Complex numbers
When teaching Junior Cert students, I would warn them that you cannot get the square root of a minus number. If you type any negative number into your calculator and get its square root, the calculator would return an error message. Now, in Senior Cycle, we can deal with this issue through the usage of complex numbers. Complex numbers are used to overcome any issues involving the square roots of negative numbers. We use , where i represents an imaginary number. From this, we can see that
What is a complex number?
A complex number z is a number that can be written in the form z = a + bi, where a is called the real part of the complex number and b is called the imaginary part of the complex number. In simple terms, a complex number = real part + (imaginary part)i.
The following are the five most important parts of complex numbers:
1. Addition, subtraction and multiplication of complex numbers
2. Division of complex numbers and the complex conjugate
3. The Argand diagram
4. The modulus
5. Equality of complex numbers.
Addition, subtraction and multiplication of complex numbers
These questions are student friendly. All you have to do is group the real parts together and then group the imaginary parts together. Simply add the reals with reals and the imaginaries with the imaginaries. When multiplying complex numbers, ensure that you multiply out i2, i3, i4, …using i2= –1.
Division of complex numbers and the complex conjugate
Dividing by a complex number requires a lot more work when compared to addition, subtraction and multiplication. In order to divide a complex number by another complex number, we need to use the complex conjugate. Division of complex numbers is a popular exam question. Know how to do it because there are no formulas for it in the mathematical tables.
Method for dividing complex numbers
Multiply above and below by the complex conjugate of the denominator. Our objective is to ensure that there is no imaginary numbers in the denominator.
What is a complex conjugate?
The conjugate of a complex number is the same complex number except that you have to change the sign of the imaginary part. We use z to denote the conjugate. If z = a + bi is a complex number, then z = a + bi is called the complex conjugate of z.