What is the real part of the complex number shown?
In this question, we’re given the point 𝑧 on an Argand diagram, and we need to determine the real part of this complex number 𝑧. To answer this question, let’s start by recalling what we mean by an Argand diagram. In an Argand diagram, the horizontal coordinate of our complex number represents the real part of this complex number and the vertical coordinate of this number represents the imaginary part of this complex number. And since we want to find the real part of this complex number, that means we want to know its horizontal coordinate on an Argand diagram.
And we know how to find both its horizontal and vertical coordinate. To find its horizontal coordinate, we go in a straight line down to our horizontal axis. And we see that this is at zero. Similarly, if we went horizontally, we would see its vertical coordinate is six. So its coordinates are zero, six. The one we’re interested in is the horizontal coordinate, which is zero. This is the real part of our complex number. And this is enough to answer our question. We’ve shown the real part of this complex number is equal to zero. However, we’ve actually shown more than this.
Remember, we can represent any complex number in algebraic form. That’s in the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real constants. And in particular, the value of 𝑎 will be the real part of our complex number and the value of 𝑏 will be the imaginary part of our complex number. So, in fact, not only have we shown that 𝑧 has real part zero, we’ve also shown that 𝑧 is the complex number zero plus six 𝑖, which of course we can simplify to be six 𝑖.
Therefore, we were able to find the real part of a complex number given in an Argand diagram by finding its horizontal coordinate. Since its horizontal coordinate was equal to zero, we were able to show the real part of this complex number was equal to zero.