Does √-1 evaluate to i or -i? (1 Viewer)

[fan96]

When we use take the square root using the radical sign √ in the reals we take only the positive root.

So and not .

But what does the radical sign √ actually mean when used on complex numbers?

What does evaluate to? or ?

 

[ProdigyInspired]

The former, so sqrt(-1) = i, and 1 = -i

 

[fan96]

Why is it and not ?

 

[ProdigyInspired]

Why is it and not ?

I'm not familiar at all with the complex mathematical background of it, but the general idea is that it's impossible to square root a negative number i.e. with the definition of a square root where the result squared itself is the number inside of the square root (a bit confusing, but it leads), you therefore need imaginary numbers, which we use to describe. Don't think too much into it, there is not always a logic to why it happens, rather it is a convenient definition used.

 

[ProdigyInspired]

There is no purpose to make sqrt(1) a definition as it is clearly already = 1. It's more convenient to use and makes more sense to do that. Since sqrt(-1) doesn't exist without the use of imaginary numbers, can be used to define and introduce an impossible number as a variable and therefore allow expansion across equations that use imaginary numbers.

 

[Trebla]

When we use take the square root using the radical sign √ in the reals we take only the positive root.

So and not .

But what does the radical sign √ actually mean when used on complex numbers?

What does evaluate to? or ?

A lot of textbooks seem to define . However, the actual definition is just (that 'i' is this mysterious thing that squares to -1). Radical signs should ideally be avoided when denoting 'i' as square roots of negative numbers do not follow the standard properties of radicals/surds (treating 'i' like a pronumeral in arithmetic operations typically avoids this).

 

[InteGrand]

When we use take the square root using the radical sign √ in the reals we take only the positive root.

So and not .

But what does the radical sign √ actually mean when used on complex numbers?

What does evaluate to? or ?

Basically the convention with the "√" symbol for complex numbers is to take the root that has its principal argument in the range (-pi/2, pi/2]. (For any non-zero complex number z, there is a unique square root of z with principal argument in this range. This square root is called the principal square root of z.)

I don't know if the HSC adheres to this convention though, and you don't really need to know it for the HSC either I think.

 

[darkk_blu]

The radical symbol (√) is just square root i.e. to the power of 1/2. In the complex field, √-1, its just i, otherwise i.e. in the real number field, it is impossible to square root negative numbers, hence we write no solution. As for -√1, its -i, much like 1 and -1. In reality, we use i was invented because in the past, imaginary numbers were taboo and people didn't like the thought of square rooting a negative number, pioneers of mathematics and complex numbers decided to make i denote √-1. Nowadays its just to follow convention plus its easier and much neater to write -8i than √-64. In any case, we only really convert from √-1 to i when simplifying the square root of a number like after using the quadratic equation and to put numbers in a+bi form.