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w = x x 2 + y 2 + - y x 2 + y 2 i .

We then have

z × w = ( x + y i ) × ( x x 2 + y 2 + - y x 2 + y 2 i = x 2 x 2 + y 2 - - y 2 x 2 + y 2 + ( x - y x 2 + y 2 + y x x 2 + y 2 ) i = x 2 + y 2 x 2 + y 2 + 0 x 2 + y 2 i = 1 + 0 i = 1 ,

as desired.

Prove parts (1) and (2) of [link] .

One might think that these kinds of improvements of the real numbers will go on and on. For instance, we might next have to create and adjoin another object j so that the number i has a square root; i.e., so that the equation i - z 2 = 0 has a solution. Fortunately and surprisingly, this is not necessary,as we will see when we finally come to the Fundamental Theorem of Algebra in [link] .

The subset of C consisting of the pairs x + 0 i is a perfect (isomorphic) copy of the real number system R . We are justifiedthen in saying that the complex number system extends the real number system, and we will say that a real number x is the same as the complex number x + 0 i . That is, real numbers are special kinds of complex numbers. The complex numbers of the form 0 + y i are called purely imaginary numbers. Obviously, the only complex number that is both real and purely imaginary is the number 0 = 0 + 0 i . The set C can also be regarded as a 2-dimensional space, a plane, and it is also helpful to realize that the complex numbers form a 2-dimensional vector space over the fieldof real numbers.

If z = x + y i , we say that the real number x is the real part of z and write x = ( z ) . We say that the real number y is the imaginary part of z and write y = ( z ) .

If z = x + y i is a complex number, define the complex conjugate z ¯ of z by z ¯ = x - y i .

The complex number i satisfies i 2 = - 1 , showing that the negative number - 1 has a square root in C , or equivalently that the equation 1 + z 2 = 0 has a solution in C . We have thus satisfied our initial goal of extending the real numbers. But what about other complex numbers?Do they have square roots, cube roots, n th roots? What about solutions to other kinds of equations than 1 + z 2 ?

  1. Prove that every complex number has a square root. HINT: Let z = a + b i . Assume w = x + y i satisfies w 2 = z , and just solve the two equations in two unknowns that arise.
  2. Prove that every quadratic equation a z 2 + b z + c = 0 , for a , b , and c complex numbers, has a solution in C . HINT: If a = 0 , it is easy to find a solution. If a 0 , we need only find a solution to the equivalent equation
    z 2 + b a z + c a = 0 .
    Justify the following algebraic manipulations, and then solve the equation.
    z 2 + b a z + c a = z 2 + b a z + b 2 4 a 2 - b 2 4 a 2 + c a = ( z + b 2 a ) 2 - b 2 4 a 2 + c a .

What about this new field C ? Does every complex number have a cube root, a fourth root, does every equation have a solution in C ? A natural instinct would be to suspect that C takes care of square roots, but that it probably does not necessarily have higher order roots.However, the content of the Fundamental Theorem of Algebra, to be proved in [link] , is that every equation of the form P ( z ) = 0 , where P is a nonconstant polynomial, has a solution in C . This immediately implies that every complex number c has an n th root, for any solution of the equation z n - c = 0 would be an n th root of c .

The fact that the Fundamental Theorem of Algebra is true is a good indication that the field C is a “good” field. But it's not perfect.

In no way can the field C be made into an ordered field. That is, there exists no subset P of C that satisfies the two positivity axioms.

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
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salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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