Question

Imaginary part of \((2+i 3)(3-i 4)\) is

Short answer

The imaginary part is 1.

Step-by-step solution

Expand the Expression

To find the imaginary part, first expand the expression \((2+i 3)(3-i 4)\) using the distributive property (FOIL method, First, Outer, Inner, Last).Calculate each term as follows:- First: \(2 \times 3 = 6\)- Outer: \(2 \times (-i 4) = -8i\)- Inner: \(i 3 \times 3 = 9i\)- Last: \(i 3 \times (-i 4) = -12\)Combine these results to write the expression as\(6 - 8i + 9i - 12\).

Combine Like Terms

Combine the real and imaginary terms from the expanded expression \(6 - 8i + 9i - 12\).- Real terms: \(6 - 12 = -6\)- Imaginary terms: \(-8i + 9i = i\)So the expression simplifies to \(-6 + i\).

Identify the Imaginary Part

The expression has been simplified to \(-6 + i\).The imaginary part of this expression is the coefficient of the imaginary unit \(i\), which is \(1\).

Key concepts

Imaginary Part

Complex numbers have two components: a real part and an imaginary part. The imaginary part of a complex number is associated with the imaginary unit, denoted as 'i', which is defined as the square root of -1. This means that any multiplication involving the imaginary unit will follow this definition.
In a complex number, denoted in the form \(a + bi\), 'a' is the real part, while 'bi' is the imaginary part. Here, 'b' is the coefficient of the imaginary unit. For example, in the expression \(-6 + i\), the imaginary unit is 'i' and its coefficient is 1. Therefore, the imaginary part is 1.
Understanding the imaginary part is essential because it allows us to work with complex numbers in mathematical operations like addition, subtraction, multiplication, and division. This concept is crucial when working with quadratic equations and other higher-degree polynomials that sometimes have no real solutions but do have complex solutions.

FOIL Method

The FOIL method is a technique used to expand the product of two binomials. It stands for First, Outer, Inner, Last, describing the order in which you multiply the terms in the binomials. This method ensures every term in the first binomial is multiplied by every term in the second binomial, a necessary step to fully expand the expression.
In our exercise, the expression \(2 + i3\) is multiplied by \(3 - i4\). By applying the FOIL method:

• First: Multiply the first terms: \(2 \times 3 = 6\)
• Outer: Multiply the outer terms: \(2 \times (-i4) = -8i\)
• Inner: Multiply the inner terms: \(i3 \times 3 = 9i\)
• Last: Multiply the last terms: \(i3 \times (-i4) = -12\) (remembering that \(i^2 = -1\))

Using the FOIL method ensures that no terms are left out during multiplication. It is a systematic approach, especially helpful in managing complex expressions involving brackets.

Distributive Property

The distributive property is a fundamental mathematical rule applied in arithmetic and algebra. It states that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). This property extends beyond simple numbers to expressions involving variables or even complex numbers.
In our exercise, the distributive property helps in expanding the expression \( (2+i3)(3-i4) \). Each term in the first complex number is distributed across each term in the second complex number using this property, similar to the FOIL method. This results in all possible pairs of terms being multiplied, as shown:

• Distributing involves multiplying each part of one expression by each part of the other and then combining similar terms.
• The distributive property is important because it allows us to simplify expressions and solve equations that might otherwise be more challenging to handle.

Recognizing and applying the distributive property can significantly simplify problems involving complex expressions, like the multiplication of complex numbers or other polynomial expressions.