Math Problem Statement

1-i / 2i

Solution

The given expression is 1i2i\frac{1 - i}{2i}, where ii represents the imaginary unit. Let's simplify it step by step.

Step 1: Multiply by the conjugate of the denominator

To eliminate the imaginary part in the denominator, multiply both the numerator and denominator by the conjugate of 2i2i, which is 2i-2i:

1i2i×ii=(1i)(i)2i(i)\frac{1 - i}{2i} \times \frac{-i}{-i} = \frac{(1 - i)(-i)}{2i \cdot (-i)}

Step 2: Simplify the denominator

2i(i)=2i2=2(1)=22i \cdot (-i) = -2i^2 = -2(-1) = 2

So the denominator becomes 22.

Step 3: Simplify the numerator

Now, expand the numerator:

(1i)(i)=1(i)i(i)=i+i2(1 - i)(-i) = 1 \cdot (-i) - i \cdot (-i) = -i + i^2

Since i2=1i^2 = -1, the numerator simplifies to:

i1=(1+i)-i - 1 = -(1 + i)

Step 4: Combine the results

Now, the expression is:

(1+i)2=1i2\frac{-(1 + i)}{2} = \frac{-1 - i}{2}

Thus, the simplified form of 1i2i\frac{1 - i}{2i} is:

1i2\boxed{\frac{-1 - i}{2}}

Would you like a more detailed explanation or have any questions?


Here are 5 related questions to explore further:

  1. How do you multiply complex numbers?
  2. What is the geometric interpretation of multiplying by ii?
  3. How do you find the conjugate of a complex number?
  4. How do you divide complex numbers in polar form?
  5. Can you explain the properties of imaginary numbers?

Tip: When dividing complex numbers, always multiply by the conjugate to simplify the expression.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Conjugates
Imaginary Unit

Formulas

i^2 = -1
Conjugate of a complex number: a + bi -> a - bi
Multiplying complex numbers: (a+bi)(c+di) = ac + adi + bci + bdi^2

Theorems

Properties of the Imaginary Unit
Conjugate Multiplication

Suitable Grade Level

Grades 10-12