Write Each Of The Given Numbers In The Form A+bi

Write Each of the Given Numbers in theForm a + bi

Understanding how to express numbers as a + bi is a fundamental skill in algebra and pre‑calculus. Whether you are working with pure real numbers, imaginary numbers, or numbers given in polar or radical form, converting them to the standard complex form makes addition, subtraction, multiplication, and division straightforward. This guide walks you through the concept, the step‑by‑step process, detailed examples, common pitfalls, and practice problems to help you master the technique.

Introduction

Complex numbers extend the real number system by introducing the imaginary unit i, defined by the property i² = ‑1. Any complex number can be written uniquely as

[ z = a + bi, ]

where a is the real part and b is the imaginary part (both real numbers). The phrase “write each of the given numbers in the form a + bi” appears frequently in homework assignments and exams because it tests your ability to recognize and manipulate the two components of a complex number.

Understanding the Components

Before diving into conversion techniques, clarify what each symbol represents:

• a (real part): the coefficient that sits without i. It can be any real number, including zero.
• b (imaginary part): the coefficient that multiplies i. It is also any real number; if b = 0, the number is purely real; if a = 0, the number is purely imaginary.
• i: the imaginary unit, satisfying i² = ‑1.

When a number is already given as a sum of a real term and an imaginary term (e.g., 3 − 4i), it is already in a + bi form. The challenge arises when the number appears in other guises: radicals, fractions, polar coordinates, or expressions that need simplification.

Step‑by‑Step Process to Write Numbers in a + bi Form

Follow these systematic steps for any given number:

1. Identify the type of representation

   • Pure real (e.g., 7)
   • Pure imaginary (e.g., 5i)
   • Radical expression (e.g., √‑9)
   • Fraction with imaginary denominator (e.g., (2 + 3i)/(1 ‑ i))
   • Polar or exponential form (e.g., 4 cis 30°)
   • Combination of the above.

2. Simplify radicals involving negative numbers
   Replace √‑k with i√k (for k > 0).
   Example: √‑16 = 4i.

3. Rationalize denominators that contain i
   Multiply numerator and denominator by the complex conjugate of the denominator.
   The conjugate of c + di is c ‑ di.

4. Distribute and combine like terms
   After multiplication, separate real terms (those without i) from imaginary terms (those with i).

5. Express the result as a + bi
   Write the real part as a, the imaginary coefficient as b, and attach i to b.

6. Check for further simplification
   Reduce fractions, combine like terms, and ensure a and b are in simplest form.

Detailed Examples

Example 1: Pure Real Number

Given:  12

Since there is no imaginary part, b = 0.

[ 12 = 12 + 0i. ]

Example 2: Pure Imaginary Number

Given:  ‑7i

Here a = 0, b = ‑7.

[ -7i = 0 - 7i. ]

Example 3: Square Root of a Negative Number

Given:  √‑25

Step 2: √‑25 = i√25 = 5i.

Thus a = 0, b = 5 → 0 + 5i (or simply 5i).

Example 4: Fraction with Imaginary Denominator

Given:  (3 + 4i)/(2 ‑ i)

1. Identify conjugate of denominator: 2 + i.
2. Multiply numerator and denominator by the conjugate:

[ \frac{(3+4i)(2+i)}{(2-i)(2+i)}. ]

3. Compute denominator: (2‑i)(2+i) = 2² ‑ i² = 4 ‑ (‑1) = 5.
4. Expand numerator:

[ (3+4i)(2+i) = 3·2 + 3·i + 4i·2 + 4i·i = 6 + 3i + 8i + 4i². ]

Since i² = ‑1, 4i² = ‑4.

Combine: 6 ‑ 4 + (3i + 8i) = 2 + 11i.

5. Put over denominator:

[ \frac{2 + 11i}{5} = \frac{2}{5} + \frac{11}{5}i. ]

Thus a = 2/5, b = 11/5 → (\frac{2}{5} + \frac{11}{5}i).

Example 5: Polar Form (Magnitude‑Angle)

Given:  6 cis 45° (where cis θ = cos θ + i sin θ)

1. Compute cosine and sine:

[ \cos45° = \frac{\sqrt{2}}{2},\quad \sin45° = \frac{\sqrt{2}}{2}. ]

2. Multiply by magnitude 6: [ 6\left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = 3\sqrt{2} + 3\sqrt{2}i. ]

Thus a = 3√2, b = 3√2 → (3\sqrt{2} + 3\sqrt{2}i).

Example 6: Combination of Radicals and Fractions

Given:  (\frac{5}{\sqrt{-9}})

1. Simplify radical: √‑9 = 3i.
2. Expression becomes (\frac{5}{3i}).
3. Rationalize: multiply numerator and denominator by i (since i · i = ‑1):

[ \frac{5}{3i} \cdot \frac{i}{i} = \frac{5i}{3i^2} = \frac{5i}{3(-1)} = -\frac{5}{3}i. ]

So (a = 0), (b = -\frac{5}{3}) → (0 - \frac{5}{3}i) or simply (-\frac{5}{3}i).

Conclusion

Converting any complex expression to the standard form (a + bi) is a systematic process of simplifying radicals, rationalizing denominators, and separating real and imaginary parts. Whether starting from a real number, a pure imaginary, a radical involving negatives, a fraction with complex terms, or a polar representation, the steps above guide you to an unambiguous (a + bi) format. This form is essential for addition, subtraction, multiplication, division, and deeper analysis of complex numbers in algebra and beyond.

Such precision underpins advancements in technology and mathematics.

The mastery of these principles bridges theoretical understanding with practical application, ensuring clarity and efficacy across disciplines. Continued refinement remains essential for progress. Thus, embracing such knowledge secures its enduring relevance.

Applications in Engineering and Physics

Complex numbers extend beyond abstract mathematics into practical domains. In electrical engineering, they model alternating current (AC) circuits, where impedance combines resistance and reactance into a complex quantity. For example, a capacitor's reactance is represented as (-\frac{1}{\omega}i), enabling calculations for voltage and current. Quantum mechanics relies on complex wave functions, where the probability density is derived from the squared magnitude of (\psi(x) = a + bi). Signal processing uses Fourier transforms, decomposing signals into complex exponentials (e^{i\omega t}) to analyze frequency components. These applications underscore the real-world impact of mastering complex number operations.

Advanced Operations: Powers and Roots

Beyond arithmetic, complex numbers enable deeper explorations. For instance, raising (1 + i) to the third power:
[ (1 + i)^3 = (1 + i)^2(1 + i) = (1 + 2i + i^2)(1 + i) = (1 + 2i - 1)(1 + i) = 2i(1 + i) = 2i + 2i^2 = -2 + 2i. ]
Roots are equally revealing. The square roots of (-4) are found by solving (z^2 = -4), yielding (z = \pm 2i). For (z^2 = 1 + i), express (1 + i) in polar form:
[ \sqrt{2} \text{ cis } 45^\circ \implies \text{Roots: } \sqrt[4]{2} \text{ cis } 22.5^\circ, \sqrt[4]{2} \text{ cis } 202.5^\circ. ]
These operations reveal the multi-valued nature of complex functions, essential for fields like control theory.

Conclusion

The systematic conversion of complex expressions to (a + bi) form—whether from radicals, fractions, or polar coordinates—serves as the foundation for advanced mathematical and scientific inquiry. This standardization ensures clarity in operations, from basic arithmetic to multi-valued functions, enabling breakthroughs in engineering, physics, and computing. As technology evolves, the ability to manipulate complex numbers remains critical for solving real-world problems, from designing efficient power grids to modeling quantum behaviors. Mastery of these principles not only enhances analytical precision but also unlocks innovative solutions across disciplines. Thus, the study of complex numbers is not merely an academic exercise but a vital tool for progress in an increasingly interconnected world.