Simplify The Expression:$ \left(5 + 6i^{11}\right) + \left(8i^3 + I^5\right) + \left(i^2 - I^4\right) $
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Introduction
In mathematics, complex numbers are a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, geometry, and calculus. 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. In this article, we will simplify the expression (5 + 6i^11) + (8i^3 + i^5) + (i^2 - i^4) using the properties of complex numbers.
Understanding Complex Numbers
Complex numbers are of the form a + bi, where a and b are real numbers, and i is the imaginary unit. The imaginary unit i satisfies the equation i^2 = -1. This means that i can be expressed as the square root of -1, denoted by √(-1). Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but with some additional rules.
Simplifying the Expression
To simplify the expression (5 + 6i^11) + (8i^3 + i^5) + (i^2 - i^4), we need to evaluate each term separately. We will start by simplifying the terms involving powers of i.
Simplifying i^11
To simplify i^11, we can use the property of i that i^2 = -1. This means that i^4 = (i2)2 = (-1)^2 = 1. Therefore, i^11 = i^(4*2 + 3) = (i4)2 * i^3 = 1^2 * i^3 = i^3.
Simplifying i^3
To simplify i^3, we can use the property of i that i^2 = -1. This means that i^3 = i^2 * i = -1 * i = -i.
Simplifying i^5
To simplify i^5, we can use the property of i that i^2 = -1. This means that i^4 = (i2)2 = (-1)^2 = 1. Therefore, i^5 = i^4 * i = 1 * i = i.
Simplifying i^2 and i^4
To simplify i^2 and i^4, we can use the property of i that i^2 = -1. This means that i^4 = (i2)2 = (-1)^2 = 1.
Substituting the Simplified Terms
Now that we have simplified the terms involving powers of i, we can substitute them back into the original expression.
(5 + 6i^11) + (8i^3 + i^5) + (i^2 - i^4)
= (5 + 6i^3) + (8i^3 + i^5) + (i^2 - i^4)
= (5 + 6(-i)) + (8(-i) + i) + (-1 - 1)
= (5 - 6i) + (-8i + i) - 2
= (5 - 6i) - 7i - 2
= 5 - 13i - 2
= 3 - 13i
Conclusion
In this article, we simplified the expression (5 + 6i^11) + (8i^3 + i^5) + (i^2 - i^4) using the properties of complex numbers. We started by simplifying the terms involving powers of i, and then substituted them back into the original expression. The final simplified expression is 3 - 13i.
Frequently Asked Questions
Q: What is the imaginary unit i?
A: The imaginary unit i is a number that satisfies the equation i^2 = -1.
Q: How do you simplify powers of i?
A: To simplify powers of i, you can use the property of i that i^2 = -1. This means that i^4 = (i2)2 = (-1)^2 = 1.
Q: How do you add complex numbers?
A: To add complex numbers, you can add the real parts and the imaginary parts separately.
Q: How do you multiply complex numbers?
A: To multiply complex numbers, you can use the distributive property and the fact that i^2 = -1.
Further Reading
- Complex Numbers: A Comprehensive Guide
- Simplifying Complex Expressions
- Properties of Complex Numbers
References
- "Complex Numbers" by Math Open Reference
- "Simplifying Complex Expressions" by Khan Academy
- "Properties of Complex Numbers" by Wolfram MathWorld
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Introduction
Complex numbers are a fundamental concept in mathematics that plays a crucial role in various branches of mathematics, including algebra, geometry, and calculus. In this article, we will answer some of the most frequently asked questions about complex numbers.
Q&A
Q: What is a complex number?
A: 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 imaginary unit i?
A: The imaginary unit i is a number that satisfies the equation i^2 = -1. It is often represented by the letter i.
Q: How do you simplify powers of i?
A: To simplify powers of i, you can use the property of i that i^2 = -1. This means that i^4 = (i2)2 = (-1)^2 = 1. You can also use the following properties:
- i^3 = i^2 * i = -1 * i = -i
- i^5 = i^4 * i = 1 * i = i
- i^6 = i^4 * i^2 = 1 * (-1) = -1
- i^7 = i^4 * i^3 = 1 * (-i) = -i
- i^8 = i^4 * i^4 = 1 * 1 = 1
Q: How do you add complex numbers?
A: To add complex numbers, you can add the real parts and the imaginary parts separately. For example:
(3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i
Q: How do you subtract complex numbers?
A: To subtract complex numbers, you can subtract the real parts and the imaginary parts separately. For example:
(3 + 4i) - (2 + 5i) = (3 - 2) + (4i - 5i) = 1 - i
Q: How do you multiply complex numbers?
A: To multiply complex numbers, you can use the distributive property and the fact that i^2 = -1. For example:
(3 + 4i) * (2 + 5i) = (3 * 2) + (3 * 5i) + (4i * 2) + (4i * 5i) = 6 + 15i + 8i + 20i^2 = 6 + 23i + 20(-1) = -14 + 23i
Q: How do you divide complex numbers?
A: To divide complex numbers, you can multiply the numerator and denominator by the conjugate of the denominator. For example:
(3 + 4i) / (2 + 5i) = ((3 + 4i) * (2 - 5i)) / ((2 + 5i) * (2 - 5i)) = ((3 * 2) + (3 * -5i) + (4i * 2) + (4i * -5i)) / ((2 * 2) + (2 * -5i) + (5i * 2) + (5i * -5i)) = (6 - 15i + 8i - 20i^2) / (4 - 10i + 10i - 25i^2) = (6 - 7i + 20) / (4 + 25) = 26 - 7i / 29 = (26/29) - (7/29)i
Conclusion
In this article, we answered some of the most frequently asked questions about complex numbers. We covered topics such as simplifying powers of i, adding and subtracting complex numbers, multiplying and dividing complex numbers, and more.
Frequently Asked Questions
Q: What is the difference between a real number and a complex number?
A: A real number is a number that can be expressed in the form a, where a is a real 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.
Q: How do you represent complex numbers in the complex plane?
A: Complex numbers can be represented in the complex plane using the x-axis and the y-axis. The x-axis represents the real part of the complex number, and the y-axis represents the imaginary part.
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number is a complex number with the same real part and the opposite imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.
Further Reading
- Complex Numbers: A Comprehensive Guide
- Simplifying Complex Expressions
- Properties of Complex Numbers
References
- "Complex Numbers" by Math Open Reference
- "Simplifying Complex Expressions" by Khan Academy
- "Properties of Complex Numbers" by Wolfram MathWorld