Find the value of two 𝑧 plus 𝑖𝑧 bar given 𝑧 on the Argand diagram below.
We know that any complex number 𝑧 can be written in the form 𝑎 plus 𝑏𝑖, where 𝑎 is the real part and 𝑏 the imaginary part of a complex number. On an Argand diagram, the real part or component is denoted by the horizontal axis and the imaginary part by the vertical axis. In this question, the point 𝑧 has coordinates four, five. Therefore, 𝑧 is the complex number four plus five 𝑖.
The notation 𝑧 bar denotes the complex conjugate. We know that if 𝑧 is equal to 𝑎 plus 𝑏𝑖, then 𝑧 bar is equal to 𝑎 minus 𝑏𝑖. The imaginary part of our complex number has the opposite sign. This means that 𝑧 bar in our question is equal to four minus five 𝑖. This can be shown on the Argand diagram as the point with coordinates four, negative five. Note that the complex conjugate is a reflection in the real axis.
We were asked to find the value of two 𝑧 plus 𝑖𝑧 bar. This is equal to two multiplied by four plus five 𝑖 plus 𝑖 multiplied by four minus five 𝑖. Distributing the first set of parentheses gives us eight plus 10𝑖. Distributing the second set gives us four 𝑖 minus five 𝑖 squared. Two 𝑧 plus 𝑖𝑧 bar is equal to eight plus 10𝑖 plus four 𝑖 minus five 𝑖 squared.
From our knowledge of complex numbers, we know that 𝑖 squared is equal to negative one. Negative five multiplied by negative one is equal to five. Therefore, the right-hand side simplifies to eight plus 10𝑖 plus four 𝑖 plus five. We can then group or collect the real and imaginary parts separately. Eight plus five is equal to 13, and 10𝑖 plus four 𝑖 is 14𝑖. Two 𝑧 plus 𝑖𝑧 bar is equal to 13 plus 14𝑖.