Problem 53 - College Algebra

Question

Which one of the following is true? a. Some irrational numbers are not complex numbers. b. \((3+7 i)(3-7 i)\) is an imaginary number. c. \(\frac{7+3 i}{5+3 i}=\frac{7}{5}\) d. In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)

Step-by-Step Solution

Answer

d. In the complex number system, \(x^{2} + y^{2}\) (the sum of two squares) can be factored as \((x+yi)(x-yi)\)

Step 1: Evaluate statement a

In mathematics, any number, whether it be real, imaginary or irrational, is also considered a complex number. Therefore, it is false to state that some irrational numbers are not complex numbers.

Step 2: Evaluate statement b

To verify this, you must multiply the two complex numbers together using the distributive law or FOIL (First, Outside, Inner, Last). After simplifying the expression, you obtain \(3*(3) + 3*(-7i) + 7i*(3) + 7i*(-7i)\), which simplifies into \(9-21i+21i-49\). After adding, you get \(-40\), which is a real number, not an imaginary one, so this statement is false.

Step 3: Evaluate statement c

Here, you have a complex number divided by another complex number. To simplify this, you should multiply the numerator and the denominator by the conjugate of the denominator. After simplifying the fractions, you get the result \( \frac{23}{34} + \frac{9}{34}i \), which is not equal to \(\frac{7}{5}\), making this statement false.

Step 4: Evaluate statement d

To verify this statement, you should distribute \(x+y i\) and \(x-y i\) using the distributive law or FOIL method. In doing this, you get \(x*x + x*(-yi) + yi*x + yi*(-yi)\), which simplifies into \(x^{2} + y^{2}\), confirming that this statement is true.

Key Concepts

Irrational Numbers

To start off, let's dive into irrational numbers. These are numbers that cannot be expressed as a simple fraction of two integers. A classic example is the square root of 2, which cannot be exactly written as a fraction. Irrational numbers have decimal expansions that go on forever without repeating.

• Examples include: \(\pi\), \(e\), and \(\sqrt{3}\).
• They are a part of the real number system but not rational (where rational are numbers that can be expressed as a fraction).

Despite their peculiar nature, irrational numbers are still considered complex numbers. This is because a complex number is any number that can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Thus, even irrational numbers fit into the broader family of complex numbers.

Imaginary Numbers

Imaginary numbers come from the necessity of having numbers that square to a negative result. Regular real numbers squared give us positive results; however, numbers like \(i\), the imaginary unit defined as \(\sqrt{-1}\), allow for a new dimension of numbers.

• The imaginary unit \(i\) satisfies the equation \(i^2 = -1\).
• Complex numbers combine real and imaginary parts, such as \(3 + 4i\).

This means that although often linked to complex numbers, imaginary numbers themselves are crucial for expressions in the form \(a + bi\), providing the "imaginary part". They allow us to solve equations that would otherwise have no solution in the real number system.

Distributive Law

The distributive law is a handy algebraic tool that helps simplify expressions, especially when dealing with complex polynomials. It states that for any three numbers \(a, b,\) and \(c\), the equation \(a(b+c) = ab + ac\) holds true.When applied to the product of two binomials, such as in the expression \((3+7i)(3-7i)\), this law helps us to distribute each term:

• First: \(3 \times 3\)
• Outside: \(3 \times (-7i)\)
• Inside: \(7i \times 3\)
• Last: \(7i \times (-7i)\)

Using distribution, we simplify expressions and reach real results, even starting with complex components. This is demonstrated in the simplification of the result \(-40\).

Factoring

Factoring is a method used to express a polynomial as a product of its roots, often used to simplify complex mathematical expressions. In the context of complex numbers, factoring involves finding components that, when multiplied, provide an original polynomial form.For example, in the expression \((x+yi)(x-yi)\), when this is expanded, using either the distributive law or the FOIL method, it results in \(x^2 + y^2\).

• This demonstrates how the sum of two squares can be factored neatly into the product of complex conjugates.

Moreover, understanding factoring is crucial for simplifying expressions and solving equations in mathematics. It's a fundamental skill, particularly when analyzing and solving quadratic expressions within the complex number system.