In Exercises 21–28, divide and express the result in standard form. 2i/(1 + i)

Verified step by step guidance

1. Identify the problem: You need to divide the complex number \$2i\( by the complex number \)(1 + i)\( and express the result in standard form \)a + bi\(, where \)a\( and \)b$ are real numbers.
2. To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \((1 + i)\) is \((1 - i)\). So, multiply numerator and denominator by \((1 - i)\):
3. \[\frac{2i}{1 + i} \times \frac{1 - i}{1 - i}\]
4. Use the distributive property (FOIL) to expand both numerator and denominator:
5. - Numerator: \$2i \times (1 - i) = 2i - 2i^2$
6. - Denominator: \((1 + i)(1 - i) = 1 - i^2\)
7. Recall that \(i^2 = -1\), so simplify both numerator and denominator using this fact, then combine like terms to write the expression in the form \(a + bi\).

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form makes it easier to perform arithmetic operations and interpret the number geometrically.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression into a standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part from the denominator, resulting in a real number denominator, which simplifies division of complex numbers.