Problem 70 - Algebra 2

Question

Simplify each expression. $$ (7+3 i)-(2+i) $$

Step-by-Step Solution

Answer

The simplified expression is \(5 + 2i\).

Step 1: Identify the Real and Imaginary Parts

Recognize that the first number, 7, and the number being subtracted, 2, are real numbers. Similarly, identify that the numbers related to the imaginary unit 'i', which are +3 and -1 (implicit in '-i'), are the imaginary parts of the complex numbers.

Step 2: Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number: 7 - 2.

Step 3: Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number: 3 - 1.

Step 4: Combine the Results

Combine the result of the subtraction of the real parts with the result of the subtraction of the imaginary parts to get the simplified expression.

Key Concepts

Complex Numbers

A complex number is an extension of the familiar real numbers that we use in everyday arithmetic. It comprises a real part and an imaginary part, which is represented by the imaginary unit, denoted as 'i'. In general, a complex number is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. For example, in the complex number \(7+3i\), \(7\) is the real part and \(3\) is the coefficient of the imaginary part.

The concept of complex numbers is fundamental in various branches of mathematics and are widely applied in fields such as engineering, physics, and computer science. They are useful in solving equations that have no real solutions, like \(x^2+1=0\).

Subtracting Complex Numbers

Subtracting complex numbers is similar to subtracting polynomials. To perform the subtraction, you break down each complex number into its real and imaginary parts. For instance, if you have the expression \( (7+3i) - (2+i) \), you'd take the real parts \(7\) and \(2\), and subtract them to get \(7 - 2\).

Following this, you'd subtract the coefficients of the imaginary parts, \(3\) from \(1\), to get \(3 - 1\). Finally, you combine these results to obtain the simplified form of the complex number after subtraction. The process is straightforward, and keeping track of the real and imaginary parts separately simplifies the calculation.

Imaginary Unit i

The concept of the imaginary unit \(i\) is foundational in the arithmetic of complex numbers. It is defined by the property that \(i^2 = -1\). This peculiar entity allows for the existence of complex numbers, as it symbolizes the square root of negative one, which doesn't have a real solution.

When performing operations with complex numbers, the imaginary unit plays a crucial role in simplifying expressions. For instance, if you multiply \(i\) by itself, the result is \(i^2=-1\), which can help reduce expressions that might seem complex at first glance. Understanding how to manipulate the imaginary unit is essential for working with complex expressions.

Real and Imaginary Parts

Every complex number can be split into two distinct parts: the real part and the imaginary part. In the expression \(a + bi\), \(a\) represents the real part and \(b\) represents the imaginary part, which is always accompanied by the imaginary unit \(i\).

When simplifying complex numbers, it's crucial to treat these parts separately. For example, when adding, subtracting, or multiplying complex numbers, combine only the real parts with real parts and the imaginary parts with imaginary parts. This ensures accurate arithmetic and prevents confusion between the distinct components of complex numbers.