Subtract $(5-2i)-(1+8i)$.Which Property Allows You To Write The Expression As $5-2i-1-8i$?A. AssociativeB. CommutativeC. DistributiveD. Identity
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and engineering. When working with complex numbers, it's essential to understand the properties that govern their operations. In this article, we will explore the property that allows us to write the expression $(5-2i)-(1+8i)$ as $5-2i-1-8i$.
Understanding Complex Numbers
A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2=-1$. The real part of a complex number is $a$, and the imaginary part is $b$.
Subtracting Complex Numbers
When subtracting complex numbers, we need to subtract the real parts and the imaginary parts separately. This is where the property comes in that allows us to write the expression $(5-2i)-(1+8i)$ as $5-2i-1-8i$.
The Commutative Property
The commutative property of addition states that the order of the numbers being added does not change the result. In other words, $a+b=b+a$. This property also applies to subtraction, which means that $a-b=b-a$.
Applying the Commutative Property
Now, let's apply the commutative property to the expression $(5-2i)-(1+8i)$. We can rewrite it as $(5-2i)+(-1-8i)$, since the order of the numbers being subtracted does not change the result.
Distributing the Negative Sign
When we distribute the negative sign to the terms inside the parentheses, we get $5-2i-1-8i$. This is the final result of the subtraction.
Conclusion
In conclusion, the property that allows us to write the expression $(5-2i)-(1+8i)$ as $5-2i-1-8i$ is the commutative property of subtraction. This property states that the order of the numbers being subtracted does not change the result. Understanding this property is essential when working with complex numbers and performing operations on them.
Properties of Complex Numbers
There are several properties of complex numbers that are essential to understand when working with them. These properties include:
- Commutative Property of Addition: The order of the numbers being added does not change the result.
- Commutative Property of Subtraction: The order of the numbers being subtracted does not change the result.
- Associative Property of Addition: The order in which we add complex numbers does not change the result.
- Distributive Property: We can distribute a complex number to the terms inside the parentheses.
- Identity Property: The number 0 is the additive identity, and the number 1 is the multiplicative identity.
Examples of Complex Number Operations
Here are some examples of complex number operations that demonstrate the properties we discussed earlier:
- Addition: $(3+4i)+(2+5i)=(3+2)+(4i+5i)=5+9i$
- Subtraction: $(5-2i)-(1+8i)=(5-1)+(-2i-8i)=4-10i$
- Multiplication: $(3+4i)(2+5i)=(3\cdot2+3\cdot5i+4i\cdot2+4i\cdot5i)=6+15i+8i+20i^2=6+23i-20= -14+23i$
Real-World Applications of Complex Numbers
Complex numbers have numerous real-world applications in various fields, including:
- Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
- Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
- Navigation: Complex numbers are used to represent GPS coordinates and navigate through complex terrain.
- Computer Graphics: Complex numbers are used to represent 3D objects and perform transformations on them.
Conclusion
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. In our previous article, we explored the properties of complex numbers and how to perform operations on them. In this article, we will answer some frequently asked questions about complex numbers.
Q: What is a complex number?
A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2=-1$.
Q: What is the difference between a real number and a complex number?
A real number is a number that can be expressed without any imaginary part, whereas a complex number has both real and imaginary parts.
Q: How do I add complex numbers?
To add complex numbers, we need to add the real parts and the imaginary parts separately. For example, $(3+4i)+(2+5i)=(3+2)+(4i+5i)=5+9i$.
Q: How do I subtract complex numbers?
To subtract complex numbers, we need to subtract the real parts and the imaginary parts separately. For example, $(5-2i)-(1+8i)=(5-1)+(-2i-8i)=4-10i$.
Q: How do I multiply complex numbers?
To multiply complex numbers, we need to multiply the real parts and the imaginary parts separately and then combine the results. For example, $(3+4i)(2+5i)=(3\cdot2+3\cdot5i+4i\cdot2+4i\cdot5i)=6+15i+8i+20i^2=6+23i-20=-14+23i$.
Q: What is the conjugate of a complex number?
The conjugate of a complex number $a+bi$ is $a-bi$.
Q: How do I find the conjugate of a complex number?
To find the conjugate of a complex number, we need to change the sign of the imaginary part. For example, the conjugate of $3+4i$ is $3-4i$.
Q: What is the modulus of a complex number?
The modulus of a complex number $a+bi$ is $\sqrt{a2+b2}$.
Q: How do I find the modulus of a complex number?
To find the modulus of a complex number, we need to calculate the square root of the sum of the squares of the real and imaginary parts. For example, the modulus of $3+4i$ is $\sqrt{32+42}=\sqrt{9+16}=\sqrt{25}=5$.
Q: What is the argument of a complex number?
The argument of a complex number $a+bi$ is the angle between the positive real axis and the line segment connecting the origin to the point $(a,b)$ in the complex plane.
Q: How do I find the argument of a complex number?
To find the argument of a complex number, we need to use the inverse tangent function. For example, the argument of $3+4i$ is $\tan^{-1}\left(\frac{4}{3}\right)$.
Conclusion
In conclusion, complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the properties and operations of complex numbers, we can solve problems and analyze complex systems. We hope this Q&A guide has been helpful in answering your questions about complex numbers.