The Elements of Algebra Preliminary to the Differential Calculus; And Fit for the Higher Classes of Schools in Which the Principles of Arithmetic Are Taught

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General Books - Algebra - 66 pages
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1837 Excerpt: ...x2 xxx X3 Jj Is ij Jj........ Jj and so on. Here 2, 3, 4, &c. are called exponents of x. Similarly, (a + b) x (a + b) is written (a + b)" (a + 6)x(a + 6)x(a+&) (a + b)3 and so on. The following results may now be easily found: The beginner's common mistake is, that x multiplied n times by x is the nth power. This is not correct; x multiplied once by x (xx) is the second power; x multiplied n times by x is the (n + l)th power. Now why, instead of 1, the rational answer, did we obtain j:, which has no meaning? Because we applied the preceding rule to a case which does not fall under it. That rule was, " to divide a power by another power of a less exponent," &c, and was derived from the preceding rule of multiplication, which rule of multiplication did not apply to any cases except where both factors were powers of x, and, consequently, where the exponent of the product was greater. than the exponent of either factor; that is, where X" is the product, and 3 one of the factors, that rule does not apply unless b be less than a. When we come to this symbol (x) we must do one of two things, either, 1. Consider x as shewing that a rule has been used in a case to which it does not apply, strike it out, and write 1 in its place; or, 2. Let x3 (which as yet has no meaning) stand for 1; in which case the rule does apply, and gives the true result. Therefore, we lay down the following definition. By any letter with the exponent 0, such as a, we mean 1; or every quantity raised to the power whose exponent is 0, is 1. Anomaly 2. If we apply the equation x-r-x1 = xai to a case in which b is greater than a, say b = a + 6, the mere rule gives a result which has no meaning. The reason is as before; a rule has been applied to a case to wh...

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