Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the complex number in standard form. $$7-\sqrt{-25}$$

Step-by-Step Solution

Answer

The standard form of the complex number is \(7 - 5i\).

Step 1: Identify the Real and Imaginary Parts

The number given is \(7-\sqrt{-25}\), where 7 is the real part. The \(\sqrt{-25}\) represents the imaginary part in the expression.

Step 2: Simplify the Imaginary Part

The expression \(\sqrt{-25}\) can be rewritten as \(\sqrt{-1}\) times \(\sqrt{25}\), or \(5i\). The square root of -1 is defined as the imaginary number \(i\). Thus, \(\sqrt{-25}\) simplifies to \(5i\). But since it is subtracted in the expression, it will be \(-5i\).

Step 3: Write the Complex Number in Standard Form

The standard form of a complex number is \(a + bi\). From Steps 1 and 2, we've identified that \(a = 7\) and \(b = -5\). So, the complex number in standard form would be \(7 - 5i\).

Key Concepts

Real Part

In a complex number, the real part is an essential building block. Each complex number is generally expressed in the form of \(a + bi\), where \(a\) is the real part and \(b\) is part of the imaginary term. Imagine the real part as the number you are already familiar with along the standard number line. For instance, if you have the expression \(7 - \sqrt{-25}\), the 7 stands out as the real part.
It doesn't involve any imaginary unit, \(i\), and behaves just like a number from your everyday counting and calculations. But don't underestimate its role—it's crucial when combining with the imaginary part to construct the full complex number.

Imaginary Part

The imaginary part is what truly distinguishes a complex number. While it may seem perplexing at first, it simply introduces the imaginary unit \(i\), which represents the square root of \(-1\). In our example with \(7 - \sqrt{-25}\), the term \(\sqrt{-25}\) brings the imaginary component into play.
By simplifying, \(\sqrt{-25}\) becomes \(-5i\) since \(\sqrt{-25}\) is equivalent to \(\sqrt{-1} \times \sqrt{25}\), resulting in \(5i\). This part, \(-5i\), combines with the real part to form the entire complex unit. Think of the imaginary component as a way to handle square roots of negative numbers, moving beyond real-number limits.

Standard Form of Complex Numbers

Understanding the standard form of a complex number is crucial when working with these numbers. It's the straightforward \(a + bi\) format, where \(a\) is the real part, and \(b\) is the coefficient of the imaginary unit \(i\). This arrangement helps in dealing with operations like addition, subtraction, and solving equations involving complex numbers.
In our previous example, simplifying the complex number \(7 - \sqrt{-25}\) yields \(7 - 5i\). Here, 7 is the real part, and \(-5\) acts as the coefficient of \(i\), confirming the standard form of the complex number \(7 - 5i\). Using the standard form is a widely accepted convention, making operations with complex numbers more systematic. The format ensures consistency and clarity across various complex arithmetic and algebra challenges.