Question

Determine whether each statement is true or false. If it is true, give an example. If it is false, give a counterexample. a. The sum of two imaginary numbers is an imaginary number. b. The product of two pure imaginary numbers is a real number. c. A pure imaginary number is an imaginary number. d. A complex number is a real number.

Short answer

Statement a is true with example being \(3i + 4i = 7i\). Statement b is true with example \(3i * 9i = -27\). Statement c is true as a pure imaginary number has a real part equal to 0. Statement d is false as complex numbers can have a non-zero imaginary part, which real numbers can't have. Counterexample can be \(2+3i\).

Step-by-step solution

Understanding the Concepts

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 that satisfies \(i^2 = -1\). A real number is a value of a continuous quantity that can represent a distance along a line. An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit 'i', where \(i^2 = -1\). A pure imaginary number has a real part equal to 0 and is often expressed in the form \(0+bi\) or just \(bi\) where \(b ≠ 0\).

Statement Verification - Part a

'The sum of two imaginary numbers is an imaginary number' is a true statement. An example: let's take two imaginary numbers \(3i\) and \(4i\). Their sum is \(7i\) which is an imaginary number.

Statement Verification - Part b

'The product of two pure imaginary numbers is a real number' is also a true statement. An example can be the product of \(3i\) and \(9i\) which equals \(-27\). Because \(i^2 = -1\), thus the product of any two pure imaginary numbers (\(bi\) and \(di\)) is a real number.

Statement Verification - Part c

'A pure imaginary number is an imaginary number'. This is true as well because a pure imaginary number is just an imaginary number with its real part equal to 0.

Statement Verification - Part d

'A complex number is a real number' is a false statement. A counterexample can be any complex number with a non-zero imaginary part. For instance, the complex number \(2+3i\) is not a real number because it has an imaginary component (3i). A complex number can be a real number only if its imaginary part equals 0, but as a category, complex numbers encompass more than just real numbers.

Key concepts

Imaginary Numbers

Imaginary numbers are a key concept in understanding complex numbers. These numbers extend our number system beyond real numbers. An imaginary number is made by multiplying a real number with the imaginary unit 'i', which is defined as the square root of -1. Therefore, it satisfies the equation \(i^2 = -1\). Any expression involving imaginary numbers takes the form \(bi\), where \(b\) is a real number. For example, \(3i\) and \(-5i\) are imaginary numbers.
The imaginary unit 'i' allows for computations and expression of numbers that are otherwise not possible in the realm of real numbers. Imaginary numbers are used extensively in fields like engineering and physics, where they help with understanding oscillations, waves, and other phenomena.

Real Numbers

Real numbers are the basis of our everyday number system. They include rational numbers, such as integers, and irrational numbers, like the square root of a non-perfect square. Real numbers represent quantities along a continuous line, and cover everything from negative to positive infinity, without any 'gaps' in between.
In the context of complex numbers, real numbers are simply those complex numbers that do not have an imaginary part. For instance, the number 5 is a real number because it can be written as \(5 + 0i\). Consequently, every real number is also a complex number with an imaginary component of zero. This connection between real and complex numbers is crucial for understanding the broader number system.

Pure Imaginary Numbers

Pure imaginary numbers represent a special subset of imaginary numbers where the real part is zero. This means they take the form \(0 + bi\) or simply \(bi\), with \(beq 0\). Unlike other complex numbers, pure imaginary numbers lie completely on the imaginary axis of the complex plane, where the real line intersects at zero.
As an example, \(7i\) is a pure imaginary number because its real part is zero. A key property of pure imaginary numbers is that their product results in a real number. For instance, when we multiply \(3i\) and \(9i\), we calculate \(27\times i^2 = 27\times (-1) = -27\), which is a real number. This interaction between imaginary and real components is what makes complex numbers so versatile.

Complex Number Operations

Working with complex numbers involves several operations, much like those with real numbers. Complex numbers are represented as \(a + bi\), where 'a' is the real part and 'bi' is the imaginary part. These parts allow us to perform addition, subtraction, multiplication, and division with complex numbers.
For addition or subtraction, we simply combine the real parts and the imaginary parts separately. For example, when adding \(2 + 3i\) and \(3 + 4i\), the result is \((2+3) + (3i+4i) = 5 + 7i\).

• Multiplication: Involves distributing each term, such as multiplying \((2 + 3i)\) with \( (4 + 5i)\). Expand and simplify to find \(8 + 10i + 12i + 15i^2\), giving a result of \(-7 + 22i\) since \(i^2\) equals \(-1\).
• Division: To divide by a complex number, multiply both the numerator and the denominator by the conjugate of the divisor, enabling the simplification using the property \(i^2 = -1\).

Complex number operations underlie many applications in science and engineering, where analyses rely on capturing and processing complex, multifaceted data.