Express The Following In Simplest A + Bi Form.

Expressing Complex Numbers in the Simplest a + bi Form

This comprehensive guide delves into the intricacies of expressing complex numbers in their simplest a + bi form, a fundamental concept in complex analysis. We'll explore various methods, tackle challenging examples, and equip you with the skills to confidently handle any complex number simplification problem.

Understanding Complex Numbers and the a + bi Form

A complex number is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit defined as the square root of -1 (√-1). 'a' is called the real part and 'b' is called the imaginary part of the complex number. The simplest a + bi form is the standard representation, ensuring clarity and consistency in mathematical operations.

The beauty of this form lies in its ability to represent all numbers, both real and imaginary. A purely real number (like 5) can be expressed as 5 + 0i, and a purely imaginary number (like 3i) can be written as 0 + 3i. This unified representation allows for elegant mathematical manipulation across the entire number system.

Basic Operations and Simplification Techniques

Before diving into more complex examples, let's review the fundamental operations with complex numbers and how they contribute to simplification:

Addition and Subtraction

Adding or subtracting complex numbers involves adding or subtracting their real and imaginary parts separately:

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

Example: (3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i

Multiplication

Multiplication follows the distributive property, remembering that i² = -1:

(a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i

Example: (2 + 3i)(1 - i) = (21 - 3-1) + (2*-1 + 3*1)i = 5 + i

Division

Dividing complex numbers requires a clever technique involving the complex conjugate. The complex conjugate of a + bi is a - bi. Multiplying the numerator and denominator by the conjugate of the denominator eliminates the imaginary part from the denominator:

(a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)] = [(ac + bd) + (bc - ad)i] / (c² + d²)

This results in a complex number in the a + bi form, where the denominator is a real number.

Example: (3 + 2i) / (1 + i) = [(3 + 2i)(1 - i)] / [(1 + i)(1 - i)] = (5 - i) / 2 = 5/2 - (1/2)i

Handling Radicals and Higher Powers of i

Often, simplification involves dealing with radicals and higher powers of 'i'. Recall that:

• i¹ = i
• i² = -1
• i³ = -i
• i⁴ = 1

The pattern repeats every four powers. Any higher power of 'i' can be reduced to one of these four values.

Example: Simplify i¹⁵. Since 15 divided by 4 leaves a remainder of 3, i¹⁵ is equivalent to i³ = -i.

Simplifying Expressions with Radicals

When dealing with square roots of negative numbers, remember to express them in terms of 'i' before proceeding with any other operations.

Example: Simplify √(-9) + √(-16). This becomes 3i + 4i = 7i.

Example: Simplify (√-2)(√-8). This becomes (i√2)(i√8) = i²√16 = -4

Advanced Simplification Techniques and Examples

Let's tackle some more challenging examples to solidify your understanding:

Example 1: Simplify (2 + 3i)²

Using the formula (a + b)² = a² + 2ab + b², we get:

(2 + 3i)² = 2² + 2(2)(3i) + (3i)² = 4 + 12i + 9i² = 4 + 12i - 9 = -5 + 12i

Example 2: Simplify (1 + i)³

Expanding (1 + i)³ gives:

(1 + i)³ = (1 + i)²(1 + i) = (1 + 2i + i²)(1 + i) = (1 + 2i - 1)(1 + i) = 2i(1 + i) = 2i + 2i² = 2i - 2 = -2 + 2i

Example 3: Express (3 + 2i) / (1 - 2i) in the form a + bi

Multiplying the numerator and denominator by the conjugate of the denominator (1 + 2i):

[(3 + 2i)(1 + 2i)] / [(1 - 2i)(1 + 2i)] = (3 + 6i + 2i + 4i²) / (1 - 4i²) = (3 + 8i - 4) / (1 + 4) = (-1 + 8i) / 5 = -1/5 + (8/5)i

Example 4: Simplify (1 + i) / (2 - i) + (3 - 2i) / (1 + i)

First, simplify each fraction individually:

(1 + i) / (2 - i) = [(1 + i)(2 + i)] / [(2 - i)(2 + i)] = (2 + 3i + i²) / (4 - i²) = (2 + 3i - 1) / (4 + 1) = (1 + 3i) / 5 = 1/5 + (3/5)i

(3 - 2i) / (1 + i) = [(3 - 2i)(1 - i)] / [(1 + i)(1 - i)] = (3 - 5i + 2i²) / (1 - i²) = (3 - 5i - 2) / (1 + 1) = (1 - 5i) / 2 = 1/2 - (5/2)i

Now, add the simplified fractions:

(1/5 + (3/5)i) + (1/2 - (5/2)i) = (1/5 + 1/2) + ((3/5) - (5/2))i = (7/10) + (-19/10)i = 7/10 - 19/10i

Conclusion: Mastering the a + bi Form

Expressing complex numbers in the simplest a + bi form is crucial for performing various mathematical operations and solving complex equations. Through consistent practice and a solid understanding of the basic operations and simplification techniques outlined in this guide, you'll develop the confidence to tackle even the most challenging complex number problems. Remember to systematically apply the rules of arithmetic, paying close attention to the manipulation of the imaginary unit 'i' and its powers. By mastering these skills, you'll lay a strong foundation for further explorations in complex analysis and related mathematical fields. Consistent practice is key – the more problems you work through, the more proficient you'll become.