Question

Which of the following is correct?
a) 2 + 3i > 1 + 4i
b) 2 + 2i > 3 + 3i
c) 5 + 8i > 5 + 7i
d) None of these

Answer 1

Hint: We can relate inequalities between those numbers which can be represented on the number line. We cannot compare those numbers which don’t exist in reality on the number line. So, use this basic property of numbers in mathematics.

Complete step-by-step answer:
As we know that any complex number is represented in the form of (a + ib), where a is the real part and b is the imaginary part and the value of ‘i’ is $\sqrt{-1}$ . Now, if b is equal to 0 then the number (a + ib) will become ‘0’ i.e. it will be a real number. Hence, we can represent this number ‘a’ (if b = 0) i.e. real number, on a number line and can relate it with other numbers in form of inequalities. Example: 2 > 1, 3 > 0 etc.
Now, if $b\ne 0$ in the complex number ‘a + ib’, then it cannot be represented on the number line as the number is not existing in reality. So, we cannot compare any complex number with another complex number given that the imaginary part of any of them is not zero. So, we cannot compare any complex number ${{a}_{1}}+i{{b}_{1}}$ with other ${{a}_{2}}+i{{b}_{2}}$ (where ${{b}_{1}},{{b}_{2}}$ will not equal to zero). So, we cannot determine the greater or lower number between them. Hence, all the given options with inequalities are not true.
Two complex numbers ${{a}_{1}}+i{{b}_{1}},{{a}_{2}}+i{{b}_{2}}$ can be equal if ${{a}_{1}}={{a}_{2}},{{b}_{1}}={{b}_{2}}$ but cannot be represented in form of inequalities in any case.

Hence option (d) is correct as the other options are in the form of inequalities of two complex numbers where the imaginary part is not zero.

Note: One may go wrong if he or she relates the given inequalities with respect to their magnitudes. As the magnitude of any complex number z = x + iy is given as $\sqrt{{{x}^{2}}+{{y}^{2}}}$ . So, as example of option (i), we get magnitude or modulus of both the complex numbers (2 + 3i) and (1 + 4i) are $\sqrt{4+9}=\sqrt{13},\sqrt{1+16}=\sqrt{17}$ respectively. So, one may give his or her answer as 2 + 3i < 1 + 4i as $\sqrt{13}<17$ , which is wrong because we cannot compare complex numbers (with imaginary parts). Hence, be clear with this property to solve these types of questions.
One may correct option (c) given as 5 + 8i > 5 + 7i or 8i > 7i
As 8 is a higher number than 7, one may think that 8i > 7i but the option is wrong because we cannot compare 8i and 7i as both cannot be shown on the number line. So, we can compare only those numbers which can be represented on the number line. One may confuse with this option as well, so be careful.