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Real numbers are a fundamental part of mathematics, and they encompass all the numbers we are typically used to. This includes both positive and negative integers, fractions, and decimal numbers. What distinguishes real numbers is that they can be found on the number line. This means any number from infinitely negative to infinitely positive that you can think of, including zero, is a real number.

• Examples: -5, 0, 4.8, 3/2, etc.
• They can be positive or negative, whole numbers, fractions, or decimals.

The real numbers play a crucial role in operations with complex numbers. When multiplying a real number by a complex number, you're essentially scaling the complex number. This affects both the real component and the imaginary component of the complex number.

The distributive property is one of the most useful properties in mathematics, especially when dealing with expressions involving both real and complex numbers. It allows us to distribute or "spread out" a multiplication operation over an addition or subtraction inside parentheses.
For example, in the expression \(3(8 - 5i)\), the distributive property lets us multiply the 3 by each term inside the parentheses independently:

• First, multiply 3 by 8 to get 24.
• Then, multiply 3 by \(-5i\) to get \(-15i\).

By applying the distributive property, we make the calculation more manageable and systematic. This process results in a simpler expression \(24 - 15i\), helping us navigate complex algebraic expressions more efficiently.

Complex numbers consist of a real part and an imaginary part, generally expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. When we multiply complex numbers, we need to employ various arithmetic properties, including the distributive property, to combine these components correctly.
When multiplying a real number by a complex number, such as \(3(8 - 5i)\), we distribute the real number to both the real and imaginary parts of the complex number. It's important to follow through each multiplication:

• Multiply the real number by the real part, \(3 \times 8 = 24\).
• Multiply the real number by the imaginary part, \(3 \times -5i = -15i\).

These operations give us the new complex number formed, \(24 - 15i\). Understanding this multiplication helps in simplifying complex expressions and illustrates the foundational principles of working with complex numbers.