Which equation illustrates the identity property of multiplication?
$(a+b i) \cdot(c+d i)=(c+d i) \cdot(a+b i)$
$(a+b i) \cdot 0=0$
$(a+b i) \cdot 1=(a+b i)$
$(a+b i) \cdot c=(a c+b c i)$
Answer (2)
The identity property of multiplication states that any number multiplied by 1 equals itself.
Examine each equation to determine which one demonstrates the identity property.
The equation ( a + bi ) ⋅ 1 = ( a + bi ) shows that a complex number multiplied by 1 results in the same complex number.
Therefore, the equation that illustrates the identity property of multiplication is ( a + bi ) ⋅ 1 = ( a + bi ) .
Explanation
Understanding the Identity Property The question asks us to identify which equation demonstrates the identity property of multiplication. The identity property states that any number multiplied by 1 equals itself. We need to examine each option to see which one fits this property.
Analyzing Each Option Let's analyze each option:
( a + bi ) ⋅ ( c + d i ) = ( c + d i ) ⋅ ( a + bi ) : This equation shows that changing the order of multiplication doesn't change the result, which illustrates the commutative property of multiplication, not the identity property.
( a + bi ) ⋅ 0 = 0 : This equation shows that any number multiplied by zero equals zero, which illustrates the zero property of multiplication, not the identity property.
( a + bi ) ⋅ 1 = ( a + bi ) : This equation shows that when we multiply the complex number ( a + bi ) by 1, we get the same complex number ( a + bi ) . This perfectly illustrates the identity property of multiplication.
( a + bi ) ⋅ c = ( a c + b c i ) : This equation shows the distributive property of multiplication over addition.
Identifying the Correct Equation The equation that illustrates the identity property of multiplication is ( a + bi ) ⋅ 1 = ( a + bi ) .
Examples
The identity property of multiplication is a fundamental concept in mathematics. It's used in various applications, such as simplifying expressions, solving equations, and understanding the structure of number systems. For example, when balancing chemical equations, we use the identity property to ensure that the number of atoms of each element remains the same on both sides of the equation. Similarly, in computer science, the identity property is used in designing algorithms and data structures.
The equation that illustrates the identity property of multiplication is ( a + bi ) o d o t 1 = ( a + bi ) . This equation shows that multiplying a complex number by 1 yields the same complex number. Therefore, the correct answer is the third option.
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