Inequalities of real and complex numbers

[Char. Limit]

I was considering complex numbers (independently) and I came across an interesting question. Are any of these statements true?

$$1<i$$

$$1>i$$

$$i<-i$$

$$-1<i$$

$$-i<i<-1$$

 

[HallsofIvy]

The complex numbers are not a "ordered field". Neither "1< i" nor "i< 1" is true. In fact, you simply cannot write "a< b" for general complex numbers a and b.

 

[Gerenuk]

Indeed you cannot order complex numbers.
I guess, the order relations need to obey some sensible rules. With complex numbers you would always get contradictions. You could try to define the order by the modulus of the complex number. But then the universal law
$$a<b\qquad\Rightarrow\qquad a+c<b+c$$
would not hold for all complex numbers. Therefore you wouldn't be able to use the most basic properties for rearranging equations with variables.

 

[HallsofIvy]

Specifically, in an ordered field, the order must obey
1) if a< b and c is any third member of the field then a+ c< b+ c.

2) if a< b and 0< c, then ac< bc

3) For any two a, b in the field one and only one must be true:
i) a= b
ii) a< b
iii) b< a.

Suppose the Complex number were an ordered field with order "<". Clearly i is not equal to 0 so by (iii) either i< 0 or 0< i.

If i< 0 then, adding -i to both sides, by (i) 0< -i. So by (ii) we must have i(-i)< o(-i) or $-i^2= 1< 0$. Since this is not necessarily the "usual" order on real numbers that is not yet a contradiction. However, that implies, as before, that 0< -1 and then, multiplying i< 0 by -1, we have -i< 0, contradicting 0< -i above.

Suppose 0< i. Then 0(i)< i(i)= -1. But then 0(-1)< i(-1) so that 0< -i and, adding i to both sides, i< 0, again a contradiction.

 

[CellCoree]

The statements 1<i and 1>i are not true because the imaginary unit i is defined as the square root of -1, so it cannot be greater than or less than a real number like 1.

The statement i<-i is also not true because the imaginary unit i and its negative -i have the same absolute value, so they are neither greater than nor less than each other.

The statement -1<i is true because the imaginary unit i is greater than -1 on the complex plane, since it has a positive imaginary component.

The statement -i<i<-1 is also true because -i is less than i, which is less than -1 on the complex plane.

In general, inequalities involving complex numbers are not as straightforward as those involving real numbers because complex numbers have both real and imaginary components. When comparing complex numbers, we need to take into account both the real and imaginary parts.