- The equation \(a + bi + 0 + 0i = a + bi\) represents the addition of a complex number and the complex number zero.
- The identity property of addition states that adding the additive identity element (which is \(0+0i\) for complex numbers) to any element results in the original element.
- Since \(z + 0 = z\), the equation demonstrates the identity property.
- The correct option is \(\boxed{b}\).
Explanation
- Analysis of the Equation and Properties of Addition
The equation \(a + bi + 0 + 0i = a + bi\) involves a complex number \(z = a + bi\) and the complex number \(0 = 0 + 0i\). In the set of complex numbers, the identity property of addition states that there exists an additive identity element, denoted by \(0\), such that for any complex number \(z\), the sum of \(z\) and the identity element is the complex number itself. That is, \(z + 0 = z\). Since \(0 + 0i = 0\), the given equation \(a + bi + (0 + 0i) = a + bi\) is a direct application of this property.
- Evaluating Alternative Properties
To confirm that this is the identity property, we can briefly examine the other properties provided:
- The inverse property of addition states that for any complex number \(z\), there exists an additive inverse \(-z\) such that \(z + (-z) = 0\). This is not shown here.
- The commutative property of addition states that for any two complex numbers \(z_1\) and \(z_2\), \(z_1 + z_2 = z_2 + z_1\). This is not shown here.
- The associative property of addition states that for any three complex numbers \(z_1, z_2,\) and \(z_3\), \((z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)\). This is not shown here.
Thus, the equation clearly demonstrates the identity property of addition.
- Conclusion
The equation \(a + bi + 0 + 0i = a + bi\) shows that adding the complex number \(0 = 0 + 0i\) to any complex number \(a + bi\) leaves the complex number unchanged. This confirms that the property shown is the identity property. The final answer is b.
Examples
The identity property is fundamental in many real-world applications, especially in computer science and finance. For instance, if a person has a bank account balance represented by a complex-valued financial model (like a profit-loss forecast), adding a zero-value transaction (i.e., a transaction of \(\$0 + \$0i\)) does not change the account's status. Similarly, in programming, initializing a variable to zero ensures that any subsequent addition of data to that variable correctly reflects the total, as adding zero does not modify the original value. Recognizing these algebraic structures helps in simplifying complex mathematical models used in engineering and economics.