Can Complex Numbers be Compared Using Greater-Than and Less-Than Relations?

In summary, the greater-than and less-than relations are not applicable among complex numbers. This is because complex numbers, together with the standard operations of addition and multiplication, form a field but not an ordered field. This is due to the difficulty in determining whether the imaginary number i is positive or negative, leading to contradictions when trying to establish an ordering.
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quawa99
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Are the less than (<) and greater than(>) relations applicable among complex numbers?
By complex numbers I don't mean their modulus, I mean just the raw complex numbers.
 
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The short answer is "no". The greater-than and less-than relations do not apply.

A longer answer is that the complex numbers together with the standard operations of addition and multiplication form a "field". But there is no greater-than relation that can be used to make it an "ordered field". http://en.wikipedia.org/wiki/Ordered_field

The problem comes when you try to decide whether i is positive or negative. i is different from zero, so it has to be either positive or negative. If it is positive then i*i must be positive. But i*i=-1 and -1 is negative. If i is negative then -i must be positive. So -i*-i must be positive. But -i*-i=-1 and -1 is negative.
 
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1. What is the definition of "inequality" for complex numbers?

The inequality of complex numbers refers to the comparison of two complex numbers and determining which one is larger or smaller. This is similar to comparing real numbers, where the greater number is to the right on the number line and the smaller number is to the left.

2. How do you represent inequality of complex numbers on a graph?

Inequality of complex numbers can be represented on a graph by plotting the numbers as points on the complex plane. The point with the greater magnitude (distance from the origin) will be further from the origin than the point with the smaller magnitude.

3. Can two complex numbers be equal even if their real and imaginary parts are different?

No, two complex numbers cannot be equal if their real and imaginary parts are different. Complex numbers are equal only if their real and imaginary parts are equal.

4. How do you compare the magnitudes of two complex numbers?

To compare the magnitudes of two complex numbers, you can use the absolute value or modulus of the complex numbers. The complex number with the larger absolute value will have a greater magnitude.

5. Is there a specific rule for comparing two complex numbers?

Yes, there is a specific rule for comparing two complex numbers. The rule states that if the real parts of the complex numbers are equal, then the one with the larger imaginary part is considered greater. If the real parts are not equal, then the one with the larger real part is considered greater.

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