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Chapter 1: Problem 5

True or False The conjugate of \(2+5 i\) is \(-2-5 i\).

Short Answer

Expert verified
False

Step by step solution

01

Understanding the Conjugate

The conjugate of a complex number is formed by changing the sign of the imaginary part.
02

Identifying the Complex Number

The given complex number is:a = 2 + 5i.
03

Forming the Conjugate

To find the conjugate of this complex number, change the sign of the imaginary part:Conjugate of a = 2 - 5i.
04

Comparing with the Given Conjugate

The provided potential conjugate is -2 - 5i. Comparing this with the correct conjugate (2 - 5i), we see they are not the same.
05

Conclusion

The statement 'The conjugate of 2 + 5i is -2 - 5 i' is False.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conjugate of a Complex Number
The concept of a conjugate is essential when dealing with complex numbers. A complex number is expressed as: \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The conjugate of a complex number \(a + bi\) is formed by changing the sign of its imaginary part. This means the conjugate is given by \(a - bi\). Understanding this is crucial for many operations involving complex numbers, such as division, simplifying expressions, and solving equations.
Imaginary Part
The imaginary part of a complex number is the term that contains the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). In a complex number expressed as \(a + bi\), the term \(bi\) represents the imaginary part. Here are some key points about the imaginary part:
  • It is used to represent numbers that extend beyond the real number line.
  • Changing the sign of the imaginary part helps in forming the conjugate.
For instance, in \(2 + 5i\), the imaginary part is \(+5i\). Understanding the imaginary part helps in grasping the full nature of complex numbers.
True or False Statements
True or false statements are a fundamental part of learning and assessing understanding. In this exercise, examining the statement: 'The conjugate of \(2 + 5i\) is \(-2 - 5i\)'. To solve this, follow these steps:
  • Find the conjugate of \(2 + 5i\). As established, it is \(2 - 5i\).
  • Compare this with the given potential conjugate \(-2 - 5i\).
Since \(-2 - 5i\) does not match \(2 - 5i\), the statement is false. Using logical reasoning to confirm or refute such statements is crucial in mathematics.
Algebraic Expressions
Complex numbers often appear within algebraic expressions. Understanding how to manipulate and simplify these expressions is essential. An algebraic expression involving complex numbers may require you to:
  • Simplify by combining like terms.
  • Form and use conjugates to simplify or solve equations.
  • Apply operations such as addition, subtraction, multiplication, and division correctly.
For example, given the expression \((a + bi) + (c + di)\), you add the real parts and the imaginary parts separately: \((a + c) + (b + d)i\). Recognizing and simplifying algebraic expressions involving complex numbers is a valuable skill in both pure and applied mathematics.

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Most popular questions from this chapter

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