To identify which equation illustrates the identity property of multiplication, we should first understand what the identity property of multiplication states:
The identity property of multiplication states that any number multiplied by 1 remains unchanged. In mathematical terms, for any number [tex]\( x \)[/tex]:
[tex]\[ x \times 1 = x \][/tex]
Now, let's examine each equation to determine which one matches this property:
1. [tex]\( (a + bi) \times c = (ac + bci) \)[/tex]
This equation shows the distribution of two complex numbers when multiplied but doesn't illustrate the identity property of multiplication. It doesn’t state that the number has been multiplied by 1 to remain unchanged.
2. [tex]\( (a + bi) \times 0 = 0 \)[/tex]
This equation shows the multiplication of a complex number by 0, which results in 0. While this is true, it does not represent the identity property of multiplication because the identity property involves multiplication by 1, not 0.
3. [tex]\( (a + bi) \times (c + di) = (c + di) \times (a + bi) \)[/tex]
This equation illustrates the commutative property of multiplication, where the product of two complex numbers remains the same regardless of their order. This is not the identity property of multiplication, which involves multiplying by 1.
4. [tex]\( (a + bi) \times 1 = (a + bi) \)[/tex]
This equation shows a complex number [tex]\((a + bi)\)[/tex] multiplied by 1, resulting in the same complex number [tex]\((a + bi)\)[/tex]. This perfectly illustrates the identity property of multiplication since multiplying by 1 leaves the number unchanged.
Therefore, the equation that illustrates the identity property of multiplication is:
[tex]\[ (a + bi) \times 1 = (a + bi) \][/tex]
Thus, the correct answer is equation 4.
