Question

Simplify: (a) \(4 i-7 i^{3}\) (b) \((2-3 i) / 5 i\)

Short answer

(a) The simplified expression is \(11i\). (b) The simplified expression is \((-2i - 3) / 5\).

Step-by-step solution

(a) Simplify the expression \(4i - 7i^3\)

We start by observing that \(i^3 = i^2 \cdot i\). Since \(i^2 = -1\), we can rewrite the expression as: \(4i - 7(-1)i = 4i + 7i\) Now, combine the imaginary terms: \( (4 + 7)i = 11i \) Therefore, the simplified expression for (a) is \(11i\).

(b) Simplify the expression \((2 - 3i) / 5i\)

To simplify a complex number division, we usually multiply both the numerator and the denominator by the complex conjugate of the denominator. In this case, the complex conjugate of \(5i\) is \(-5i\). However, since the denominator is purely imaginary, we can simplify this expression more directly. Multiply both the numerator and the denominator by \(-i\): \((-i) \cdot (2 - 3i) / (-i) \cdot (5i)\) Now, distribute \(-i\) in both the numerator and the denominator: \(((-i) \cdot 2) + ((-i) \cdot (-3i)) / ((-i) \cdot 5i)\) This gives us: \((-2i + 3i^2) / (-5i^2)\) Recall that \(i^2 = -1\), so we can simplify the expression further: \((-2i + 3(-1)) / (-5(-1))\) \((-2i - 3) / 5\) Thus, the simplified expression for (b) is \(( -2i - 3) / 5\).

Key concepts

Understanding the Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental component in the realm of complex numbers. It is defined by the equation \(i^2 = -1\). This definition may seem unusual because no real number squared results in a negative number. However, this unique property of \(i\) allows us to describe numbers that extend beyond the real number line.

When you encounter powers of \(i\), they exhibit a cyclic pattern:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

This cycle repeats every four powers. So, if you need to simplify expressions containing higher powers of \(i\), reduce them by taking advantage of this cycle. This property helps simplify complex expressions quickly and efficiently.

Exploring Complex Conjugates

The complex conjugate of a complex number is a powerful tool for simplifying equations. For any complex number in the form \(a + bi\), its complex conjugate is \(a - bi\). By reflecting the imaginary part, we gain a mirror image of the original number across the real axis.

Complex conjugates have the wonderful property where multiplying a complex number by its conjugate results in a real number. Consider \((a + bi)(a - bi)\):

\((a^2 - (bi)^2) = a^2 + b^2\)

This transformation eliminates the imaginary part and leaves us with the sum of the squares of the real and imaginary components.

This key property of complex conjugates is notably useful in complex number division, where the conjugate is used to "rationalize" the denominator.

Simplifying Complex Expressions

To simplify complex expressions, especially those involving powers of the imaginary unit, it’s important to apply the rules and properties correctly. Let’s consider an expression like \(4i - 7i^3\). By recognizing that \(i^3 = i^2 \cdot i\) and substituting \(i^2 = -1\), we can rewrite \(i^3\) as \(-i\).

This conversion allows us to simplify the original expression:
\(4i + 7(-1)i = 4i + 7i\).

Now, by combining the like terms (in this case, the terms that include \(i\)), you achieve a simpler form, such as \(11i\) for this particular example. Employing these steps will help you tackle a variety of similar problems with ease.

Complex Number Division Made Simple

Dividing complex numbers can initially seem daunting, but it becomes straightforward with the right approach. When dealing with \(\frac{2 - 3i}{5i}\), remember that division by a complex number often requires using its conjugate to simplify the expression.

However, a simpler path, when the denominator is purely imaginary like \(5i\), is to multiply both the numerator and the denominator by \(-i\). This action leverages the property \(i^2 = -1\) to eliminate \(i\) from the denominator:

Multiply:
\(((2 - 3i)(-i)) / ((5i)(-i))\)

Upon simplifying the terms, this yields:

• The numerator: \((-2i + 3i^2)\) becomes \((-2i - 3)\)
• The denominator: \(5i \cdot (-i)\) simplifies to \(5\)

Thus, by reimagining complex division with these shortcuts, you simplify the expression to \((-2i - 3)/5\). This method demystifies the division process, making it more manageable and less intimidating!