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A rational quantity, is that which can be expressed in finite terms, or without any radical sign, or fractional index; as a, or a, or 5a, &c.

An irrational Quantity, or Surd, is that of which the value cannot be accurately expressed in numbers, as the square roots of 2, 3, 5. Surds are commonly expressed by means. of the radical sign √; as √2, √a, Va3, or a fractional in

dex; as 2

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A square or cube number, &c., is that which has an exact square or cube root, &c.

9 2

Thus, 4 and a2 are square numbers; and 64 and

16

are cube numbers, &c.

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A measure of any quantity, is that by which it can be divided without leaving a remainder.

Thus, 3 is a measure of 6, 7a is a measure of 35a, and 9ab of 27 a2b2.

Commensurable quantities, are such as can be each divided by the same quantity, without leaving a remainder.

Thus, 6 and 8, 2√2 and 3 √2, 5a3b and 7ab2, are commensurable quantities; the common divisors being 2, √2, and ab.

Incommensurable quantities, are such as have no common measure, or divisor, except unity.

Thus, 15 and 16, √2 and √3, and a + b and a2 + b2 are incommensurable quantities.

A multiple of any quantity, is that which is some exact number of times that quantity.

Thus, 12 is a multiple of 4, 15a is a multiple of 3ɑ, and 20a2b2 of 5ab.

The reciprocal of any quantity, is that quantity inverted, or unity divided by it.

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A function of one or more quantities, is an expression into which those quantities enter, in any manner whatever, either combined, or not, with known quantities.

* This definition of a Surd, or irrational Quantity, is due to Robert Adrian, LL. D., Professor of Mathematics and Natural Philosophy in the University of Pennsylvania, who had first published it in his edition of Hutton's Course of Mathematics.-ED.

Thus, a — 2x, ax +3x2, 2x — a (aa—x2)3, axTM, aa, &c., are functions of x; and axy+bx2, ay+x (ax2 — by2)2, &c., are functions of x and y.

A vinculum, is a bar, or parentheses (), made use of to collect several quantities into one.

Thus a + bxc, or (a + b) c, denotes that the compound quantity a+b. is to be multiplied by the simple quantity c; and √ab+c2, or (ab+c), is the square root of the compound quantity ab+c2.

Practical Examples for computing the numeral Values of various Algebraic Expressions, or Combinations of Letters. 1, and e=0.

Supposing a = 6,

b

1. a2+2ab c + d =

5, c = 4, d d =
Then

36 † 60 − 4 + 1 = 93.

2. 2a3 3a2bc3 = 432 540 + 64

3. a2 X (a + b).

44.

2abc = 36 X 11 240

156.

4. 2a√ (b2 ac) + √(2ac + c2)= 12 x 1 +820.

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5. 3a √ (2ac +c2), or 3a (2ac + c2) =18 √64 144. 6. ✓ [2a2 √(2ac + c2)] = √(72 — √64) = √√ (72 — 8)=√64 =8.

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Required the numeral values of the following quantities; supposing a, b, c, d, e, to be 6, 5, 4, 1, and 0, respectively, as above.

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ADDITION.

ADDITION is the connecting of quantities together by means of their proper signs, and incorporating such as are like, or that can be united into one sum; the rule for performing which is commonly divided into the three following

cases.

CASE I.

When the Quantities are like, and have like sıgns. RULE.-Add all the coefficients of the several quantities together, and to their sum annex the letter or letters belonging to each term, prefixing, when necessary, the common sign.

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* The term Addition, which is generally used to denote this rule, is too scanty to express the nature of the operations that are to be performed in it; which are sometimes those of addition, and sometimes subtraction, according as the quantities are negative or positive. It should, therefore, be called by some name signifying incorporation, or striking a balance; in which case, the incongruity here mentioned would be removed.

CASE II.

When the Quantities are like, but have unlike ṣigns. RULE.-Add all the affirmative coefficients into one suni, and those that are negative into another, when there are several of the same kind: then subtract the less of these sums from the greater, and to the difference prefix the sign of the greater, annexing the common letter or letters as before.

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When the Quantities are unlike; or some like and others unlike.

RULE. Collect all the like quantities together, by taking their sums or differences, as in the foregoing cases, and set

down those that are unlike, one after another, with their pro

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EXAMPLES FOR PRACTICE.

2

2.

1. Required the sum of (a + b) and 1⁄2 (a —b). 2. Add 5x 3a+b+7 and

4a

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Ans. a.

3x+269 togeth7a36-2.

Ans. 2x

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36+2c 10 togethAns. 7a-2c - 19.

and 6a

Ans. 10a +76

2c+ 3 to

3c -2.

4. Add 3a+26 5, a +56 — c, gether. 5. Add x3 + ax2 + bx +2 and x3 + cx2+dx Ans. 2x3+(a+c) x2 + (b + d) x + 1.

1 together.

6. Add 6xy 12x2, 4x2+3xy, 4x-2xy, and -3xy

+4a2 together.

Ans. 4xy - 8x3

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