Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in fields like vehicle design and finance. Here, we will discuss the square root of -1/2.
The square root is the inverse of the square of a number. The number -1/2 is not a positive number, and its square root is complex. The square root of -1/2 is expressed in both radical and exponential form. In radical form, it is expressed as √(-1/2), whereas (-1/2)^(1/2) is the exponential form. The square root of -1/2 can be written as (i/√2), which is a complex number because it involves the imaginary unit i, where i = √(-1).
Finding the square root of negative numbers involves using the imaginary unit. Since -1/2 is not a positive number, typical real-number methods like prime factorization or long division are not applicable. Instead, we use complex number techniques. 1. Use the property of square roots of negative numbers: √(-a) = i√a. 2. Express -1/2 as a product of -1 and 1/2. 3. Apply the property: √(-1/2) = √(-1) * √(1/2) = i * (√1/√2) = (i/√2).
To find the square root of -1/2 using complex numbers, we use the imaginary unit i, where i = √(-1).
Step 1: Express -1/2 as (-1) * (1/2).
Step 2: Use the property of square roots: √(-1/2) = √(-1) * √(1/2).
Step 3: Simplify using the imaginary unit: √(-1) = i, so √(-1/2) = i * √(1/2).
Step 4: Further simplify: √(1/2) = 1/√2, so the result is (i/√2).
Another way to find the square root of a complex number is using its polar form.
Step 1: Express -1/2 in polar form as r(cos θ + i sin θ), where r is the modulus and θ is the argument.
Step 2: For -1/2, r = 1/2 and θ = π (since it lies on the negative real axis).
Step 3: Apply the square root formula for polar forms: √r (cos(θ/2) + i sin(θ/2)).
Step 4: √(1/2) = 1/√2, and θ/2 = π/2.
Step 5: Substitute these values to get (1/√2)(cos(π/2) + i sin(π/2)) = (i/√2).
Students often make mistakes while dealing with complex square roots. Here are some common errors and how to avoid them:
Students often make mistakes when finding square roots of negative numbers. Here are some common errors and how to avoid them.
Can you help Alex find the modulus of the complex number √(-1/2)?
The modulus of the complex number is 1/√2.
The modulus of a complex number a + bi is √(a² + b²). For √(-1/2) = (i/√2), the modulus is √(0² + (1/√2)²) = 1/√2.
If the square root of -1/2 is expressed in polar form, what is the angle it makes with the positive real axis?
The angle is π/2 radians.
For -1/2, the original angle is π. Its square root in polar form is at angle π/2, since θ/2 = π/2.
Calculate √(-1/2) multiplied by 2.
The result is i√2.
√(-1/2) = i/√2. When multiplied by 2, it becomes 2 * (i/√2) = i√2.
How do you express the square root of -1/2 in terms of exponential form?
It is expressed as (1/√2)eiπ/2.
The exponential form is r * eiθ, where r = 1/√2 and θ = π/2, so it is (1/√2)eiπ/2.
Find the square of the complex number (i/√2).
The square is -1/2.
(i/√2)² = (i²/2) = -1/2, since i² = -1.
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