Questions about imaginary number and root of 4

[Shing]

I am thinking that
if the imaginary number is bigger than the other number,
is it right to say that:
i> 5 ?
7i> 3i ?
Does i has magnitude?

if
$$Z_1=4+5i$$
then
$$Z_2=1-3i$$
whether $$Z_1>Z_2 or Z_2>Z_1$$ is true?

If we say $$Z_a$$ is bigger than $$Z_b$$, does that means the absolute value of these complex number?

Thank you :)

 

[Shing]

Also
I have been wonder why $$\sqrt{4} =2$$
but why NOT -2, negative two?

 

[arildno]

I am thinking that
if the imaginary number is bigger than the other number,
is it right to say that:
i> 5 ?
7i> 3i ?
Does i has magnitude?

if
$$Z_1=4+5i$$
then
$$Z_2=1-3i$$
whether $$Z_1>Z_2 or Z_2>Z_1$$ is true?

If we say Z_a is bigger than Z_b, does that means the absooluate value of these complex number?

Thank you :)

No, "i" is not bigger than any real number.
As it happens, when we talk about complex and imaginary numbers, we cannot have an ordering relation between them.
Numerous complex numbers will have the same magnitude, so that a number's magnitude cannot serve as an identificatory trait of that number.

 

[arildno]

Also
I have been wonder why $$\sqrt{4} =2$$
but why NOT -2, negative two?

Good question!

Answer:
Because we have, by convenience, DEFINED the square root thusly.

Similarly, by convenience, we have defined 1 not to be a prime number.

 

[HallsofIvy]

I am thinking that
if the imaginary number is bigger than the other number,
is it right to say that:
i> 5 ?
7i> 3i ?
Does i has magnitude?

It is not possible to assign an "order" to the complex numbers in such a way as to have an "ordered field" (that is, so that if a< b and 0< c, then ac< bc and if a< b, then a+c< b+ c (for any c)). For example, if we were to define an order so that 0< i, then we must have 0*i< i*i or 0< -1. Since this is not necessarily "regular order" that is not a contradiction itself but multiplying by i again, 0*i< -1*i or 0< -i. Adding i to both sides, we must have 0+i< -i+ i or i< 0, contradicting 0< i. But if we try to define an order so that i< 0, we can, in the same way, show that 0< i getting the same contradiction.

if
$$Z_1=4+5i$$
then
$$Z_2=1-3i$$
whether $$Z_1>Z_2 or Z_2>Z_1$$ is true?

Neither is true- as I just showed there is no way to compare complex numbers.

If we say $$Z_a$$ is bigger than $$Z_b$$, does that means the absolute value of these complex number?

I've never seen anyone say that Z_a> Z_b for Z_a and Z_b complex numbers. If you mean to say one has larger absolute value than the other, then you must say |Z_a|> |Z_b|.

Also
I have been wonder why $$\sqrt{4} =2$$
but why NOT -2, negative two?

We define it that way because fractional powers with even denominators give real results only for positive numbers. If we want to be able to say that $\sqrt{x}= x^{1/2}$ (which is a very useful thing to do) and then combine it with other such functions, we need to stick to positive numbers.

 

[Shing]

Thank you so much, Arildno and HallsofIvy!

But I don't understand that,would you explain a bit more please?

We define it that way because fractional powers with even denominators give real results only for positive numbers. If we want to be able to say that Click to see the LaTeX code for this image (which is a very useful thing to do) and then combine it with other such functions, we need to stick to positive numbers.

imaginary number is so amusing, I am thinking of the geometry meaning of some operations of complex number.

I was thinking if these are true:
$$Z_1\times Z_2$$ produce a new complex number "vector" $$Z_3$$
$${Z_1}^{1/2}\times {Z_2}^{1/2}$$produce a new complex number$$Z_3$$ too

When I was thinking of the reciprocal of a complex number $$Z_1$$

$${1\over {Z_1}} = {{x-iy}\over {x^2+y^2}}$$

I was shocked!

Does that mean $${1\over {Z_1}}$$ is $$Z_1$$* times the reciprocal of the area of the square magnitude of $$Z_1$$($$|Z_1|^2$$)?

If so, what is the meaning of a reciprocal of an area?

Also, is $${1\over {Z_1}}$$ still a complex number?

 

[Pere Callahan]

Also, is $${1\over {Z_1}}$$ still a complex number?

Yes, it is, unless Z_1 is zero.

Try to relate the geometric meaning of 1/z to the circle inversion.

 

[nandu11]

I am thinking that
if the imaginary number is bigger than the other number,
is it right to say that:
i> 5 ?
7i> 3i ?
Does i has magnitude?

if
$$Z_1=4+5i$$
then
$$Z_2=1-3i$$
whether $$Z_1>Z_2 or Z_2>Z_1$$ is true?

If we say $$Z_a$$ is bigger than $$Z_b$$, does that means the absolute value of these complex number?

Thank you :)

"i" is nothin..as we say an imaginary number n nothin else..when we say that Za is greater than Zb..we mean to take their real part n not the imaginary parts

 

[Diffy]

You can however compare absolute values of complexes. If you have $$Z_1=4+5i$$ and $$Z_2=1-3i$$ then you can look at $$|Z_1|$$ and $$|Z_2|$$.

This will tell you the complex numbers distance away from zero in the complex plane.

In general let $$Z = a+Bi$$ then $$|Z| = \sqrt{a^2 + b^2}$$

Now you can say $$|Z_1|>|Z_2|$$.

 

[HallsofIvy]

"i" is nothin..as we say an imaginary number n nothin else..when we say that Za is greater than Zb..we mean to take their real part n not the imaginary parts

I have never seen such a statement! And, exactly what do you mean by '"i" is nothin"?

 

[HallsofIvy]

"i" is nothin..as we say an imaginary number n nothin else..when we say that Za is greater than Zb..we mean to take their real part n not the imaginary parts

I have never seen anyone say "Za> Zb" when they meant Re(Z_a)> Re(Z_b). And what do you mean by ' "i" is nothin'?