Let $x=a+b i$, $y=c+d i$, and $z=f+g i$. Which statements are true? Choose three correct answers.

$(x+y)+z=x+(y+z)$
$(x \times y) \times z=x \times(y \times z)$
$x-y=y-x$

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 associativity of addition and multiplication. ( x + y ) + z = x + ( y + z ) ; ( x × y ) × z = x × ( y × z ) ​

Explanation

Problem Analysis We are given three complex numbers x = a + bi , y = c + d i , and z = f + g i , where a , b , c , d , f , g are real numbers. We need to determine which of the following statements are true:

( x + y ) + z = x + ( y + z )

( x × y ) × z = x × ( y × z )

x − y = y − x

Associativity of Addition Statement 1: ( x + y ) + z = x + ( y + z ) . This statement tests the associative property of addition for complex numbers. Let's verify this property.


x + y = ( a + bi ) + ( c + d i ) = ( a + c ) + ( b + d ) i ( x + y ) + z = [( a + c ) + ( b + d ) i ] + ( f + g i ) = ( a + c + f ) + ( b + d + g ) i
y + z = ( c + d i ) + ( f + g i ) = ( c + f ) + ( d + g ) i x + ( y + z ) = ( a + bi ) + [( c + f ) + ( d + g ) i ] = ( a + c + f ) + ( b + d + g ) i
Since ( x + y ) + z = ( a + c + f ) + ( b + d + g ) i = x + ( y + z ) , the associative property of addition holds for complex numbers. Thus, statement 1 is true.

Associativity of Multiplication Statement 2: ( x × y ) × z = x × ( y × z ) . This statement tests the associative property of multiplication for complex numbers. Let's verify this property.

x × y = ( a + bi ) ( c + d i ) = ( a c − b d ) + ( a d + b c ) i ( x × y ) × z = [( a c − b d ) + ( a d + b c ) i ] ( f + g i ) = [( a c − b d ) f − ( a d + b c ) g ] + [( a c − b d ) g + ( a d + b c ) f ] i = ( a c f − b df − a d g − b c g ) + ( a c g − b d g + a df + b c f ) i
y × z = ( c + d i ) ( f + g i ) = ( c f − d g ) + ( c g + df ) i x × ( y × z ) = ( a + bi ) [( c f − d g ) + ( c g + df ) i ] = [ a ( c f − d g ) − b ( c g + df )] + [ a ( c g + df ) + b ( c f − d g )] i = ( a c f − a d g − b c g − b df ) + ( a c g + a df + b c f − b d g ) i
Since ( x × y ) × z = ( 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 ) , the associative property of multiplication holds for complex numbers. Thus, statement 2 is true.

Commutativity of Subtraction Statement 3: x − y = y − x . This statement tests the commutative property of subtraction for complex numbers. Let's verify this property.

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 = y − x , we must have ( a − c ) + ( b − d ) i = ( c − a ) + ( d − b ) i . This implies a − c = c − a and b − d = d − b . Thus, 2 a = 2 c and 2 b = 2 d , which means a = c and b = d . Therefore, x = y .
However, the statement x − y = y − x is not true in general. It is only true when x = y . Thus, statement 3 is false.

Conclusion Therefore, the true statements are:

( x + y ) + z = x + ( y + z )

( x × y ) × z = x × ( y × z )


Examples
Complex numbers are used extensively in electrical engineering to analyze alternating current (AC) circuits. By representing voltage and current as complex quantities, engineers can simplify calculations involving impedance, phase shifts, and power. The associative properties of addition and multiplication ensure that circuit analysis remains consistent regardless of how components are grouped or calculated, which is crucial for designing and troubleshooting complex electrical systems. For example, when analyzing a series of impedances, the total impedance can be calculated by summing the individual impedances in any order, thanks to the associative property of addition.

Answered by GinnyAnswer | 2025-07-08