L10-Mon-26-Sep-2016-Sec-A-7-Complex-Num-HW08-A-10-Radicals-HW09-Q09, page 86. L10-Mon-26-Sep-2016-Sec-A-7-Complex-Num-HW08-A-10-Radicals-HW09-Q09

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1 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 86 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09

2 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 87 We use eponents to indicate the number of factors of a base that are to be multiplied. For eample, the eponent in 3 tells us to multiply two factors of the base, 3, to get 9. In some situations we may have to do the reverse; that is, we may be given the final product and asked to find the base. For eample, find numbers whose square is 16. We call those numbers the square roots of 16 and we use the radical symbol to indicate the square root. Every positive real number has two square roots, which are opposites of each other. The square root that is positive is called the principal square root; the square root that is negative is called the negative square root. principal square root of 16 is 4 since 4 is 16. We indicate the principal square root of 16 as 16 negative square root of 16 is -4 since 4 is 16. We write the negative square root of 16 as 16 Square root of 16 is 4 and -4. Write the square root of 16 as 16. An epression that contains a radical symbol, is called a radical epression or simply a radical. Epression under radical symbol is the radicand. E.g.,, 16 is a radical with a radicand of 16.

3 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page y 7 y 333y y 1y 3yyyy yyyy y 5 5 3

4 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page y yyyyyyyyyyyyyyyy 5 5 yyyyyyyy y y yyyyyyyyyyyyyyyy y y y y A quick way to simplify a variable part is to divide the eponent by the inde. E.g., in the above: For 9 divide 9 / 3 3, to get the final eponent for outside the radical. 16 For y divide 16 / 3 5 with remainder 1. The quotient is the final eponent for y outside the radical and the remainder is the final eponent for y inside the radical Another eample: Another eample:

5 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 90 Two ways of solving: Correct way: or Incorrect way (it works by pure luck): or 4 Be sure solutions work! Be sure solution is in domain of original equation. 1 0 Since the radicand must be nonnegative, we have 1. Also, since the left side of the 1 equation is positive (it is the principal square root) we also have 0. Thus the domain is 0,1. So, X = 3 is the only solution.

6 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 91 Simplify: Another eample: y y y y y y 3 3 y y y y

7 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 9 A.7 Comple Numbers When mathematicians tried to solve the simple quadratic equation 1 0 they were stumped because is always a positive number. So, showing their usual pluck, the solved it this way: Since 1 does not eist, they defined a new number called the imaginary unit. i 1. Although ancient Greeks are known to have observed these numbers, imaginary numbers were defined in 157 AD by Rafael Bombelli.

8 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 93 However, imaginary numbers have concrete applications in sciences such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Other topics utilizing imaginary numbers include the investigation of electrical current, wavelength, liquid flow in relation to obstacles, analysis of stress on beams, the movement of shock absorbers in cars, the study of resonance of structures, the design of dynamos and electric motors, and the manipulation of large matrices used in modeling. For eample, the mathematical models that describe how AC current flows through wires use imaginary numbers. From quantum mechanics, here is a model of the propagation of a plane wave along the -ais as a function of time (Merzbacher, p17): i t 1, t e d

9 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 94 Divide 37 by 4 to get 9 with 1 left over. Thus, i i i 1 i i

10 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page i 5 4 7i 4 7i 4 7i 54 7i 16 49i 54 7i i i i i 4i 8i 1i 3i 810i i i Write in comple form: 4 3i 4 4 i 3i 3i i 4i 3i 4i i i 3

11 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 96 Simplify: i 18 i Note that: 36 i Simplify: i 5i 3i

12 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 97 Let s prove that false by proving that 1 = -1. Start with this identity i i i 1 QE.. D. Be careful with the properties!

13 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 98 Square root of i If i 1 then what is i? Do we need to invent a new number for this? No. Mathematicians have proved that for all polynomial epressions all we need is the set of comple numbers. So, i is a comple number and can be written in the form a bi. Let s see what this is: i a bi i a bi i a abi b i i a b abi In order for these to be equal, the imaginary parts must be equal and the real parts must be equal. So we can write: 0 a b 0 a ba b and 0 a b or 0 a b b a or b a When a = b we have 1 b b 1 b 1 b b When a = -b we have: 1 b b 1 b i abi 1 ab 1 a b This is a contradiction since a positive number cannot equal a negative number, so a cannot equal b. So, we have i a bi i

14 L10-Mon-6-Sep-016-Sec-A-7-Comple-Num-HW08-A-10-Radicals-HW09-Q09, page 99 We can check to see if squaring i yields i. i i i 1 i i i i 1 1 i i 1 i 1 i