233. The symbol V-1 is called the imaginary unit, and may be defined as an expression the square of which is -1. Hence, √1x √− 1 = (√ − 1)2 = − 1 ;

√-ax√ b= √ax √- 1x √b x √ - 1

234. It will be useful to form the successive powers of the imaginary unit.

(√−1).

(√− 1)2.

= + √−1;

= −1;

(√—1)3 = (√−1)2 V-I =(−1) √−1 = −√— 1; (√− 1)• = (√− 1)2 (√ — 1)2 = (− 1)(− 1) =+1; (√− 1)3 = (√ − 1)* √−1 =(+1) √−1 =+ √−1; and so on. We have, therefore, if n is any integer,

235. Every imaginary expression may be made to assume the form a+b√−1, where a and b are real numbers, and may be integers, fractions, or surds.

If b = 0, the expression consists of only the real part a, and is therefore real.

If a 0, the expression consists of only the imaginary part b√1, and is called a pure imaginary.

236. The form a+bv-1 is the typical form of imaginary expressions.

Reduce to the typical form 6+√- 8.

This may be written 6+√8x√1, or 6+2√2×√−1; here a 6, and b = 2√2.

=

237. Two expressions of the form a+b √−1, a−b√-1, are called conjugate imaginaries.

To find the sum and product of two conjugate imaginaries,

From the above it appears that the sum and product of two conjugate imaginaries are both real.

238. An imaginary expression cannot be equal to a real number.

Since band (ca) are both positive, we have a negative number equal to a positive number, which is impossible.

239. If two imaginary expressions are equal, the real parts are equal and the imaginary parts are equal.

which is impossible unless b d and a = c.

=0

240. If x and y are real and x+y√—1=0, then x=

(2) Multiply 3+2√−1 by 5 - 4√— 1.

(3+2√1)(5-4√1)

(3) Divide 14+5√−1 by 2-3√— 1.

14 +5√ 1 _ (14+ 5√= 1)(2 + 3√-1)

=

= 1 + 4√- 1.

5. √−121 – √–49. 10. √18 + √− 18 – √– 8.

Reduce to the form b√-1 and multiply:

11. 1-4 by 1 - √-4.

12. 4√3 by 4-√3.

13. √3 − 2√− 2 by √3+2√− 2.

14. √54-√2 by √54 + √− 2.

15. Va+√b by √-a-√-b.

16. avab by a√— ab3.

α

17. 2√3 – √-5 by 2√3 + √− 5.

18. V-10 by V— 2.

Reduce to the form b√-1 and divide :

19. √12 by √— 3.

20. √15 by √ 5.

21. √-5 by √– 20. 22. a by √-a.

23.√25 by √— 5.

24.√25 by √-5.

-

25. 4√— 20 by — 2√— 25.

26. 4+2 by 2√2.

CHAPTER XIX.

QUADRATIC EQUATIONS.

242. We have already considered equations of the first degree in one or more unknowns. We now proceed to the treatment of equations containing one or more unknowns to a degree not exceeding the second. An equation which contains the square of the unknown, but no higher power, is called a quadratic equation.

243. A quadratic equation which involves but one unknown number can contain only :

(1) Terms involving the square of the unknown number. (2) Terms involving the first power of the unknown number.

(3) Terms which do not involve the unknown number. Collecting similar terms, every quadratic equation can be made to assume the form

ax2 + bx + c = 0,

where a, b, and c are known numbers, and x the unknown number.

If a, b, c are numbers expressed by figures, the equation is a numerical quadratic. If a, b, c are numbers represented. wholly or in part by letters, the equation is a literal quadratic.

244. In the equation ax2+ bx+c=0, a, b, and c are called the coefficients of the equation. The third term c is called the constant term.