the cube of 2; 27 the cube of 3, &c.

15. A perfect number is that which is just equal to the sum of all its aliquot parts; as 6; for its aliquot parts are 1, 2, 3, which make 6, when added.

16. An aliquot part of a number, is that which is contained in another an exact number of times; as 2 is an aliquot part of 8; 3 an aliquot part of 15; and 7 of 28*.

17. Ratio is the relation that subsists between two numbers of the same kind, the comparison being made by considering what part, or parts, the one is of the other; as, when 6 is compared with 3, it is found to contain it twice, and the ratio of these two numbers is therefore said to be as two to one.

18. The ratio of two numbers is usually expressed by two points placed between them, thus, 8:4; and the former is called the antecedent of the ratio and the latter the consequent†.

19. Analogy, or proportion, is the similarity of ratios, when compared with one another; thus, 6 has to 3 the same ratio that 8 has to 4; therefore these numbers are proportional to each other; for 0:3 :: 8:4.

20. A multiple of a number is that which contains the number a certain number of times without a remainder; as 9 is a multiple of 3; 48 of 12; 72 of 8, &c.

*For the different species and properties of numbers, see Hutton's Mathematical Recreations, Vol. I.

† Quantities, that are not of the same kind, cannot be compared together-One line may have to another line the same ratio that one weight has to another weight, or that one portion of time has to another; but a hue has no relation, in respect of magnitude, to a weight, or, in duration, to a portion of time.

21. That number which measures another, or is contained in it any number of times exactly, is called its submultiple.

22. The greatest common measure of two or more numbers, is the greatest number which will divide those numbers without a remainder; as 8 is the greatest common measure of 16, 24, and 40.

23. The reciprocal of a number is the quotient obtained by dividing unity, or 1, by that number; as is the reciprocal of of 8;, of 28, &c.

24. Homogenious quantities are such as can be added together.

25. A small figure, placed over any number, signifies that it is multiplied as many times into itself. Thus 8-64, and 5'=

125.

NOTATION.

1. The method of writing, or expressing numbers, by proper characters, is called NOTATION; and reckoning or reading their value, when written, is called NUMERATION.

2. Various contrivances have been used to accomplish the purposes of Notation and Numeration, at different periods, and by different nations of the world.

3. It is highly probable that short lines, or dots, were the first characters employed to denote or record numbers.

4. The letters of the alphabet appear also to have been used for this purpose, but each letter represented a particular number, which is evident in the divisions of Homer's Poems, and the 119th Psalm.

5. The Greeks used the initial letters of the names of the numbers, in their language, to express any number they wished to represent.

6. The Romans likewise employed a similar method, and, besides characters, for each rank of classes, they introduced others to represent five, fifty, and five hundred. The characters which they employed, and their values, are the following: I, for one; V, for five; X, ten; C, one hundred; D, five hundred; and M, one thousand.

7. By combining these characters according to the following rules, any number may be represented.

8. When the same letter is repeated twice, or oftener, its value is represented as often. Thus II signifies two; XXX, thirty; CC, two hundred.

9. When a numeral letter, of less value, is placed after one of greater, their values are added; thus, XI signifies eleven; LXV, sixty-five; MDCXXVIII, one thousand six hundred and twenty-eight.

10. When a numeral letter of less value is placed before one of greater, the value of the less is taken from that of the greater; thus, IV signifies four; XL, forty; XC, ninety; CD, four hundred. Sometimes I is used instead of D, for five hundred; and the value is increased ten times by annexing to the right hand.

11. Sometimes thousands are represented by drawing a line over the top of the numeral; v being used for five thousand; L, for fifty thousand; Cč, two hundred thousand.