Let [tex]$x=a+b i$[/tex] and [tex]$y=c+d i$[/tex] and [tex]$z=f+g i$[/tex]. Which statements are true? Choose three correct answers.
[tex]$(x+y)+z=x+(y+z)$[/tex]
[tex]$(x \times y) \times z=x \times(y \times z)$[/tex]
[tex]$x-y=y-x$[/tex]
Answer (1)
Complex number addition is associative: ( x + y ) + z = x + ( y + z ) .
Complex number multiplication is associative: ( x × y ) × z = x × ( y × z ) .
Complex number subtraction is not commutative: x − ye q y − x in general.
The true statements are the first two: ( x + y ) + z = x + ( y + z ) and ( x × y ) × z = x × ( y × z ) .
Explanation
Problem Analysis Let x = a + bi , y = c + d i , and z = f + g i be complex numbers. We need to determine which of the given statements are true.
Statement Verification We will analyze each statement separately:
( x + y ) + z = x + ( y + z ) : This statement represents the associative property of addition. For complex numbers, addition is associative. Let's verify: ( x + y ) + z = ( a + bi + c + d i ) + f + g i = ( a + c + bi + d i ) + f + g i = ( a + c + f ) + ( b + d + g ) i x + ( y + z ) = a + bi + ( c + d i + f + g i ) = a + bi + ( c + f + d i + g i ) = ( a + c + f ) + ( b + d + g ) i Since both expressions are equal, the associative property holds for addition of complex numbers. Thus, the statement is true.
( x × y ) × z = x × ( y × z ) : This statement represents the associative property of multiplication. For complex numbers, multiplication is associative. Let's verify: ( x × y ) × z = (( a + bi ) ( c + d i )) ( f + g i ) = ( a c + a d i + b c i − b d ) ( f + g i ) = (( a c − b d ) + ( a d + b c ) i ) ( f + g i ) = ( a c − b d ) f + ( a c − b d ) g i + ( a d + b c ) f i − ( a d + b c ) g = ( a c f − b df − a d g − b c g ) + ( a c g − b d g + a df + b c f ) i x × ( y × z ) = ( a + bi ) (( c + d i ) ( f + g i )) = ( a + bi ) ( c f + c g i + df i − d g ) = ( a + bi ) (( c f − d g ) + ( c g + df ) i ) = a ( c f − d g ) + a ( c g + df ) i + b ( c f − d g ) i − b ( c g + df ) = ( a c f − a d g − b c g − b df ) + ( a c g + a df + b c f − b d g ) i Since both expressions are equal, the associative property holds for multiplication of complex numbers. Thus, the statement is true.
x − y = y − x : This statement represents the commutative property of subtraction. For complex numbers, subtraction is generally not commutative. Let's verify: x − y = ( a + bi ) − ( c + d i ) = ( a − c ) + ( b − d ) i y − x = ( c + d i ) − ( a + bi ) = ( c − a ) + ( d − b ) i For x − y to be equal to y − x , we must have a − c = c − a and b − d = d − b . This implies 2 a = 2 c and 2 b = 2 d , so a = c and b = d . This means x = y . However, the statement is not generally true for all complex numbers x and y . Thus, the statement is false.
Conclusion The first two statements are true, and the third statement is false.
Examples
Complex numbers are used in electrical engineering to represent alternating current (AC) circuits. The voltage, current, and impedance in an AC circuit can be represented as complex numbers. The associative properties of addition and multiplication are crucial when analyzing complex circuits with multiple components in series or parallel. For example, when calculating the total impedance of several components connected in series, the associative property allows engineers to group and combine impedances in any order, simplifying the analysis and ensuring accurate results. This ensures that the order in which components are combined does not affect the final result, which is essential for designing and troubleshooting electrical systems.
