Problem 15 - Elementary and Intermediate Algebra

Question

Complete each solution. Determine whether each statement is true or false. a. \(\sqrt{6} i=i \sqrt{6}\) b. \(\sqrt{8} i=\sqrt{8 i}\) c. \(\sqrt{-25}=-\sqrt{25}\) d. \(-i=i\)

Step-by-Step Solution

Answer

a. True; b. False; c. False; d. False.

Step 1: Analyze Statement (a)

Consider the equation \( \sqrt{6} i = i \sqrt{6} \). Both sides of the equation involve multiplying \( i \) with \( \sqrt{6} \). The commutative property of multiplication ensures that \( a \cdot b = b \cdot a \). Therefore, this statement is true as both sides are equivalent.

Step 2: Analyze Statement (b)

Consider the equation \( \sqrt{8} i = \sqrt{8i} \). The left side represents the imaginary number \( \sqrt{8} \cdot i \). The right side suggests taking the square root of the product \( 8i \). According to the rules of square roots, \( \sqrt{ab} eq \sqrt{a} \cdot \sqrt{b} \) for negative \( b \). Hence, this statement is false.

Step 3: Analyze Statement (c)

Consider the equation \( \sqrt{-25} = -\sqrt{25} \). The left side represents an imaginary number \( 5i \) because \( \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \). The right side simplifies to \(-5\), a real number. Since \( 5i eq -5 \), this statement is false.

Step 4: Analyze Statement (d)

Consider the equation \( -i = i \). \(-i\) and \(i\) are imaginary numbers with identical magnitudes but different signs. Since they represent different numbers in the complex plane, this equation is false.

Key Concepts

Imaginary Unit

In mathematics, particularly within the realm of complex numbers, the imaginary unit is a fundamental concept. It's represented by the symbol \(i\) and defined by the property that \(i^2 = -1\). This means that \(i\) is the square root of \(-1\), a concept that cannot be understood through the context of real numbers. Here's why it's essential:

• It allows for the expression and manipulation of complex numbers, which take the form \(a + bi\), where \(a\) and \(b\) are real numbers.


• The imaginary unit helps in solving equations that do not have solutions within the real number system, such as \(x^2 + 1 = 0\).


• It is crucial in various fields like engineering, physics, and computer science, where complex numbers are frequently applied.

American mathematician Benjamin Peirce once humorously said about the imaginary unit that "it is an idea which, when perceived, will produce great delight." Embrace the idea of \(i\) to widen your mathematical horizons.

Square Roots

Square roots are central in understanding complex numbers and many algebraic principles. The square root of a number \(x\) is such that when squared, it results in \(x\). Written as \(\sqrt{x}\), it poses unique challenges when dealing with negative numbers.

• If \(x\) is positive, \(\sqrt{x}\) represents the positive root.


• For negative numbers, we cannot determine a real number root under the conventional system, hence the introduction of the imaginary unit \(i\). For example, \(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i\).


• It's important to note that \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\) only holds true when both \(a\) and \(b\) are positive or zero.

Understanding square roots is essential for further exploration of both real and complex numbers. It teaches us how to extend meanings from the real number system into a broader context, leading to a deeper grasp of mathematics as a whole.

Commutative Property of Multiplication

The commutative property of multiplication is a fundamental property of arithmetic, stating that the order in which two numbers are multiplied does not affect the product. In formal terms, this property asserts that for any two numbers \(a\) and \(b\), the equation \(a \cdot b = b \cdot a\) holds.

• This property applies to all real and complex numbers, allowing flexibilities in computation and algebraic manipulation.


• It simultaneously simplifies calculations and assures that rearranging terms won't affect the end result.


• In complex numbers, this concept remains valid, such as in \(\sqrt{6}i = i\sqrt{6}\), showcasing the versatility of the property.

The commutative property is a cornerstone that lays the foundation for more complex mathematical operations and proofs. It highlights mathematics' inherent ability to work consistently across different structures and systems. Understanding it well prepares students for tackling higher-level math challenges seamlessly.