Fsc Part 1 Mathematics (Complete Solution)

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Q13

Prove that the sum as well as the product of any two conjugate complex numbers is a real number.

**SOLUTION:**

Let the two conjugate complex numbers are a+bι , a-bι

Sum of two conjugate complex numbers | = | (a+bι) + (a-bι) | which is real number |

= | a + ιb + a -ι b = 2a |

Product of two conjugate complex numbers = (a+bι)(a-bι)

= (a)^{2}-(bι)^{2}

= a^{2}-b^{2}ι^{2}

= a^{2}-b^{2}(-1)

= a^{2}+b^{2}

Both 2a and a^{2}+b^{2} are real numbers.