- The commutative property of addition states that \(a+b = b+a\).
- To simplify complex number addition \((-1+i)+(21+5i)\), we rearrange real and imaginary parts.
- Option (b) \((-1+21)+(i+5i)\) represents the regrouping step facilitated by the commutative and associative properties of addition.
- Final answer: \(\boxed{b}\)
Explanation
- Analysis of the Commutative Property
The expression to simplify is \((-1+i) + (21+5i)\). The commutative property of addition states that for any two numbers \(a\) and \(b\), the order of addition does not affect the sum, meaning \(a + b = b + a\). Applying this to the given expression, we would swap the two terms: \((-1+i) + (21+5i) = (21+5i) + (-1+i)\). We need to examine which of the provided options demonstrates an application of this property or related rearrangement properties.
- Evaluating the Options
Let's evaluate the provided options:
A\(-1+(i+21)+5i\): This involves regrouping terms, likely using the associative property, but it does not represent the simplest form of commutative property application.
B\((-1+21)+(i+5i)\): This rearrangement collects the real parts together (\(-1\) and \(21\)) and the imaginary parts together (\(i\) and \(5i\)). This process requires applying both the commutative property (to swap \(-1+i\) and \(21+5i\) or rearrange the inner terms) and the associative property to group the like terms. This is a standard first step in simplifying complex numbers.
C\(-(1-i)+(21+5i)\): This merely factors a negative out of the first term and does not demonstrate the commutative property.
D\((-1+i)+(21+5i)+0\): This uses the identity property of addition, not the commutative property.
While the strict commutative property \((-1+i) + (21+5i) = (21+5i) + (-1+i)\) is not explicitly listed as an option, option (b) is the standard algebraic first step that uses the commutative property to rearrange terms so that like terms are adjacent, facilitating the use of the associative property to group them.
- Conclusion
The standard procedure for adding complex numbers is to group the real parts and the imaginary parts together. Given \((-1+i) + (21+5i)\), we use the commutative property to reorder the terms and the associative property to change the grouping to \((-1+21) + (i+5i)\). Thus, option (b) best represents the first logical step in simplifying the expression using these fundamental properties of addition.
Examples
In electrical engineering, complex numbers are used to represent impedance in AC circuits. When components are connected in series, their total impedance is the sum of individual impedances. If you have an inductor with impedance \(Z_1 = 2+3i\) and a resistor with impedance \(Z_2 = 10+0i\), the total impedance is \(Z_1 + Z_2 = (2+3i) + (10+0i)\). To simplify this, you rearrange the terms as \((2+10) + (3i+0i) = 12+3i\) using the commutative and associative properties, ensuring the circuit analysis is straightforward and accurate.