L10-Mon-26-Sep-2016-Sec-A-7-Complex-Num-HW08-A-10...

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


We use exponents to indicate the number of factors of a base that are to be multiplied. For example, the exponent in 23 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 example, 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 24 is 16. We indicate the principal square root of 16 as 16

negative square root of 16 is -4 since 24 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 expression that contains a radical symbol, is called a radical expression or simply a radical.

Expression under radical symbol is the radicand. E.g.,, 16 is a radical with a radicand of 16.


320 2 2 5 2 5 2 5x xxx x x x x

5 5 2

3 23

27 27 3 3 3 3 3 32 2 3 2 212 212

x y x y xxxxxy xxxx xxyyyy yyyyxy yxy


9 16

4 8

250 2 5 5 5

2 5 5

5 10

x y xxxxxxxxxyyyyyyyyyyyyyyyyxxxx x yyyyyyyy

x y x

3 3

33

3

16

5

5

9

3 1

3

250 2 5 5 5

2 5

5 2

x y xxxxxxxxxyyyyyyyyyyyyyyyy

x y yx y y

A quick way to simplify a variable part is to divide the exponent by the index. E.g., in the above:

For 9x divide 9 / 3 3 , to get the final exponent for x outside the radical.

For 16y divide with remai/ n1 536 1der . The quotient is the final exponent for y outside

the radical and the remainder is the final exponent for y inside the radical.

2

3 1 7 3 1 7 3 1 73 3 1 7 1 71 7 6 21 7 1 7 1 7 1 7 7 49

Another example:

3 12 2 3 5 18 3 2 2 3 2 3 5 2 3 3

3 2 3 2 3 5 3 2

6 3 2 3 15 2

4 3 15 2

Another example:

6 2 6 62 26 33 6 3 6 6


Two ways of solving:

Correct way:

2 2

3 2 4 5

2 4 2

4 1

4 1

4 13

xxx

x

xx

•

•

or

Incorrect way (it works by pure luck):

2 2

4 5

4 5

4

3 2

1

4 13

5

1

4

xxx

x

xx

•

•

22

2

2

12

12

120 120 3 4

3 or 4

x x

x x

x xx xx x

x x

Be sure solutions work! Be sure solution is in domain of original equation.

Since the radicand must be nonnegative, we have

12 012

12

xxx

. Also, since the left side of the

equation is positive (it is the principal square root) we also have 0x . Thus the domain is 0,12 .

So, X = 3 is the only solution.


Simplify: 3 2

4 3

x xx

3 2 2 3 1 2122 3 1 2 3 4 8 12 6 12 9 12 5 12 5

3 44 3

x x x x x x x x x x x xxx

Another example:

2 3 2 31 3 1 31 2 1

3 2 3 21 1

4 3 2 3 2 92 3 3 24 3 2 9 3 2

3 2 3 2

4 3 13 64 6 9 64 3 4 18 27 18 4 3 13 6 23 18

23 18

4 2

2 2

22 2

x y x y

x y x y

x y x yx y

xx y x yy


A.7 Complex Numbers When mathematicians tried to solve the simple quadratic equation

2 1 0x they were stumped because 2x is always a positive number. So, showing their usual pluck, the solved it this way:

2

2

2

1 01

1

1

xx

xx

Since 1 does not exist, they defined a

new number called the imaginary unit.

1i .

Although ancient Greeks are known to have observed these numbers, imaginary numbers were defined in 1572 AD by Rafael Bombelli.


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 example, 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 x-axis as a function of time (Merzbacher, p17):

21 2 2,

2x

i x t

x xx t e d


Divide 37 by 4 to get 9 with 1 left over. Thus, 937 4 1 91i i i i i


2

5 4 75 4 74 7 4 75 4 716 495 4 716 495 4 7

654 7

134 7

13 13

iii i

iii

i

i

i

22 3 4 8 2 12 38 10 3 18 10 311 10

i i i i iiii

Write in complex form: 43i

2

4 43 3

43

43 1

43

403

ii i i

ii

i

i

i


Simplify: 2 18

2

2 18 2 18

36

36 16 16

i ii

Note that:

2 18 36 6

Simplify: 9 16 25

9 16 25 3 4 53

i ii


Let’s prove that false by proving that 1 = -1.

2

Start with this identity 1 1

1 1 1 1 1 1 1 . . .Q Ei i i D

Be careful with the properties!


Square root of i

If 1i then what is i ? Do we need to invent a new number for this? No. Mathematicians

have proved that for all polynomial expressions all we need is the set of complex numbers. So, i

is a complex number and can be written in the form a bi . Let’s see what this is:

2 2

2 2 2

2 2

22

i a bi

i a bi

i a abi b ii 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:

2 20

00 0

a ba b a b

a b or a bb a or b a

and

21 2

12

i abiab

ab

When a = b we have

2

12

12

1222

bb

b

b

b

When a = -b we have:

2

12

12

bb

b

This is a contradiction since a positive number cannot equal a negative number, so a cannot equal –b.

So, we have 2 22 2

i a bi i


We can check to see if squaring 2 22 2

i yields i.

2

2

2 2 2 2 2 22 2 2 2 2 2

2 21 1

2 22 141 1 2 121 22

i i i

i i

i i i

i

i

i

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