Results 1 to 8 of 8
Like Tree3Likes
  • 1 Post By ProdigyInspired
  • 1 Post By Trebla
  • 1 Post By InteGrand

Thread: Does √-1 evaluate to i or -i?

  1. #1
    617 pages fan96's Avatar
    Join Date
    May 2017
    HSC
    2018
    Gender
    Male
    Location
    NSW
    Posts
    417
    Rep Power
    2

    Does √-1 evaluate to i or -i?

    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 ?
    HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]

    ATAR: 97.40 | Uni Course: B Advanced Mathematics (Hons) / B Engineering (Hons) (Computer) at UNSW

  2. #2
    Tafe Advocate ProdigyInspired's Avatar
    Join Date
    Oct 2014
    HSC
    2016
    Gender
    Male
    Posts
    630
    Rep Power
    4

    Re: Does √-1 evaluate to i or -i?

    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

  3. #3
    617 pages fan96's Avatar
    Join Date
    May 2017
    HSC
    2018
    Gender
    Male
    Location
    NSW
    Posts
    417
    Rep Power
    2

    Re: Does √-1 evaluate to i or -i?

    Why is it and not ?
    HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]

    ATAR: 97.40 | Uni Course: B Advanced Mathematics (Hons) / B Engineering (Hons) (Computer) at UNSW

  4. #4
    Tafe Advocate ProdigyInspired's Avatar
    Join Date
    Oct 2014
    HSC
    2016
    Gender
    Male
    Posts
    630
    Rep Power
    4

    Re: Does √-1 evaluate to i or -i?

    Quote Originally Posted by fan96 View Post
    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.
    fan96 likes this.
    Bachelor of Applied Finance and Bachelor of Commerce (Professional Accounting) at MQ.

    HSC 2016: English Advanced, Math 3U, Math 4U, Chemistry, Business Studies

  5. #5
    Tafe Advocate ProdigyInspired's Avatar
    Join Date
    Oct 2014
    HSC
    2016
    Gender
    Male
    Posts
    630
    Rep Power
    4

    Re: Does √-1 evaluate to i or -i?

    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

  6. #6
    Administrator Trebla's Avatar
    Join Date
    Feb 2005
    HSC
    2006
    Gender
    Male
    Posts
    6,229
    Rep Power
    20

    Re: Does √-1 evaluate to i or -i?

    Quote Originally Posted by fan96 View Post
    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).
    fan96 likes this.

  7. #7
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,075
    Rep Power
    7

    Re: Does √-1 evaluate to i or -i?

    Quote Originally Posted by fan96 View Post
    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.
    fan96 likes this.

  8. #8
    Cadet
    Join Date
    Jan 2018
    HSC
    2018
    Gender
    Male
    Posts
    71
    Rep Power
    1

    Re: Does √-1 evaluate to i or -i?

    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.
    Last edited by darkk_blu; 2 Feb 2018 at 6:59 PM.

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •