0
Note, first, that
i2 = -1; also, that
(1 + i)2 = 1 + 2i - 1 = 2i; and, finally, that
[±½(√2) ]2 = ½.
Thus, [±½(√2)(1 + i)]2 = i.
Therefore, [±½ (√2)(i - 1)]2 = -i, and
±½ (√2)(i - 1) = √-i.
* * *
In case calculus is familiar, you may prefer to find the desired roots in this manner:
eiθ = cosθ + i sinθ; thus, when
θ = ½ π, eiθ = i; and, when
θ = ¼ π, eiθ = +√i.
Then, +√i = cos ¼ π + i sin ¼ π
= (½√2) + i ½√2
= (½√2) (1 + i).
This gives,
+√-i = (+√i)(+√-1)
= i(+√i)
= (½√2) i(1 + i)
= (½√2) (i - 1).
Therefore the two roots are:
±√-i = ±(½√2)(i - 1).
* * *
The first method is simpler, for proving the result; the second method, for finding it.
What else can I help you with?
How do you find the two square roots in a number?
Use a calculator (if you need) to find the principal square root. The second square root is the negative of the number.
Which two square roots of integers can you find pie?
The two square roots of integers that can be associated with π (pi) are 0 and 1. The square root of 0 is 0, while the square root of 1 is 1. These values are significant because they represent the fundamental square roots of integers, and while π itself is an irrational number, these square roots are foundational in mathematics.
How do you find the two square roots of 23?
Find the square root of 23. The second one is the same number with a negative sign.
What are the two square roots of 0.0256?
√0.0256 = ± 0.16 The two square roots are 0.16 and -0.16.
How do you find the two square roots of i imaginary number Answer in rectangular form?
The two square roots of i are (k, k) and (-k, -k) where k = sqrt(2)/2 = 1/sqrt(2).
How do you find the two square roots in a number?
Use a calculator (if you need) to find the principal square root. The second square root is the negative of the number.
Which two square roots of integers can you find pie?
The two square roots of integers that can be associated with π (pi) are 0 and 1. The square root of 0 is 0, while the square root of 1 is 1. These values are significant because they represent the fundamental square roots of integers, and while π itself is an irrational number, these square roots are foundational in mathematics.
How do you find the two square roots of 23?
Find the square root of 23. The second one is the same number with a negative sign.
What are the two square roots of 0.0256?
√0.0256 = ± 0.16 The two square roots are 0.16 and -0.16.
How do you find the two square roots of i imaginary number Answer in rectangular form?
The two square roots of i are (k, k) and (-k, -k) where k = sqrt(2)/2 = 1/sqrt(2).
What is two square roots of the number 169?
The two square roots of the number, '169', are +13 and -13 .
What are the square roots of 256?
256 has two square roots: 16 and -16.
What are the two square roots for 196?
The square roots of 196 are 14 and -14.
What are the 2 square roots of 18?
The two square roots of 18 are: 4.242641 and -4.242641
What are the 2 square roots of 4900?
The two square roots are +70 and -70 .
Can you name two square roots of 0.64?
0.64 has square roots {0.8, -0.8}.
What is the two square roots of 81?
Both 9 and -9 are square roots of 81
