Cannot two imaginary numbers with equal imaginary parts and unequal real parts be compared in size?

Complex numbers cannot be compared in size because we cannot define a complex number as a self-consistent ordered domain that makes it additionally and multiplicatively compatible.

Real numbers can be compared in size, but those who have studied complex numbers will find that we can't compare the size of two complex numbers, and we don't even know which is bigger, the imaginary number unit "i" or "0".

Any two numbers in a number field should be relatively large, first of all, this number field is an ordered field, that is, we can establish a set of rules so that all numbers in the number field form an ordered relationship, and are compatible in addition and multiplication.

Mathematically, for a number field q, if we can define a full order relation such that q is an ordered domain, then the following two conditions must be satisfied (a, b, and c belong to q):

Condition 1: When a>b, there are a+c>b+c;

Condition 2: When A>B and C>0, there is AC>BC;

For integer and real fields, these two conditions are obviously satisfied, so both integers and real numbers are ordered domains, and any two elements in them can be compared in size.

Complex numbers are extensions of real numbers, and with the introduction of the imaginary unit "i", we can think of complex numbers as two-dimensional numbers, but no matter how we define them, we can't make complex numbers satisfy the two conditions of an ordered field.

The full order relation requires that any two elements in the number field can be compared, so if we take the imaginary unit "i" as an example, it must satisfy any of i>0, i<0, or i=0.

(1) Assume i>0

According to condition 2, we let a=i, b=0, then there is:

i*i>0*i

That is, the -1>0 contradiction.

(2) Assume i<0

Explain that i is a negative element, so -i is a positive element, there is -i>0, and also according to condition 2, there is:

i)*(i)>0*(-i)

That is, the -1>0 contradiction.

(3) Assuming i=0

Then there's no play!

We can't even compare the magnitudes of the imaginary units "i" and "0", let alone the complex numbers. But each complex number corresponds to a modulus, and the modulus belongs to the real number, so the modulus of the complex number can be compared in size, and the geometric meaning of the complex modulus is the distance from the complex number to the origin.

Geometrically, we can understand that all real numbers can be arranged sequentially from left to right, because real numbers are one-dimensional; However, two-dimensional complex numbers cannot be arranged sequentially, because two-dimensional numbers are already more complex than one-dimensional numbers, and we cannot arrange two-dimensional elements one by one in one dimension.

In this case, it is not possible to compare the size, because their structure is not symmetrical.

It can be compared to size, real and complex numbers are not just sets, they are also algebraic systems that define addition and multiplication operations, mathematically called fields, which can be calculated.

It is not possible to compare the size, because the imaginary numbers must be structurally the same before they can be compared.

This statement is correct

Any two numbers in the set of real numbers can determine the size relationship, and for any two (real) numbers a, b, a b, a b, a b, these three cases have and only one auspicious preparation is true; In the complex number c, we cannot specify the magnitude relation, because the infraction is an imaginary unit iIf we specify i>0 and multiply both sides by i at the same time, giving i 2>0 i.e. -1>0, this is obviously contradictory. The same goes for i

is not comparable to the size.

The relationship between two imaginary numbers can only be equal or unequal, and when they are equal, the real part and the imaginary part are equal respectively.

Imaginary numbers are only equal to or not equal to.