Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-4-\sqrt{3} i$$

Step-by-Step Solution

Answer

The complex conjugate of \(-4-\sqrt{3}i\) is \(-4+\sqrt{3}i\). The result of their multiplication is 19.

Step 1: Writing the Complex Conjugate

The complex number given is \(-4-\sqrt{3} i\). The complex conjugate is obtained by changing the sign of the imaginary part. So, the complex conjugate of \(-4-\sqrt{3} i\) is \(-4 + \sqrt{3} i\).

Step 2: Multiply the Number by its Complex Conjugate

The multiplication of a complex number with its conjugate is carried out as follows: \((-4-\sqrt{3} i) \cdot (-4 +\sqrt{3} i) = (-4)^2 -(\sqrt{3}i)^2 = 16 -(-3) = 16 + 3 = 19 \)

Key Concepts

Complex Numbers

Complex numbers are a sophisticated concept, expanding upon the more familiar real numbers. They consist of two parts: a real part and an imaginary part. Formally, a complex number is expressed as \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary coefficient, and \(i\) is the imaginary unit. The imaginary unit \(i\) is the cornerstone of this system, defined by the property that \(i^2 = -1\).

This extension of the number system allows for solutions to equations that don't have solutions in the realm of real numbers alone. For instance, an equation like \(x^2 + 1 = 0\) has no solution if we only accept real numbers, since no real number squared will yield a negative result. However, in the world of complex numbers, this equation has two solutions: \(x = i\) and \(x = -i\).

Complex numbers find applications in many fields including engineering, physics, and applied mathematics. They are particularly useful in describing oscillations, waves, and other phenomena oscillating in two dimensions.

Imaginary Numbers

Imaginary numbers might sound like a concept taken right out of a fantasy novel, but they are a very real part of mathematics. An imaginary number is basically a real number multiplied by the imaginary unit \(i\), which we noted before is defined by the equation \(i^2 = -1\).

The beauty of imaginary numbers is that they provide a way to take the square root of a negative number. For example, while the square root of \(-1\) is not a real number, it can be expressed as \(i\) in the complex number system. More complex examples include numbers like \(-\tfrac{3}{5}\), whose square root would be \(\tfrac{\tfrac{3}{5}}{\tfrac{5}{5}} i\) or simplified to \(i\tfrac{\tfrac{3}{5}}{\tfrac{5}{5}}\) which is \(\tfrac{\tfrac{3}{5}}{\tfrac{5}{5}}\) times \(i\).

When dealing with imaginary numbers, it's important to remember that they behave just like real numbers in many arithmetic operations, such as addition, subtraction, and multiplication by a real number. However, when multiplying two imaginary numbers, the result can be a real number because of the property that \(i^2 = -1\).

Multiplying Complex Numbers

Let's dive into how multiplying complex numbers works. To multiply two complex numbers, you apply the distributive property, much like you would when multiplying binomials. When you multiply \((a+bi)\) by \((c+di)\), you distribute each part of the first complex number with each part of the second, resulting in a sum of four terms: \(ac + adi + bci + bdi^2\).

But remember, since \(i^2 = -1\), the last term, rather than being an imaginary number, becomes a real number! This results in: \(ac - bd + (ad + bc)i\).

Let's look at an example. Suppose we multiply \(3 + 2i\) and \(1 + 4i\). Using our method, we get \((3)(1) + (3)(4i) + (2i)(1) + (2i)(4i)\), which simplifies to \(3 + 12i + 2i - 8\) (remembering that \(i^2 = -1\)). This further simplifies to \(-5 + 14i\). This process is key to understanding operations with complex numbers and paves the way for more advanced topics, like complex conjugates and division of complex numbers.