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A.Imaginary Quantity
Imaginary Quantities are the root of negative numbers.
e.g.: √–1, √ –3, √ – 8...
Real quantities are all the other roots, except imaginary quantities.
The unity quantity is √–1 and is represented by the letter i => i = √–1
The power of the imaginary quantity is:
(√–1)1 => i1 = (√–1)
(√–1)2 => -1
(√–1)3 = (√–1)2 * (√ –1) = (-1) * (√–1) = - (√–1) => i3 = - (√–1)
(√–1)4 = (√–1)2 * (√–1)2 = (-1)*(-1) = 1 => i4 = 1
(√–1)5 = (√–1)4 * (√–1) = 1 * (√–1) = (√–1)
(√-1)6 = (√-1)4 * (√-1)2 = 1 * (-1) = -1, => i6 = -1
etc.
As you can see, there is a certain order. It starts with (root –1), -1, and then again, root –
1, -1... every even potent has –1, every odd potent (root –1).
1. Imaginary Pures
All expression of nte √–a ( or -a 1/n), where n is even and –a is a real negative quantity, is
imaginary pure.
e.g.: (√–2), (√–5) are from this kind.
-
How to SIMPLIFY imaginary pure?
All imaginary roots can be reduces to the form of the real quantity multiplied by the unity
quantity (√–1).
e.g. : (√–b2) = (√ b2 * (-1)) = (√b2 )* (√–1) = b * (√ –1) = b* i
√- 4 = (√4 * (-1)) = (√4 ) * (√-1) = 2 *(√–1) = 2 * i
√-3 = (√3 * (-1)) = (√3) * (√-1) = (√3) * (√-1) = i * (√3)
2. Add & Substract Imaginary Pures
They are reduced to the real quantity by (√-1) and this is reduced to similar radicals.
e.g.:
(√ -4) + (√-9)
for the term (√-4) you could say:
(√-4) = (√4 * (-1) = 2 * (√-1)
for the term (√-9) you could say:
(√-9) = (√9 * (-1) = 3 * (√-1)
So, instead of (√ -4) + (√-9) you could say:
2 * (√-1) + 3 * (√-1) = (2+3) * (√-1) = 5 (√-1) = 5 * i
e.g. 2:
2 * (√-36) – (√-25) + (√-12)
for 2 * (√-36) you could say:
2 * 6 * (√-1) = 12 * (√-1)
for (√-25) you could say:
5 * (√-1)
for (√-12) you could say:
(√12) * (√-1) = 2 * (√3) * (√-1)
so instead of 2 * (√-36) – (√-25) + (√-12) you could say:
12 * (√-1) – 5 * (√-1) + 2 * (√3) * (√-1) = (12 – 5 + 2 * (√3) * (√-1)
= (7+2 * (√3) * (√-1)= [7+2 * (√3)] * i
3. Multiply
You reduce the imaginary number to the formula a*(√-1). You do it, changing the power
of the imaginary unity.
e.g.1.:
(√-4) * (√-9) = 2 * (√-1) * 3 * (√-1) = 2 * 3 * (√-1)2 = 6 * (-1) = -6
e.g. 2:
(√-5) * (√-2) = [(√5) * (√-1)] * [(√2) * (√-1)] = (√10) * (√-1)2
= (√10) * (-1) = -10
e.g. 3:
(√-16) * (√-25) * (-81) = (√-16) * (√-25) * (√-81)
= [4 * (√-1)]*[5 * (√-1)]* 9 * (√-1)] = 180 * (-√-1) = -180 * (√-1) = -1802
4. Divide
You reduce the imaginary number to the formula a * (√-1).
e.g.:
(√-84) / (√-7) = [(√84) * (√-1)] / [(√7) * (√-1)] = (√84) / (√7) = (√(7 / 84) = (√12)
= 2 * (√3)
B. Complex Quantities
They are expressions that have one part real and on part imaginary.
Complex Quantities are of the type a + b * (√-1), so a + b * i, where a and b are real
quantities.
1. How to add
For add complex quantities you add the real parts between themselves and the imaginary
quantities between themselves.
The complex number is build for even numbers, which they give you in order, where one
is real and the other one could be imaginary.
We got the theory of this numbers from Bombelli ( XVI century, Italian). Then Descrates
counted the imaginary number to the non-real numbers, which is part of an complex. But
this theories were not really accepted in the mathematics world until Euler punished the
use. But the main point in the theory gave C.Wessler (1745-1818, Danish). He developed
a geometric interpretation of complex numbers. Today this help us to show complex
numbers in a graph. With complex numbers we can define all arithmetic and algebraic
equations.
e.g.1:
[2+5 * (√-1)] + [3-2 * (√-1)] = 2+3+5 * (√-1) – 2 * (√-1) =
(2+3)+(5-2) * (√-1) = 5+3 * (√-1) = 5+3 * i
e.g.2:
Add 5-6 * (√-1), -3+(√-1) and 4-8 * (√-1) =>
5 – 6 * (√-1)
+(-3)+0 * (√-1)
+ 4 - 8 * (√-1)
= 6 - 13 * (√-1)
2. Add of conjugate complex quantities
Add of conjugate complex quantities is a real quantity.
( a + b * (√-1)) + (a – b * (√-1)) = (a+a) + (b-b) * (√-1) = 2a
e.g.:
( 5+3 * (√-1)) + (5-3 * (√-1)) = 2 * 5 = 10
3. Substract
To subtract complex quantities you have to subtract the real numbers between themselves
and the imaginary numbers between themselves also.
e.g. 1:
[5+7 * (√-1)] – [4+2 * (√-1)] = 5+7 * (√-1) – 4-2 * (√-1)
= (5-4) + (7-2) * (√-1) = 1+5 * (√-1) = 1 + 5 *i
e.g.2:
[8-11 * (√-1)] - [-3-7 * (√-1)] = [8-11 * (√-1)] +[3+7 * (√-1)]=
(negative brackets: sign in bracket must be changed!)
8 – 11 * (√-1)
+ 3 + 7 * (√-1)
=11- 4 * (√-1)
Subtract two conjugate complex quantities:
When you subtract two combine complex quantities you get one imaginary pure.
(a + b * (√-1)) – (a-b * (√-1)) = a+b * (√-1) – a+b * (√-1)
= (a-a)+(b+b) * (√-1) = 2b * (√-1) = 2 * b* i
4.Multiply
Complex quantities are multyply like a compose equation and as you know ,( √-1)2 = -1
e.g.:
[3+5 * (√-1)] * [4-3 * (√-1)]
= 3 + 5 * (√-1)
* 5 - 3 * (√-1)
=12 +20 *(√-1)
- 9 * (√-1) – 15 * (√-1)2
12+11 * (√-1) – 15 * (-1) = 12 +11 * (√-1) + 15 = 27+11 * (√-1)
Multiply conjugate complex quantities
The answer of multiplying tow conjugate complex quantities is one real quantity. So, if
you multiply the add by the subtract of to quantities you are going to get a difference of
squares.
[a+b * (√-1)] * [a-b * (√-1)] = a2-[b * (√-1)2 ]= a2- [b2 * (√-1)2]
= a2-[b2(-1)] = a2-(-b2) = a2+b2
e.g.1:
[8-3 * (√-1)] * [8+3 * (√-1)] = 82-(3 * (√-1)2) = 64+9=73
e.g.2:
[(√3)+5 * (√-1)] * [(√3)-5 * (√-1)] = (√3)2- (5 * (√-1))2 = 3+25 = 28
C. Examples
 3 * (√(-b4))
= 3 * (√(b4-1))
= 3 * b2 * (√-1)
= 3 * b2 * i
 (√-12)
= [√(12 * (-1))]
= [√(22 * 3 * (-1))]
= 2 * [√( 3 * (√-1))]
= 2 * (√3) * (√-1)
= 2 * i * (√3)
 2 * [√(9 * (-1))]+3 * [√(100 * (-1))]
= 2 * [√(32 * (-1))]+3 * [√102 * (-1))]
= 2 * 3 * (√-1)+3 * 10 * (√-1)
= (6+30) * (√-1)
= 36 * i
 3 * (√-64) – 5 * (√-49)+3 * (√121)
= 3 * [√(64 * (-1))]-5 * [√(49 * (-1))]+3 * [√(121 * (-1))]
= 3 * [√(82 * (-1))]-5 * [√(72* (-1))]+3 * [√(112 * (-1))]
= 3 * 8 * (√-1)-5 * 7 * (√-1)+3 * 11 * (√-1)
= (24-35+33) * (√-1)
= 22 * i
 5 * (√-36) * 4 * (√-64)
= 5 * [√(36 * (-1))] * 4 * [√(64 * (-1))]
= 5 * [√(62 * (-1))] * 4 * [√(82 * (-1))]
= 5 * 6 * (√-1) * 4 * 8 * (√-1)
= 30 * 32 * (√-1)2
= 960 * (-1)
= -960
 (√-3) * (√-2)
= [√(3 * (-1))] * [√(2 * (-1))]
= [√(6 * (-1)2)]
-(√6)
 (√-10) / (√-2)
= [√(10 * (-1))] / [√(2 * (-1))]
= [√(5 * ((-1)/ (-1)))]
= (√5)
 (√-81) / (√-3)
= [√(81 * (-1))] / [√(3 * (-1))]
= (√27)
= [√(32 * 3)]
= 3 * (√3)
 [5+(√-1)]+[7+2 * (√-1)]+[9+7 * (√-1)]
= 12-11 * (√-1)
+ 8+7 * (√-1)
= 20-4 * (√-1)
= 20-4 * i
 [7-2 * (√-1)]+[7+2 * (√-1)]
= 7-2 * (√-1)
+ 7+2 * (√-1)
=14+0 * (√-1)
= 14
 [-5-3 * (√-1)]+[-5+3 * (√-1)]
= -5-3 * (√-1)
+ -5- 3 * (√-1)
= -10+0 * (√-1)
= -10
 [-1-1 * (√-1)] – [-7-8 * (√-1)]
= -1-1 * (√-1)
+ 7+8 * (√-1)
= 6+7 * (√-1)
= 6+7 * i
 [4-7 * (√-1)]-[5-3 * (√-1)]
= 4-7 * (√-1)
-5+3 * (√-1)
= -1-4 * (√-1)
= -1-4 * i
 [-5+1 * (√-2)]-[-5-1 * (√-1)]
= -5+1 * (√-2)
+ 5+1 * (√-2)
= 2 * [√(2 * (-1))]
= 2 * [√(2 * i)]
 [(√2)+(√-3)]-[(√2)-( √-3)]
= 1 * (√2) + (√-3)
- 1 * (√2) + (√-3)
=
0 +2 * (√-3)
=2*i
 [4+7 * (√-1)] * [-3-2 * (√-1)]
= 4+7 * (√-1)
+-3-2 * (√-1)
= -12 –21 * (√-1)
+
-8 * (√-1) – 14 * (√-1)2
= -12 –29 * (√-1)-14 * (-1)
= -12-29 * (√-1)+14
= 2 – 29 * i
 [8-(√-9)] * [11+(-25)]
=[8-(√(9 * (-1)))] * [11+(√(25 * (-1)))]
=[8-(√(32* (-1)))] * [11+(√(52 * (-1)))]
= [8-3 * (√-1)] * [11+5 * (√-1)]
= [8-3 * (√-1)]
* [11+5 * (√-1)]
= 88-33 * (√-1)
40 * (√-1)-15 * (√-1)2
= 88+7 * (√-1) +15
= 103+7 * i
 (1-i) * (1 + i)
= 1-(√-1)
* 1+(√-1)
= 1-(√-1)
+ (√-1)-(-1)
= 1 + 0+1
=2
 [3+2 * (√-1)] * [3-2 * (√-1)]
= [3+2 * (√-1)]
* [3-2 * (√-1)]
= 9+6 * (√-1)
-6 * (√-1)-4 * (-1)
=9+0+4
=13
Some interesting WebPages for this theme:
http://whatis.techtarget.com/definition/0,,sid9_gci283974,00.html
http://www.jimloy.com/algebra/imaginar.htm
http://staff.jccc.net/swilson/mathtopics/complex/sqrootsimag.htm
http://www.arthuryoung.com/mathexc.HTML
D. Miniquiz
1.) Reduce to the real quantity, multiplied by (√-1)or i
a.) (√-a2)
b.) (√-2)
c.) 2 * (√-9)
d.) (√-81)
e.) (√-6)
2.) Simplify the pure imaginary quantities
a.) (√-4)+(√-16)
b.) (√-26)+(√-81)-(√-49)
c.) 2*[√(-a2)]+[√(-a4)]+[ √(-a6)]
d.) (√-18)+( √-8)+2 * (√-50)
e.) 3 * (√-20)-2 * (√-45)+3 * (√-125)
3.) Multiply the pure imaginary quantities
a.) (√-16) * (√-25)
b.) (√-81) * (√-49)
c.) (√-3) * (√-79)
d.) 2 * (√-7) * 3* (√-28)
e.) (√-49) * (√-4) * (√-9)
4.) Divide the pure imaginary quantities
a.) (√-16) / (√-4)
b.) (√150) / (√-3)
c.) 10 * (√-36) / 5 * (√-4)
d.) 2 * (√-18) / (√-6)
e.) (√-515) / (√-7)
5.) Add the complex imaginary quantities
a.) [2+3 * (√-1)] + [5-2 * (√-1)]
b.) [-4-5 * (√-1)] + [-2+8 * (√-1)]
c.) [3-2 * i ] + [5-8 * i ] + [-10+13 * i ]
d.) [1- i ] + [4+3 * i ] + [(√2) +13 * i ]
e.) [2+(√-2)]+[4-(√-3)]
6.) Add the conjugate complex quantities
a.) [-7-5 * (√-1)] + [-7+5 * (√-1)]
b.) [8-3 * (√-2)] + [8+3 * (√-2)]
c.) [(√2)+ i *(√3)] + [(√2)- i * (√3)]
d.) [7-2 * (√-1)]+[7+2 * (√-1)]
e.) [-5-3 * (√-1)] + [-5+3 * (√-1)]
7.)
a.)
b.)
c.)
d.)
e.)
[3-2 * (√-1)] – [5+3 * (√-1)]
[8+4 * (√-1)] – [3-10 * (√-1)]
[15-4 * (√-1)] – [8-7 * (√-1)]
[11+80 * (√-1)] – [3-50 * (√-1)]
[5-(√-25)] – [3+6 * i ]
8.) Subtract the following conjugate complex quntities
a.) [2-(√-1)] – [2+(√-1)]
b.) [7+3 * (√-1)] – [7-3 * (√-1)]
c.) [-3-7 * (√-1)] – [-3+7 * (√-1)]
d.) [-(√5)-4 8 (√-2)] – [-( √5)-4 * (√-2)]
e.) [-5 * (√-2)] – [-5- (√-2)]
9.) Multiply the complex quantities
a.) [3-4 * (√-1)] * [5-3 * (√-1)]
b.) [7- (√-4)] * [5+ (√-9)]
c.) [3+(√-2)] * [5-(√-2)]
d.) [(√2)+( √5)] * [(√3)+( √-2)]
e.) [4+7 * (√-1)] * [-3-2 * (√-1)]
10.) Multiply the conjugate complex quantities
a.) [(√-2)-5 * i] * [(√2) 5 * i]
b.) [2 * (√3)+4 * i] * [2 * (√3)-4 * i ]
c.) [5-( √-2)] * [5+(√-2)]
d.) [-9-(√-5)] * [-9+(√-5)]
e.) [1+ i ] * [1- i ]
E. Answers of the quiz
1.)
a.)
b.)
c.)
d.)
e.)
a*i
i * (√2)
6*i
9*i
i * (√6)
2.)
a.)
b.)
c.)
d.)
e.)
6*i
7*i
(2a+a2+a3) * i
15 * (√2i)
15 * (√5i)
3.)
a.)
b.)
c.)
d.)
e.)
–20
–63
–15
–84
–42 * i
4.)
a.)
b.)
c.)
d.)
e.)
2
5 (√ 2)
6
2 * (√3)
3 * (√5)
5.)
a.)
b.)
c.)
d.)
e.)
7+i
–6 + 3 * i
–2 + 3 * i
(5+(√2)+7 * i
6+((√2)-(√3)) * i
6.)
a.)
b.)
c.)
d.)
e.)
–14
16
2 * (√2)
14
–10
7.)
a.)
b.)
c.)
d.)
e.)
–2-5 * i
5+14 * i
7+3 * i
8 + 130 * i
2 – 11 * i
8.)
a.)
b.)
c.)
d.)
e.)
–2 * i
6*i
–14 * i
–8 * (√(2 * i))
2 * (√(2 * i))
9.)
a.)
b.)
c.)
d.)
e.)
3-29 * i
41 +11 * i
17 + 2 * ((2 * i))
[[(√6)-(√10)]+[2+(√15)]] * i
2-29 * i
10.)
a.) 27
b.) 28
c.) 27
d.) 86
e.) 2