- The goal is to simplify \((-1+i) + (21+5i)\) by grouping real and imaginary parts.
- The associative property allows grouping terms, and the commutative property allows reordering terms: \(-1 + i + 21 + 5i = -1 + 21 + i + 5i\).
- Regrouping these leads to \((-1+21) + (i+5i)\).
- Option (b) correctly represents the rearrangement that uses these properties to group like terms.
- The final answer is \(\boxed{b}\).
Explanation
- Analysis of the Expression
The expression we are simplifying is \((-1+i) + (21+5i)\). To add complex numbers, we combine the real parts and the imaginary parts separately. The complex number expression is given by the sum of two complex numbers in the form of \((a+bi) + (c+di) = (a+c) + (b+d)i\). Our current expression is \((-1+i) + (21+5i)\). To group the real parts \((-1 \text{ and } 21)\) and the imaginary parts \((i \text{ and } 5i)\), we utilize the properties of addition.
- Applying Properties of Addition
The properties involved are:
- Associative property of addition: \((a+b)+c = a+(b+c)\).
- Commutative property of addition: \(a+b = b+a\).
Starting from \((-1+i) + (21+5i)\), we first apply the associative property to remove the parentheses: \(-1 + i + 21 + 5i\).
Next, we want to group the real terms together and the imaginary terms together. Using the commutative property of addition, we can swap the order of the terms: \(-1 + 21 + i + 5i\).
Looking at the provided options, we need to identify the one that best reflects the rearrangement of these terms. Option (a) shows \(-1 + (i + 21) + 5i\), which uses the associative property. Option (b) shows \((-1 + 21) + (i + 5i)\), which is the regrouped result. Option (c) and (d) do not represent the standard reordering steps.
Given the standard procedure for rearranging terms to group real and imaginary parts, the step that directly demonstrates the commutation to prepare the regrouping is the swap that brings the real part \(21\) next to the real part \(-1\). Thus, rearranging the expression to \((-1+21)+(i+5i)\) effectively demonstrates the reordering intended by the commutative property.
- Conclusion
The first step in simplifying the sum of two complex numbers is to regroup the real components and the imaginary components. Applying the commutative property of addition allows us to rearrange the terms so that real and imaginary parts are adjacent. Among the choices, the expression \((-1+21)+(i+5i)\) represents the logical grouping result of this process, and rearranging the terms to achieve this is the fundamental first step in addition of complex numbers using these properties. Thus, (b) is the correct expression demonstrating the regrouping via the commutative and associative properties.
Examples
In electronics and electrical engineering, complex numbers are frequently used to represent impedance in AC circuits. For instance, when adding two impedances \(Z_1 = R_1 + jX_1\) and \(Z_2 = R_2 + jX_2\), we combine the resistive parts (real) and the reactive parts (imaginary) separately to find the total impedance: \(Z_{total} = (R_1 + R_2) + j(X_1 + X_2)\). Knowing how to reorder and group these complex numbers using commutative and associative properties allows engineers to analyze circuit components efficiently and calculate total impedance accurately.