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SUM OF COMPLEX NUMBERS
SUM OF COMPLEX NUMBERS
Let z1 and z2 be complex numbers, where z1 = a + bi and z2 = s + ti, which are represented by vectors z1 = <a,b> and z2 = <s,t> respectively.
Algebraically, the sum z1 + z2 = (a + s) + (b + t)i.
Geometrically, z1 + z2 is represented by the resultant vector z1 + z2.
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MULTIPLYING COMPLEX NUMBERS
MULTIPLYING COMPLEX NUMBERS
Explore multiplying complex numbers by considering what happens when you multiply complex and real numbers together. There are different rules to consider when using complex, imaginary, and real numbers in products.
Algebraically, when you multiply a complex number by a real number, the product is a complex number. Graphically, the modulus of z1 acts as a scalar. There is no rotation in this case.
Algebraically, when you multiply a complex number by an imaginary number, the product is a complex number. Graphically, the point is moved out from the origin and a rotation occurs. There is a rotation of 90°
Algebraically, when you multiply a complex number by a complex number, the product is a complex number. The easiest way to remember this formula is to multiply the moduli, which are the r-values, and add the arguments, which are the angles.
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COMPLEX CONJUGATE
COMPLEX CONJUGATE
Every complex number has a partner called the complex conjugate. When a complex number is multiplied by its conjugate, the product is a real number. Geometrically, the conjugate is a reflection across the real axis. The complex conjugate of a + bi is a – bi.
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