- The identity property of multiplication states that any number multiplied by 1 equals itself, or \(x \cdot 1 = x\).
- Option a, \((a + bi) \cdot 1 = (a + bi)\), correctly applies this property to complex numbers.
- Other options represent different properties: (b) is scalar distribution, (c) is commutativity, and (d) is the zero property.
- The correct choice is option a: \(\boxed{a}\).
Explanation
- Understanding the Identity Property of Multiplication
The identity property of multiplication states that for any real or complex number \(x\), multiplying that number by 1 results in the number itself. Mathematically, this is expressed as:
\[x \cdot 1 = x\]
or
\[1 \cdot x = x\]
This means that 1 is the multiplicative identity. In this problem, we are looking for the equation that exemplifies this property using a complex number \(z = a + bi\).
- Evaluating the Given Options
Let's evaluate each provided option based on algebraic properties:
A\((a + bi) \cdot 1 = (a + bi)\): Here, a complex number is multiplied by 1, returning the original complex number. This perfectly matches the definition of the identity property of multiplication.
B\((a + bi) \cdot c = (ac + bci)\): This demonstrates scalar multiplication or the distribution of a scalar over the components of a complex number.
C\((a + bi) \cdot (c + di) = (c + di) \cdot (a + bi)\): This illustrates the commutative property of multiplication, which states that the order of the factors does not change the product.
D\((a + bi) \cdot 0 = 0\): This is the zero property of multiplication, which states that any number multiplied by 0 results in 0.
- Conclusion
After comparing the options against the standard algebraic definitions, it is clear that only the first option demonstrates the multiplicative identity property. Therefore, the equation that illustrates this property is \((a + bi) \cdot 1 = (a + bi)\), which corresponds to option a.
Examples
The identity property of multiplication is fundamental in simplifying algebraic expressions. For instance, if you have an expression like \(\frac{x+y}{z}\), you can multiply by \(1 = \frac{k}{k}\) to change the form of the expression without altering its value: \(\frac{x+y}{z} \cdot \frac{k}{k} = \frac{k(x+y)}{kz}\). This technique is essential for finding common denominators or rationalizing denominators in complex fractions.