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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 NumbersProperties of Complex Number Arithmetic

Algebraic Identities for Complex numbers

The algebraic identities of complex number is same as that for real numbers.

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Which of the following are examples of algebraic identities?

  • `(a+b)^2 = a^2+2ab+b^2`
  • `(a-b)^2 = a^2-2ab+b^2`
  • `a^2-b^2 = (a+b)(a-b)`
  • all the above
  • all the above

The answer is 'all the above'

If an algebraic identity is true for real numbers, will that be true for complex numbers?

  • No, it will not be true at all
  • Yes, for all identities with four fundamental operations
  • Yes, for all identities with four fundamental operations

The answer is 'Yes, for all identities with four fundamental operations'.

The four fundamental operations like
 •  addition,
 •  subtraction,
 •  multiplication, and
 •  division define the properties
 •  associative,
 •  commutative
 •  distributive.

The algebraic identities employ the properties of the operations given above. Between real numbers and complex numbers the properties are same. So the algebraic identities will hold true.

 •  Algebraic identities are same as that of real numbers.

Algebraic Identities: `(z_1+z_2)^2 = z_1^2+ 2z_1z_2+z_2^2`
`(z_1-z_2)^2 = z_1^2 - 2z_1z_2+z_2^2`
`(z_1+z_2)(z_1-z_2) = z_1^2 - z_2^2`
`(z_1+z_2)^3 = z_1^3+z_2^3+ 3z_1^2z_2+ 3z_1z_2^2`
and so on.

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