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nent of the power; instead of aaaaa, we write a', where 5 is the exponent of the power, etc.

When no exponent is written over a quantity, the exponent 1 is always understood. Thus, ał and a signify the same thing.

Exponents may be attached to figures as well as letters. Thus the product of 3 by 3 may be written 32, which equals 9. 3x3x3

33,

27. 3x3x3x3x3 16

243.

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17. The sign of evolution, or the radical sign, is the character V. When placed over a quantity, it indicates that a root of that quantity is to be extracted. The name or index of the required root is the number written above the radical sign. Thus,

Vg, or simply Vo, denotes the square root of 9, which is 3. V64 denotes the cube root of 64, which is 4.

V16 denotes the fourth root of 16, which is 2.
So, also,

Va, or simply Va, denotes the square root of a.
Va denotes the fourth root of a.

va denotes the nth root of a, where n may represent any number whatever.

When no index is written over the sign, the index 2 is un. derstood. Thus, instead of Vab, we usually write Vab.

Symbols which indicate Relation. 18. The sign of equality consists of two short horizontal lines, =. When written between two quantities, it indicates that they are equal to each other.

Thus, 7+6=13 denotes that the sum of 7 and 6 is equal to 13.

In like manner, a=b+c denotes that a is equal to the sum of b and c; and a+b=c-d denotes that the sum of the numbers designated by a and b, is equal to the difference of the numbers designated by c and d.

19. The sign of inequality is the angle > or <. When placed between two quantities, it indicates that they are unequal, the opening of the angle being turned toward the greater number. When the opening is toward the left, it is read greater than ; when the opening is toward the right, it is read less than. Thus, 5>3 denotes that 5 is greater than 3; and 6<11 denotes that 6 is less than 11. So, also, a>b denotes that a is greater than b; and <y+z denotes that w is less than the sum of y and z.

20. A parenthesis, ( ), or a vinculum, - is employed to connect several quantities, all of which are to be subjected to the same operation.

Thus the expression (a+b+c) xx, or a+b+cxx, indicates that the sum of a, b, and c is to be multiplied by x. But a+b+cxx denotes that c only is to be multiplied by x.

When the parenthesis is used, the sign of multiplication is generally omitted. Thus, (a +b+c) x x is the same as (a+b+c)x.

21. The sign of ratio consists of two points like the colon : placed between the quantities compared. Thus the ratio of a to b is written a:b.

22. The sign of proportion consists of a combination of the sign of ratio and the sign of equality, thus, : = :; or a combination of eight points, thus, : :: :.

Thus, if a, b, c, d, are four quantities which are proportional to each other, we say a is to b as c is to d; and this is expressed by writing them thus:

a:b=c:d,
a:b::c:d.

or

23. The sign of variation is the character . When written between two quantities, it denotes that both increase or diminish together, and in the same ratio. Thus the expression so tv denotes that s varies in the same ratio as the product of t and v.

24. Three dots ., are sometimes employed to denote there. fore, or consequently.

A few other symbols are employed in Algebra, in addition to those here enumerated, which will be explained as they occur.

Combination of Algebraic Quantities. 25. Every number written in algebraic language--that is, by aid of algebraic symbols—is called an algebraic quantity, or an algebraic expression.

Thus, 3ais the algebraic expression for three times the square of the number a.

7a%b4 is the algebraic expression for seven times the third power of a, multiplied by the fourth power of b.

26. An algebraic quantity, not composed of parts which are separated from each other by the sign of addition or subtraction, is called a monomial, or a quantity of one term, or simply a term.

Thus, 3a, 5bc, and 7xy, are monomials.

Positive terms are those which are preceded by the sign plus, and negative terms are those which are preceded by the sign minus, When the first term of an algebraic quantity is positive, the sign is generally omitted. Thus a+b-c is the same as +a+b-c. The sign of a negative term should never be omitted.

27. The coefficient of a quantity is the number or letter prefixed to it, showing how often the quantity is to be taken.

Thus, instead of writing ata+a+a+a, which represents 5 a's added together, we write 5a, where 5 is the coefficient of

1 a. In 6(x+y), 6 is the coefficient of x+y. When no coefficient is expressed, 1 is always to be understood. Thus, la and a denote the same thing.

The coefficient may be a letter as well as a figure. In the ex. pression nx, n may be considered as the coefficient of x, because x is to be taken as many times as there are units in n. If n stands for 5, then nx is 5 times x. When the coefficient is a number, it may be called a numerical coefficient; and when it is a letter, a literal coefficient.

In 7ax, 7 may be regarded as the coefficient of ax, or 7a may be regarded as the coefficient of 2.

28. The coefficient of a positive term shows how many times the quantity is taken positively, and the coefficient of a negative term shows how many times the quantity is taken negatively. Thus, +4x=+*++*+;

-4x=-*—X—X—X.

29. Similar terms are terms composed of the same letters, affected with the same exponents. The signs and coefficients may differ, and the terms still be similar.

Thus, 3ab and 7ab are similar terms.
Also, 5a%c and --3ac are similar terms.

30. Dissimilar terms are those which have different letters or exponents.

Thus, axy and axz are dissimilar terms.
Also, 3ab2 and 4aob are dissimilar terms.

31. A polynomial is an algebraic expression consisting of more than one term; as, a+b; or a+2b-5c+x.

A polynomial consisting of two terms only is usually called a binomial; and one consisting of three terms only is called a trinomial. Thus, 3a +5b is a binomial; and 5a-3bc+xy is a trinomial.

32. The degree of a term is the number of its literal factors. Thus, 3a is a term of the first degree. bab

second " 6a2bc3

sixth In general, the degree of a term is found by taking the sum of the exponents of all the letters contained in the term.

Thus the degree of the term 5ab2cd is 1+2+1+3, or 7; that is, this term is of the seventh degree.

33. A polynomial is said to be homogeneous when all its terms are of the same degree. Thus, 3a2-4ab +62 is of the second degree, and homogeneous,

2a +3ac-4cd". third But 5a-2ab+c is not homogeneous.

34. The reciprocal of a quantity is the quotient arising from dividing a unit by that quantity.

Thus the reciprocal of 2 is ; the reciprocal of a is á

35. A function of a quantity is any expression containing that quantity. Thus,

ax2 + is a function of x.
ay®+cy+d is a function of y.
ax- by2 is a function of x and y.

Excercises in Algebraic Notation. 36. In the following examples the pupil is simply required to express given relations in algebraic language.

Ex. 1. Give the algebraic expression for the following statement: The second power of a, increased by twice the product of a and b, diminished by c, and increased by d, is equal to fifteen times x.

Ans. a2 + 2ab-c+d=15x. Ex. 2. The quotient of three divided by the sum of x and four, is equal to twice b diminished by eight.

Ex. 3. One third of the difference between six times x and four, is equal to the quotient of five divided by the sum of a and b.

Ex. 4. Three quarters of x increased by five, is equal to three sevenths of b diminished by seventeen.

Ex. 5. One ninth of the sum of six times w and five, added to one third of the sum of twice x and four, is equal to the product of a, b, and c.

Ex. 6. The quotient arising from dividing the sum of a and b by the product of c and d, is greater than four times the sum of m, n, x, and y.

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