Drag Each Expression To The Correct Location On The Table.Determine Which Expressions Represent Purely Real Numbers And Which Expressions Represent Non-real Complex Numbers.Expressions:- $2-7i^2$- $-i^2+i^3$- $7-5i$-
Introduction
In mathematics, complex numbers are a fundamental concept that plays a crucial role in various branches of mathematics and science. 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 explore the concept of purely real and non-real complex numbers and determine which expressions represent purely real numbers and which expressions represent non-real complex numbers.
What are Purely Real Numbers?
Purely real numbers are numbers that can be expressed without any imaginary part. In other words, a number is purely real if it can be written in the form a, where a is a real number and there is no imaginary part. Examples of purely real numbers include 3, -2, and 0.5.
What are Non-Real Complex Numbers?
Non-real complex numbers, on the other hand, are numbers that have an imaginary part. In other words, a number is non-real complex if it can be written in the form a + bi, where a and b are real numbers and b is not equal to zero. Examples of non-real complex numbers include 3 + 4i, -2 - 3i, and 0.5 + 0.2i.
Evaluating the Expressions
Now that we have a good understanding of purely real and non-real complex numbers, let's evaluate the given expressions and determine which ones represent purely real numbers and which ones represent non-real complex numbers.
Expression 1:
To evaluate this expression, we need to simplify the term i^2. Since i^2 = -1, we can substitute this value into the expression:
This expression represents a purely real number.
Expression 2:
To evaluate this expression, we need to simplify the terms i^2 and i^3. Since i^2 = -1 and i^3 = i^2 * i = -1 * i = -i, we can substitute these values into the expression:
This expression represents a non-real complex number.
Expression 3:
This expression already has a real part (7) and an imaginary part (-5i). Therefore, it represents a non-real complex number.
Conclusion
In conclusion, we have evaluated three expressions and determined which ones represent purely real numbers and which ones represent non-real complex numbers. Expression 1 represents a purely real number, while expressions 2 and 3 represent non-real complex numbers. Understanding the difference between purely real and non-real complex numbers is essential in various branches of mathematics and science.
Key Takeaways
- Purely real numbers are numbers that can be expressed without any imaginary part.
- Non-real complex numbers are numbers that have an imaginary part.
- To determine whether an expression represents a purely real or non-real complex number, we need to simplify the expression and check if it has an imaginary part.
Further Reading
For further reading on complex numbers, we recommend the following resources:
- Khan Academy: Complex Numbers
- MIT OpenCourseWare: Complex Analysis
- Wolfram MathWorld: Complex Numbers
References
- Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
- Lang, S. (1999). Complex Analysis. Springer-Verlag.
- Ahlfors, L. V. (1979). Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill.
Frequently Asked Questions (FAQs) about Purely Real and Non-Real Complex Numbers ====================================================================================
Q: What is the difference between a purely real number and a non-real complex number?
A: A purely real number is a number that can be expressed without any imaginary part, while a non-real complex number is a number that has an imaginary part.
Q: How do I determine whether an expression represents a purely real or non-real complex number?
A: To determine whether an expression represents a purely real or non-real complex number, you need to simplify the expression and check if it has an imaginary part. If the expression can be written in the form a, where a is a real number, then it represents a purely real number. If the expression can be written in the form a + bi, where a and b are real numbers and b is not equal to zero, then it represents a non-real complex number.
Q: What is the imaginary unit i?
A: The imaginary unit i is a mathematical concept that satisfies the equation i^2 = -1. It is used to represent the square root of -1 and is a fundamental component of complex numbers.
Q: Can a complex number have a zero imaginary part?
A: Yes, a complex number can have a zero imaginary part. In this case, the complex number is equivalent to a purely real number.
Q: Can a purely real number be expressed as a complex number?
A: Yes, a purely real number can be expressed as a complex number with a zero imaginary part. For example, the purely real number 3 can be expressed as the complex number 3 + 0i.
Q: What are some examples of purely real numbers?
A: Some examples of purely real numbers include 3, -2, 0.5, and 1.
Q: What are some examples of non-real complex numbers?
A: Some examples of non-real complex numbers include 3 + 4i, -2 - 3i, and 0.5 + 0.2i.
Q: Can a non-real complex number have a zero real part?
A: Yes, a non-real complex number can have a zero real part. In this case, the complex number is equivalent to the imaginary unit i.
Q: Can a complex number have a negative imaginary part?
A: Yes, a complex number can have a negative imaginary part. For example, the complex number 3 - 4i has a negative imaginary part.
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.
Q: Why are complex numbers important in mathematics and science?
A: Complex numbers are important in mathematics and science because they provide a powerful tool for solving equations and representing periodic phenomena. They are used in a wide range of applications, including electrical engineering, signal processing, and quantum mechanics.
Q: Can complex numbers be used to represent physical quantities?
A: Yes, complex numbers can be used to represent physical quantities such as voltage, current, and impedance in electrical engineering. They can also be used to represent quantities such as frequency and amplitude in signal processing.
Q: What are some common applications of complex numbers?
A: Some common applications of complex numbers include:
- Electrical engineering: Complex numbers are used to represent voltage, current, and impedance in electrical circuits.
- Signal processing: Complex numbers are used to represent frequency and amplitude in signal processing.
- Quantum mechanics: Complex numbers are used to represent wave functions and operators in quantum mechanics.
- Control theory: Complex numbers are used to represent transfer functions and stability in control theory.
Q: Can complex numbers be used to solve equations?
A: Yes, complex numbers can be used to solve equations. In fact, complex numbers provide a powerful tool for solving equations that cannot be solved using real numbers alone.
Q: What are some common techniques for solving equations using complex numbers?
A: Some common techniques for solving equations using complex numbers include:
- Substitution: Substituting complex numbers into an equation to solve for the unknown.
- Elimination: Eliminating complex numbers from an equation to solve for the unknown.
- Quadratic formula: Using the quadratic formula to solve quadratic equations with complex coefficients.
Q: Can complex numbers be used to represent periodic phenomena?
A: Yes, complex numbers can be used to represent periodic phenomena such as sound waves, light waves, and electrical signals.
Q: What are some common applications of complex numbers in representing periodic phenomena?
A: Some common applications of complex numbers in representing periodic phenomena include:
- Signal processing: Complex numbers are used to represent frequency and amplitude in signal processing.
- Electrical engineering: Complex numbers are used to represent voltage, current, and impedance in electrical circuits.
- Acoustics: Complex numbers are used to represent sound waves and their properties.
- Optics: Complex numbers are used to represent light waves and their properties.