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HIGHER ALGEBRA.

CHAPTER I.

DEFINITIONS.

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1. Units. In counting separate objects the standards by which we count are called units ; and in measuring continuous magnitudes the standards by which we measure are called units.

Thus, in counting the boys in a school, the unit is a boy; in selling eggs by the dozen, the unit is a dozen eggs; in selling bricks by the thousand, the unit is a thousand bricks; in measuring short distances, the unit is an inch, a foot, or a yard; in measuring long distances, the unit is a rod or a mile.

2. Numbers. Repetitions of the unit are expressed by numbers. If a man, in sawing logs into boards, wishes to keep a count of the logs, he makes a straight mark for every log sawed, and his record at different times will be as follows:

// /// //// Ne NW MV // MU /// NU MII NUN NU These representative groups are named one, two, three, four, five, six, seven, eight, nine, ten, etc., and are known collectively under the general name of numbers. It is obvious that these representative groups will have the same meaning, whatever the nature of the unit counted.

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34 3x3 x3 = 81. The gocondi pwop of a number is generally called the square of that firmabat; the, 2 is called the square of a, because if a denotes that fremheap of suite of length in the side of a square, a? denotes the timber of time of surface in the square. The third power of a numline in poprally worthlal the rule of that number; thus, a’ is called the ohitit, line make it creates the number of units of length in the Alpeil mudin, ligt noget at the number of units of volume in the cube.

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20. Roots. The root of a number is one of the equal factors of that number; the square root of a number is one of the two equal factors of that number; the cube root of a number is one of the three equal factors of that number; and so on. The sign , called the radical sign, indicates that a root is to be found. Thus, V4, or VĂ, means that the

square root of 4 is to be taken ; V8 means that the cube root of 8 is to be taken; and so on. The figure written above the radical sign is called the index of the root.

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21. Algebraic Expressions. An algebraic expression is a number written with algebraic symbols; an algebraic expression consists of one symbol, or of several symbols connected by signs of operation.

A term is an algebraic expression the parts of which are not separated by the sign of addition or subtraction. Thus, 3 ab, 5 xy, 3 ab x 5 xy, 3 ab : 5 xy are terms.

A monomial or simple expression is an expression with but one term.

A polynomial or compound expression is an expression of two or more terms. A binomial is a polynomial of two terms; a trinomial, a polynomial of three terms.

Like terms or similar terms are terms which have the same letters, and the corresponding letters affected by the same exponents. Thus, 7 aʼcx and — 5 a'cxi are like terms.

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22. The degree of a term is the sum of the exponents of its literal factors. Thus, 3 xy is of the second degree.

A polynomial is said to be homogeneous when all its terms are of the same degree. Thus, 7.23 — 5 xạy + xyz is homogeneous of the third degree.

A polynomial is said to be arranged according to the powers of some letter when the exponents of that letter either descend or ascend in order of magnitude.

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