Diaries and Letters (Paperback, Primary Source ed.)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1895 edition. Excerpt: ...squares gives 24. These are yTT and Vl3. Hence the square root of 24--2 V143 = Vll--Vl3. Again: (2) Find the square root of 11--6 V2. Eeducing to the form a + 2Vb, we have, 11--2Vl8. It remains to find, therefore, two surds whose product gives Vl8, and the sum of whose squares gives 11. They are V9 and V2. Hence the square root of 11-6V2=V9-V2, = 3-V2. EXERCISE 87. Extract the square root of the following: 1 8 + 2V15. 6. 117 + 8VI01. 2. 21-UV2. 7. 57-12VI5. 3. 16 + 2V15. 8. 4+VIS. 4. 30 + 2V216. 9. ab(a+b) + 2VaW. 5. 30-VT16. 10. i(a2x + b2y)8VaWxl/. 11. 9 I ma + b)+ n(a--b)--18 mnVc?-b CHAPTER XXIV. IMAGINARY EXPRESSIONS. 237. The indicated even root of a negative quantity is called an Imaginary Expression." 238. Although the symbol for the even root of a negative quantity represents an impossibility, imaginary expressions are frequently useful in mathematical investigations, as we shall hereafter show. Let m be any real quantity; then the square roots of--m2 will be V--m2. If we regard--m2 as the product of m2 and--1, we may further simplify the expression V--m2 into mV--1. 239. Every imaginary expression may be made to assume the form m + W-1-If in the expression m + nV--1, in and n are real quantities, the expression is still regarded as an imaginary. If in this expression m + nV--1, n = 0, the real part only remains; if m = 0, the imaginary part only remains. 240. m + n'--1 is regarded as the typical form for an imaginary expression. n V--/ is called a Pure Imaginary. m+flV-/ and m--nf--1 are called Conjugate Imaginarios. 241. As the symbol V--1, the imaginary unit, is the square root of--1, we have, VixVI =-i, and (Vi)2 x Vl =-V% and (V)2 x (Vl)2 = +1. 242. Hence, (V=iy =-i, (V=T) =-V=T, (Vi)4 = +i....