Question

Describe and correct the error in performing the operation and writing the answer in standard form. \((3+4 i)-(7-5 i)+2 i(9+12 i)\)

Short answer

The corrected result of the operation and the answer in the standard form is \( -28+27i \).

Step-by-step solution

Operation

Subtract the complex numbers together: \( (3+4i)-(7-5i) = -4+9i \). Since subtraction is commutative, the difference between each correspondent part of the complex numbers is formed by subtracting their respective real and imaginary components.

Multiplication

Multiply the two complex numbers: \( 2i(9+12i) = -24+18i \). The distributive law is applied to multiply each part of the first complex number by each part of the second one.

Sum of Complex Numbers

Aggregate the results from the previous steps: \( -4+9i -24+18i = -28+27i \). The complex numbers are added together by summing up both their real and imaginary parts.

Key concepts

Standard Form of Complex Numbers

Complex numbers are expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). This notation is known as the standard form of complex numbers. The real component \(a\) represents the x-coordinate in the complex plane, while \(b\) pairs with \(i\) to represent the imaginary component or the y-coordinate.

• Real Part: The 'a' in \(a + bi\), indicating the horizontal axis.
• Imaginary Part: The 'b' associated with 'i', representing the vertical axis.

This form is essential for performing operations such as addition and multiplication, as it separates the real and imaginary parts, making calculations more straightforward.

Addition of Complex Numbers

Adding complex numbers is akin to adding vectors, where corresponding components are summed. For two complex numbers \((a+bi)\) and \((c+di)\), addition follows this pattern:

• Add real parts: \(a + c\)
• Add imaginary parts: \(bi + di\)

The resulting expression is cast into the standard form \((a+c) + (b+d)i\).
An example is shown in the operation where \((-4+9i) + (-24+18i)\) results in a sum of \(-28 + 27i\). Real numbers \(-4\) and \(-24\) add to \(-28\), and imaginary numbers \(9i\) and \(18i\) add to \(27i\).
This straightforward process maintains the structure of the standard form throughout the operation.

Multiplication of Complex Numbers

To multiply complex numbers, utilize the distributive property whereby each part of one complex number is multiplied by every part of the other. For example, consider multiplying \((2i)\) by \((9+12i)\).

• Distribute \(2i\) across \(9 + 12i\).
• Calculate \(2i \times 9 = 18i\).
• Calculate \(2i \times 12i = 24i^2\).

Recall that \(i^2 = -1\), so \(24i^2\) becomes \(-24\). The final expression is \(-24 + 18i\).
By rewriting and factoring, the complex multiplication maintains clarity and resolves into the standard form, ensuring the real and imaginary components can be clearly identified.

Error Correction in Mathematics

Mathematics is an exact science that requires attention to detail, especially with complex numbers. Error correction involves identifying and rectifying mistakes in calculations or expressions. Errors can originate from simple miscalculations or a misunderstanding of operations. A clear example here was in step two initially performed improperly by overlooking the correct application of algebraic rules during multiplication.
To effectively correct an error:

• Review all calculations thoroughly.
• Make sure that operations adhere to mathematical rules, like \(i^2 = -1\).
• Cross-check both the process and resulting forms for accuracy against standard mathematical principles.

A careful audit of each computational step ensures any failures in logic or arithmetic are rectified, preserving the mathematical integrity and arriving at the correct solution.