Question

Describe and correct the error in performing the operation and writing the answer in standard form. $$ \begin{aligned} (4+6 i)^2 &=(4)^2+(6 i)^2 \\ &=16+36 i^2 \\ &=16+(36)(-1) \\ &=-20 \end{aligned} $$

Short answer

The correct square of the complex number \(4+6i\) is \(-20 + 48i\).

Step-by-step solution

Correct Formula for the Square of a Complex Number

The correct formula for the square of a complex number \((a+bi)^2\) is \(a^2 - b^2 + 2abi\). So, let's calculate the square of \(4+6i\) using this formula.

Calculating the Real Part for the Square of the Complex Number

The real part of the square is \(a^2 - b^2\). Here \(a=4\) and \(b=6\). Substituting these values we get \(4^2 - 6^2 = 16 - 36 = -20\). So, the real part of the complex number is \(-20\).

Calculating the Imaginary Part for the Square of the Complex Number

The imaginary part of the square is \(2abi\). Substituting \(a=4\) and \(b=6\), we get \(2*(4)*(6)i = 48i\). So, the imaginary part of the complex number is \(48i\).

Writing the Result in Standard Form

The square of the complex number \(4+6i\) is found by combining the real and imaginary parts computed in the previous steps, i.e., the result is \(-20 + 48i\).

Key concepts

Standard Form of Complex Numbers

Understanding complex numbers begins with recognizing their standard form, which is expressed as \(a + bi\), where \(a\) represents the real part and \(bi\) denotes the imaginary part. Here, \(i\) is the imaginary unit, defined by the property that \(i^2 = -1\). The numbers \(a\) and \(b\) are real numbers. In the context of our exercise, the standard form is essential for correctly expressing the square of a complex number.

In the given problem, the correct standard form for the square of the complex number \(4+6i\) should be \(-20 + 48i\) and not \(-20\). This highlights that every part of the complex number is significant in reaching the correct form which is vital for further calculations or applications of complex numbers.

Squaring Complex Numbers

Squaring complex numbers involves more than applying the square to both the real and imaginary parts separately. The erroneous belief that \((a + bi)^2 = a^2 + b^2i^2\) fails to consider the cross-term resulting from multiplication.

For a proper expansion, we should use the distributive property of multiplication over addition, like expanding \((a + b)^2\) for real numbers. Thus, the correct formula is \((a + bi)^2 = a^2 + 2abi + b^2i^2\), taking into account the real parts \(a^2 - b^2\) (because \(i^2 = -1\)) and the imaginary part \(2abi\). An example using the number \(4 + 6i\) yields \(16 + 48i + 36(-1)\), simplifying to the correct answer of \(-20 + 48i\) in standard form.

Imaginary Numbers Algebra

The algebra involving imaginary numbers relies on the fundamental definition that \(i^2 = -1\). This defines how imaginary numbers behave under various operations such as addition, subtraction, multiplication, and squaring. When performing operations with complex numbers, this property should always be at the forefront.

Key Rules of Imaginary Numbers Algebra:

• \(i^2 = -1\), by definition.
• When multiplying two imaginary terms, apply the property of \(i^2\) to simplify.
• The sum or difference of two imaginary numbers is another imaginary number.
• The product of a real number and an imaginary number is an imaginary number.

In our exercise, the mistake was made by not correctly applying the algebraic rules of imaginary numbers, leading to the omission of the cross-term during squaring. Utilizing these rules correctly will aid students in manipulating complex numbers without errors.