# How do we know when one complex number is greater than another?

## Main Question or Discussion Point

How do we know when one complex number is greater than another?

For instance, if $$a+bi > c+di$$, must $$a>c$$ and $$b>d$$?

We can't consistently order the complex numbers, to show this assume i<0 and derive a contradiction, and then the next logical assumption would be 0<i, from which you can derive yet another contradiction. I think the only way that we really assign any order to the complex numbers is through the modulus ( the modulus of z is denoted |z|, and if z=x+i*y, then |z|=(x2+y2)1/2).

Homework Helper
How do we know when one complex number is greater than another?

For instance, if $$a+bi > c+di$$, must $$a>c$$ and $$b>d$$?
There is no natural ordering in the set of complex numbers, so you can't compare them that way.

d_leet said:
We can't consistently order the complex numbers, to show this assume i<0 and derive a contradiction, and then the next logical assumption would be 0<i, from which you can derive yet another contradiction.
how can i derive the contradictions? i have tried the following. are those correct?

assume,
$$0 < i$$

$$i < i+i$$

$$i < 2i$$

$$i^2 < 2i^2$$

$$-1 < -2$$ (contadiction)

assume,
$$i < 0$$

$$i - i < -i$$

$$0 < -i$$

$$0 < (-i)^2$$

$$0 < -1$$ (contadiction)

am i right?

Last edited:
cristo
Staff Emeritus
how can i derive the contradictions? i have tried the following. are those correct?

assume,
$$0 < i$$

$$i < i+i$$

$$i < 2i$$

$$i^2 < 2i^2$$

$$-1 < -2$$ (contadiction)

assume,
$$i < 0$$

$$i - i < -i$$

$$0 < -i$$

$$0 < (-i)^2$$

$$0 < -1$$ (contadiction)

am i right?
Yes, they both look fine.

matt grime
Homework Helper
We can't order the complex numbers in a way such that the ordering respects the algebraic structure as the ordering on the real numbers does. If you're going to state such a result and prove it you had better put down the conditions and the correct statements.

It is trivial to order them lexicographically, though.

HallsofIvy
Homework Helper
Strictly speaking, just showing that "-1< -2" is not a contradiction- there is no requirement that the order you place on the complex numbers be the same as the standard order when restricted to the reals.
What you can do is:
Assume 0< i. Then 0(i)< i2 so 0< -1 (NOT a contradiction by iteslf) so 0(i)< -1(i) or 0< -i. Now add i to both sides so that i< 0. That DOES contradict 0< i.

Strictly speaking, just showing that "-1< -2" is not a contradiction- there is no requirement that the order you place on the complex numbers be the same as the standard order when restricted to the reals.
What you can do is:
Assume 0< i. Then 0(i)< i2 so 0< -1 (NOT a contradiction by iteslf) so 0(i)< -1(i) or 0< -i. Now add i to both sides so that i< 0. That DOES contradict 0< i.
An order is not implicit but defined.
For instance we can define order for natural numbers and then extend it to real numbers. Consequently we could do the same thing for imaginary numbers.

It would be an error IMHO to disprove we can define order for imaginary numbers by demonstrating that this order is not valid for real numbers.
And by analogy if we were to accept an order for imaginary numbers to disprove we can define order for real numbers by demonstrating that this order is not valid for imaginary numbers.