Why can t complex numbers compare sizes?

The size of the complex numbers can certainly be compared! However, complex numbers have vector-like directions in addition to size, which means that simple comparison of sizes cannot distinguish between two complex numbers.

There is no need for so-called university knowledge for this question.

1.Complex numbers have a one-to-one correspondence with points on a plane in a geometric sense;

2.Complex numbers are algebraic representations of physical vectors;

3.Complex numbers are the development of real numbers in the number system, and they are the only possibility of development without changing the rules of operation of real numbers.

4.Complex numbers, real numbers, and all number systems, and even mathematics as a whole, are the product of the human mind, and there is no objectivity to speak of, so that all philosophical foundations that deal with why will come down to why people think the way they do!

Complex numbers are equivalent to a vector quantity, which can point in any direction in the complex plane system, while real numbers are on the real axis, and the direction is fixed, so they can be compared in size.

Let's understand it this way! Each complex number is a point in the plane of a Cartesian coordinate system. We can't find a size relationship between two points on a plane!

Plural. z=a+bi

a, b are real numbers).

When b=0, z is a real number, and the magnitude can be compared;

When b is not zero, z is.

Imaginary number. (a=0.)

Pure imaginary numbers, no comparison of sizes.

Mathematics. on the so-called size.

Definition. Yes, the ratio on the right side of the (real) number line.

Left. Great. And the representation of complex numbers is to introduce the imaginary number axis, in.

Plane. so it doesn't fit in with the about.

Big and small. Definition. And it doesn't seem to make much sense to define the size of the plural number.

Because complex numbers have directions, they cannot be compared in size.

In fact, complex numbers are the positions of points in the plane.

The size of the location does not exist.

Complex numbers cannot be compared in size, because they are called real parts, b is called imaginary parts, and the imaginary part is not an imaginary number, but a real number.

is an imaginary unit, =-1. If two complex numbers are equal, then their real part must be equal to the real part, and the imaginary part must be equal to the imaginary part.

Numbers are those symbols that can be arranged from smallest to largest, and in this sense, complex numbers are indeed not numbers. This is not the exception, because no pair of numbers (including vectors) can be compared in the usual sense of the word. However, a complex set contains a set of real numbers, because it is only necessary to make the coefficient in front of the imaginary number i 0 in the complex number.

Complex numbers can define operations.

The magnitude of complex numbers is called modulus length, which is consistent with the method of calculating vectors. If a complex number is a real number, it means that its imaginary part should be zero; If a complex number is a pure imaginary number, then its real part must be zero, and its imaginary part must not be zero.

Because a complex number cannot be defined as a self-consistent ordered field, it is compatible in addition and multiplication.

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 treat complex numbers as two-dimensional numbers, but no matter how we define them, we cannot make complex numbers satisfy the two conditions of an ordered field.

Complex numbers cannot be compared in size. Because the mathematical definition of size is that the right side is larger than the left side on the real number axis. However, the representation of complex numbers should introduce the imaginary number axis and represent it on a plane, so it does not meet the definition of large and small.

And it doesn't seem to make much sense to define the size of the plural number. We call a number of the form Z=A+Bi (both a and b are real numbers) complex numbers, where a is called the real part, b is called the imaginary part, and i is called the imaginary unit. When the imaginary part of z is equal to zero, z is often called a real number; When the imaginary part of z is not equal to zero, and the real part is equal to zero, z is often called a pure imaginary number.

A complex number field is an algebraic closure of a real number field, i.e., any complex coefficient polynomial always has roots in the complex number field.