Complex Numbers and the Multiplicative Identity Property

Oct 13, 2025 by miodeya.com 122 views

The multiplicative identity property is a fundamental concept in mathematics that states that when a number is multiplied by 1, the result is the number itself. In the context of complex numbers, this property is essential for simplifying expressions and solving equations. In this article, we will explore the multiplicative identity property and determine which equation demonstrates this concept.

What is the Multiplicative Identity Property?

The multiplicative identity property is a mathematical concept that states that when a number is multiplied by 1, the result is the number itself. In other words, any number multiplied by 1 remains unchanged. This property is denoted by the equation:

a × 1 = a

where 'a' is any number.

Complex Numbers and the Multiplicative Identity Property

Complex numbers are numbers that have both real and imaginary parts. They are denoted by the equation:

a + bi

where 'a' is the real part and 'b' is the imaginary part.

In the context of complex numbers, the multiplicative identity property can be expressed as:

(a + bi) × 1 = a + bi

Analyzing the Options

Now that we have a clear understanding of the multiplicative identity property, let's analyze the options provided:

A. ${$ (-3+5i)+0=-3+5i$}$

This equation demonstrates the additive identity property, not the multiplicative identity property. The additive identity property states that when a number is added to 0, the result is the number itself.

B. ${$ (-3+5i)(1)=-3+5i$}$

This equation demonstrates the multiplicative identity property. When the complex number (-3 + 5i) is multiplied by 1, the result is the complex number itself.

C. ${$ (-3+5i)(-3+5i)=-16-30i$}$

This equation demonstrates the multiplication of two complex numbers, but it does not demonstrate the multiplicative identity property.

D. ${$ (-3+5i)(3-5i)=16+30i$}$

This equation demonstrates the multiplication of two complex numbers, but it does not demonstrate the multiplicative identity property.

Conclusion

In conclusion, the equation that demonstrates the multiplicative identity property is:

B. ${$ (-3+5i)(1)=-3+5i$}$

This equation shows that when the complex number (-3 + 5i) is multiplied by 1, the result is the complex number itself, which is a clear demonstration of the multiplicative identity property.

Real-World Applications

The multiplicative identity property has numerous real-world applications in mathematics, science, and engineering. Some examples include:

Signal Processing: The multiplicative identity property is used in signal processing to simplify expressions and solve equations.
Control Systems: The multiplicative identity property is used in control systems to analyze and design control systems.
Electrical Engineering: The multiplicative identity property is used in electrical engineering to analyze and design electrical circuits.

Final Thoughts

In conclusion, the multiplicative identity property is a fundamental concept in mathematics that states that when a number is multiplied by 1, the result is the number itself. The equation that demonstrates this property is:

B. ${$ (-3+5i)(1)=-3+5i$}$

This equation shows that when the complex number (-3 + 5i) is multiplied by 1, the result is the complex number itself, which is a clear demonstration of the multiplicative identity property. The multiplicative identity property has numerous real-world applications in mathematics, science, and engineering, and is an essential concept to understand in complex analysis.