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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsComplex NumbersAlgebra of Complex Numbers

Conjugate of a complex Number

Conjugate of a complex number is explained.



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In numerical expressions or algebraic expressions, we can manipulate the expressions without modifying the value of the expression
 •  add and subtract : eg `3 = 3+1-1`
 •  multiply and divide : eg : `3 = 3 xx 2/2`
 •  etc.

. These manipulations help to arrive at a different form of expressions or help to solve.

In case of complex numbers, `a+ib`, one modification stands out "convert the complex number to real number". This can be achieved either by addition or multiplication with the number `a-ib`.
 •  addition `(a+ib) + (a-ib) = 2a`
 •  multiplication `(a+ib) xx (a-ib) = a^2+b^2`

For a given complex number `z= a+i b`, the connected number that give a real number on multiplication is `a-ib`. It is named as conjugate of z and represented as `bar(z)` or `bar(a+ib) = a-ib`

What does 'conjugate' mean?

  • coupled; joined ; related in reciprocal or complementary
  • coupled; joined ; related in reciprocal or complementary
  • disjoint; additional; unrelated and orthogonal

The answer is 'coupled; joined ; related in reciprocal or complementary'

Conjugate of a complex number is the number with the same real part and negative of imaginary part.

Conjugate of a Complex NumberFor a complex number `z=a+ib in CC ` the conjugate of `z` is given as `bar(z) = a-ib`.

`bar(z)` is called ... of z.

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'Conjugate'.

Solved Exercise Problem:

Find `bar(1-3i)`.

  • `1-3i`
  • `-1-3i`
  • `1+3i`
  • `1+3i`
  • `-1+3i`

The answer is '`1+3i`'

                            
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