The former, so sqrt(-1) = i, and 1 = -i
When we use take the square root using the radical sign âˆš in the reals we take only the positive root.
So and not .
But what does the radical sign âˆš actually mean when used on complex numbers?
What does evaluate to? or ?
HSC 2018 - [English Adv.] â€¢ [Maths Ext. 1] â€¢ [Maths Ext. 2] â€¢ [Chemistry] â€¢ [Software Design and Development]
_{1}âˆ«^{(3âˆš3)} t^{2} dt cos(3Ï€/9) = log(^{3}âˆše) | Integral t^{2} dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.
The former, so sqrt(-1) = i, and 1 = -i
Bachelor of Applied Finance and Bachelor of Commerce (Professional Accounting) at MQ.
HSC 2016: English Advanced, Math 3U, Math 4U, Chemistry, Business Studies
Why is it and not ?
HSC 2018 - [English Adv.] â€¢ [Maths Ext. 1] â€¢ [Maths Ext. 2] â€¢ [Chemistry] â€¢ [Software Design and Development]
_{1}âˆ«^{(3âˆš3)} t^{2} dt cos(3Ï€/9) = log(^{3}âˆše) | Integral t^{2} dt, From 1 to the cube root of 3. Times the cosine, of three pi over nine, Equals log of the cube root of e.
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.
Bachelor of Applied Finance and Bachelor of Commerce (Professional Accounting) at MQ.
HSC 2016: English Advanced, Math 3U, Math 4U, Chemistry, Business Studies
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.
Bachelor of Applied Finance and Bachelor of Commerce (Professional Accounting) at MQ.
HSC 2016: English Advanced, Math 3U, Math 4U, Chemistry, Business Studies
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).
Basically the convention with the "âˆš" 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.
The radical symbol (âˆš) is just square root i.e. to the power of 1/2. In the complex field, âˆš-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 -âˆš1, 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 âˆš-1. Nowadays its just to follow convention plus its easier and much neater to write -8i than âˆš-64. In any case, we only really convert from âˆš-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.
Last edited by darkk_blu; 2 Feb 2018 at 6:59 PM.
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