Problem 94 - Algebra A Combined Function

Question

Write each expression in the form \(a+b i .\) $$ 7-\sqrt{-25} $$

Step-by-Step Solution

Answer

The expression in the form \(a + bi\) is \(7 - 5i\).

Step 1: Identify the Complex Component

In the expression given, we have the square root of a negative number, namely \(-25\). Use the property of complex numbers to rewrite this in terms of the imaginary unit \(i\). Recall that \( \sqrt{-1} = i \), so \( \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \).

Step 2: Rewrite the Expression

Substitute the complex number expression obtained into the original equation. The original expression is \(7 - \sqrt{-25}\), which we now substitute with \(5i\), resulting in: \(7 - 5i\).

Step 3: Write in Standard Form

The expression now reads \(7 - 5i\). In standard form, where a complex number is written as \(a + bi\), the expression \(7 - 5i\) is already in the correct form, where \(a = 7\) and \(b = -5\). Therefore, the expression is \(7 + (-5)i \) which simplifies to \(7 - 5i\).

Key Concepts

Real Numbers

Real numbers are a fundamental component of the number system that we use every day. They include all the numbers that can be located on the number line. This set consists of both rational and irrational numbers, such as whole numbers, fractions, and square roots of non-negative numbers.

• Rational Numbers: These are numbers that can be expressed as a fraction of two integers, like \( \frac{3}{4} \) or \(-2\).
• Irrational Numbers: These cannot be represented as a simple fraction, such as \( \sqrt{2} \) or \( \pi \).

Real numbers are used to express a wide range of everyday quantities, such as distance, temperature, and time. In the given exercise, the number '7' is a real number and represents the real part of the complex number. Understanding real numbers is essential for dealing with other types of numbers, like complex numbers, because they often form the basis of the real component in a complex expression.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a conceptual tool used in mathematics to extend the real number system to include complex numbers. It is defined by the property:\[i^2 = -1\]This definition allows us to work with the square roots of negative numbers, which cannot be handled within the realm of real numbers. For example:

• \( \sqrt{-1} = i \)
• \( \sqrt{-4} = 2i \) because \( \sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2 \times i = 2i \)

Imaginary numbers are a crucial part of complex numbers, where they are combined with real numbers to form expressions like \(a + bi\). In the exercise, the term \(5i\) arises from taking the square root of \(-25\). Imaginary units allow us to solve equations that have no real solutions, thereby broadening the scope of mathematical and scientific inquiry.

Standard Form

Complex numbers have a special format known as the standard form, which makes it easier to work with them. The standard form is expressed as:\[a + bi\]Where:

• \(a\) is the real part of the complex number, a real number.
• \(b\) is the coefficient of the imaginary part, which is also a real number.
• \(i\), the imaginary unit, indicates the presence of the imaginary part.

Using this format helps in simplifying complex arithmetic, such as addition, subtraction, multiplication, and division of complex numbers. The standard form makes it easy to identify and separate real and imaginary components. In our exercise, after transforming and simplifying, the expression \(7 - 5i\) is already in standard form, with \(a = 7\) and \(b = -5\). Understanding the importance of this format is vital for performing complex number calculations accurately and efficiently.