Question

Write the conjugate of each complex number. $$2-3 i$$

Short answer

The conjugate of the complex number \(2 - 3i\) is \(2 + 3i\).

Step-by-step solution

Identify the Complex Number

The given complex number is in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part. In this case, the complex number is 2 - 3i, where the real part is 2 and the imaginary part is -3i.

Understanding the Conjugate

The conjugate of a complex number is formed by changing the sign of the imaginary part. This means that for a complex number a + bi, the conjugate is a - bi.

Writing the Conjugate

Change the sign of the imaginary part of the given complex number to find its conjugate. Therefore, the conjugate of 2 - 3i is 2 + 3i.

Key concepts

Complex Numbers

Complex numbers are an expansion of traditional number systems, which allows for solutions to certain equations that don’t have solutions within the realm of real numbers alone. A complex number is composed of two parts: a real part and an imaginary part, often expressed in the form \( a + bi \). Here, \( a \) is the real part, and \( bi \) is the imaginary part, where \( i \) is the imaginary unit satisfying \( i^2 = -1 \).

Complex numbers behave quite similarly to real numbers in terms of operations like addition, subtraction, and multiplication, but their graphical representation and properties are unique. For example, on the complex plane, the horizontal axis represents the real part, while the vertical axis represents the imaginary part. This allows us to visualize complex numbers as points or vectors in this two-dimensional space. Because of their deep mathematical properties, complex numbers are incredibly useful in multiple fields including engineering, physics, and computer science.

Imaginary Numbers

Imaginary numbers arise when we seek to take the square root of a negative number, an operation that doesn't have a solution within the set of real numbers. By defining the imaginary unit \( i \) as the square root of -1, we can solve equations that previously seemed unsolvable. Any number that can be expressed in the form \( bi \), where \( b \) is a real number, is considered an imaginary number.

For example, the square roots of -9 are \( \pm3i \). It’s the presence of the imaginary unit \( i \) that distinguishes imaginary numbers from real numbers. Imaginary numbers are not

Algebraic Operations on Complex Numbers

Algebraic operations on complex numbers include addition, subtraction, multiplication, and division, and they follow specific rules that preserve the structure of complex numbers. When we add or subtract complex numbers, we simply combine their real parts and their imaginary parts separately. For instance, adding \( 2 + 3i \) and \( 4 - 5i \) would yield \( (2+4) + (3i-5i) = 6 - 2i \).

Multiplication involves utilizing the distributive property and remembering that \( i^2 = -1 \). Division can be more intricate since it entails multiplying by the conjugate of the denominator to eliminate the imaginary part from the denominator. An important aspect to remember is that the conjugate of a complex number has the effect of 'mirroring' the number across the real axis on the complex plane, which is useful in simplifying the division of complex numbers.