Problem 89 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \(3(5+3 i)\)

Step-by-Step Solution

Answer

The simplified expression in standard form is \(15 + 9i\).

Step 1: Distribute the Coefficient

To simplify the expression \(3(5+3i)\), start by distributing the \(3\) to both terms inside the parentheses. Multiply \(3\) by \(5\) and \(3\) by \(3i\). Write the result of these multiplications.

Step 2: Perform the Multiplications

Calculate the individual multiplications: \(3 \times 5 = 15\) and \(3 \times 3i = 9i\). These calculations give us the terms to be summed in the next step.

Step 3: Combine the Results

Add the results from the previous step together to form the expression \(15 + 9i\). This combines the real and imaginary parts resulting from the distribution.

Step 4: Write in Standard Form

The expression \(15 + 9i\) is already in the standard form \(a + bi\). Ensure that the expression is written as the sum of a real number and an imaginary number, which confirms it's in the correct form.

Key Concepts

Simplifying Expressions

When simplifying expressions, especially involving complex numbers, one of the key steps is to distribute and combine like terms. Let's break it down using our example.First, the expression you're simplifying often contains brackets or parentheses. These suggest that each term inside should be affected by what's outside – usually a number or a variable. To "distribute" means to take this number or variable and multiply it with each term inside the brackets. For example, in the expression \(3(5 + 3i)\), the number 3 is outside the parentheses. You need to multiply 3 by both 5 and 3i, individually. This step ensures that every term inside the parentheses is updated according to multiplication, making it ready for the next stage.After distribution, you'll often get terms that are similar – like all the numbers without the imaginary unit \(i\), or terms that all have \(i\). These can and should be combined to simplify the expression further.

• Distribute: Multiply the factor outside the parentheses with each term inside.
• Combine: Add together similar terms to simplify the expression.

Breaking down the expression into its simplest form is essential as it makes solving or using these expressions in further calculations much easier.

Standard Form

In terms of complex numbers, the standard form is a way to express every complex number in the simplest and most recognizable manner. It is represented as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined as \(\sqrt{-1}\).This format helps when comparing, adding, or multiplying complex numbers because it clearly separates the real part from the imaginary part. In our provided example, after simplifying, we reached \(15 + 9i\), which is already in standard form:- **15** is the real part, visible without an imaginary component.- **9i** is the imaginary part, clearly denoted with an "i."Using the standard form ensures that calculations involving complex numbers remain straightforward and systematic. It tells you at a glance what both the real and imaginary components of your complex number are.

• Real part: The actual number without \(i\) attached. For example, 15 in our simplification.
• Imaginary part: The coefficient of \(i\), which is 9 in this case.

By always aiming for the standard form, you ensure your expressions are transparent and easy to interpret.

Imaginary Numbers

Imaginary numbers are a fascinating extension of our number system, introduced to handle expressions involving square roots of negative numbers. The base of this system is \(i\), which is defined as \(\sqrt{-1}\). Using \(i\), any negative square root can be expressed simply.When you see a term like \(3i\), it means 3 times the imaginary unit. Here, 3 is the coefficient of \(i\). Similarly, in our example, \(9i\) represents the imaginary part of the expression.Imaginary numbers allow us to solve problems across engineering, physics, and other sciences that involve equations or phenomena not easily tackled by just real numbers.

• Definition: The imaginary unit \(i\) represents the square root of -1.
• Coefficient: The numerical factor in front of \(i\), indicating its magnitude.

Understanding and working with imaginary numbers broadens the scope of mathematical problems you can solve, especially those where real number solutions are not possible.