Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The sum of two imaginary numbers is always an imaginary number.

Step-by-Step Solution

Answer

The statement 'The sum of two imaginary numbers is always an imaginary number' is false. The sum of two imaginary numbers is usually an imaginary number, but in the case where the real parts of the two imaginary numbers sum to zero, the sum is 0, which is a real number.

Step 1: Recall definition of imaginary numbers

An imaginary number is any real number multiplied by the imaginary unit \(i\), where \(i = \sqrt{-1}\). These numbers are usually represented as \(bi\), where \(b\) is a real number.

Step 2: Sum of two imaginary numbers

We need to add two general imaginary numbers, say \(a_1i\) and \(a_2i\), to see if the resultant number is always an imaginary number. So their sum will be \(a_1i + a_2i = (a_1 + a_2)i\).

Step 3: Evaluate the Sum

The result, \((a_1 + a_2)i\), can be seen as a real number, \((a_1 + a_2)\) (which could be any real number), multiplied by \(i\), so it is indeed an imaginary number, provided that \(a_1\) and \(a_2\) are not equal and opposite, i.e., their sum is not zero. If their sum is zero, the result would be \(0i\), which is not an imaginary number but a pure real number (0).

Key Concepts

Real Numbers

Real numbers are the numbers you might be most familiar with.
They include all the numbers you can find on the number line.
This includes:

• Whole numbers, like 0, 1, 2, 3...
• Rational numbers, such as fractions \( \frac{1}{2} \) or \( -\frac{3}{4} \)
• Irrational numbers, like \( \sqrt{2} \) and \( \pi \)
• Negative numbers, such as -1, -2, -3...

So, in simple terms, a real number can be positive, negative, or zero, and it can also have fractions or decimals.
In mathematics, this concept is very broad because it forms the foundation of other number systems, such as imaginary numbers.

Sum of Imaginary Numbers

When adding two imaginary numbers, let's say \(a_1i\) and \(a_2i\), you add the coefficients \(a_1\) and \(a_2\), and then multiply the sum by the imaginary unit \(i\).
This operation gives us the result \((a_1 + a_2)i\).
Breaking it down:

• Each imaginary number has a form \(bi\)
• The imaginary unit \(i\) is constant, simplifying the addition

Now, observe that if \(a_1 + a_2\) is non-zero, the sum \((a_1 + a_2)i\) is also an imaginary number.
However, if \(a_1 + a_2 = 0\), then the sum becomes \(0i\), which equals zero.
In this case, the sum is not an imaginary number, but actually a real number.

Properties of Imaginary Numbers

Imaginary numbers may seem a bit tricky, but they have identifiable properties that make working with them straightforward.
Here are some key characteristics:

• Imaginary Unit: The core of imaginary numbers is the unit \(i\), with \(i = \sqrt{-1}\).
• Multiplicative Inversion: Multiplying \(i\) by itself gives \(-1\), since \(i^2 = -1\).
• Purely Imaginary: They don't have a real component, although sometimes they result in a real number, as shown when their sum equals zero.

Knowing these properties helps in understanding more complex systems, such as complex numbers which combine real and imaginary parts.
This foundational knowledge also assists in solving equations that would otherwise have no solution in the real number system, hence broadening your mathematical toolkit.