Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the quotient in standard form. $$\frac{6}{i}$$

Step-by-Step Solution

Answer

The quotient in standard form is \(-6i\).

Step 1 - Identification

Identify the real and imaginary parts of the complex number. In our case, the real part is 6 and the imaginary part is 1 (since 'i' is equivalent to \(1i\)).

Step 2 - Conjugate

In order to transform the denominator into a real number, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(i\) is \(-i\). Therefore, we perform the operation: \(\frac{6}{i} \times \frac{-i}{-i}\).

Step 3 - Simplify

Perform the multiplication. The denominator becomes \(i \times -i = -i^2\). Remember \(i^2 = -1\). Therefore the denominator becomes -(-1) or 1. The numerator becomes \(6 \times -i\) or \(-6i\). So the expression simplifies to \(\frac{-6i}{1}\) or simply \(-6i\).

Key Concepts

Conjugate of a Complex Number

The conjugate of a complex number plays a crucial role in simplifying expressions, especially when dividing complex numbers. A complex number is usually written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The conjugate of this complex number is \(a - bi\). It is created by changing the sign of the imaginary part.

Why is the conjugate so important? When a complex number is multiplied by its conjugate, the result is always a real number. This is because the product of \(i\) and \(-i\) is \(-i^2\), which simplifies to 1, as \(i^2 = -1\). So, when you have a complex number in the denominator, like in the exercise \(\frac{6}{i}\), multiplying numerator and denominator by the conjugate \(-i\) eliminates the imaginary unit from the denominator, giving a real number. This is essential for expressing complex numbers in standard form, which avoids having an imaginary unit in the denominator.

Multiplication of Complex Numbers

The multiplication of complex numbers may seem daunting at first, but it follows the same basic principles as multiplying binomials. To multiply two complex numbers, we use the distributive property, also known as the FOIL method for binomials, which stands for First, Outer, Inner, Last.

Consider two complex numbers, \(a + bi\) and \(c + di\). Their product is \((a + bi)(c + di)\), which simplifies to \(ac + adi + bci + bdi^2\). Because \(i^2 = -1\), this further simplifies to \(ac - bd + (ad + bc)i\), combining the real parts and the imaginary parts separately. In the exercise example, the multiplication was between \(6\) and \(-i\), and since there's no real part to multiply with \(-i\), we simply get \(-6i\).

Imaginary Unit \(i\)

The imaginary unit \(i\) is the foundation of complex numbers. It is defined as the square root of \(-1\), which does not have a solution in the set of real numbers. This imaginary unit allows us to extend the real number system to include solutions to equations like \(x^2 + 1 = 0\).

In arithmetic involving \(i\), it's crucial to remember that \(i^2 = -1\). This property helps us simplify expressions involving \(i\). For instance, in our exercise, multiplying \(i\) by \(-i\) gives us \(-i^2\), which simplifies to 1, as shown in the steps above. It's also essential to keep in mind that when dealing with powers of \(i\), they follow a pattern: \(i^3 = -i\), \(i^4 = 1\), and then the cycle repeats. Understanding the behavior of \(i\) is key to working effectively with complex numbers.