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We will discuss here about the cube roots of unity and their properties.
Suppose let us assume that the cube root of 1 is z i.e., β1 = z.
Then, cubing both sides we get, z3 = 1
or, z3 - 1 = 0
or, (z - 1)(z2 + z + 1) = 0
Therefore, either z - 1 = 0 i.e., z = 1 or, z2 + z + 1 = 0
Therefore, z = β1Β±β12β4β 1β 12β 1 = β1Β±ββ32 = -12 Β± iβ32
Therefore, the three cube roots of unity are
1, -12 + iβ32 and -12 - iβ32
among them 1 is real number and the other two are conjugate complex numbers and they are also known as imaginary cube roots of unity.
Properties of the cube roots of unity:
Property I: Among the three cube roots of unity one of the cube roots is real and the other two are conjugate complex numbers.
The three cube roots of unity are 1, -12 + iβ32 and -12 - iβ32.
Hence, we conclude that from the cube roots of unity we get 1 is real and the other two i.e., 12 + iβ32 and -12 - iβ32 are conjugate complex numbers.
Property II: Square of any one imaginary cube root of unity is equal to the other imaginary cube root of unity.
(β1+β3i2)2 = 14[(- 1)^2 - 2 β 1 β β3i + (β3i)2]
= 14[1 - 2β3i - 3]
= β1ββ3i2,
And (β1ββ3i2)2 = 14[(1^2 + 2 β 1 β β3i + (β3i)2]
= 14[1 + 2β3 i - 3]
= β1+β3i2,
Hence, we conclude that square of any cube root of unity is equal to the other.
Therefore, suppose Ο2 is one imaginary cube root of unity then the other would be Ο.
Property III: The product of the two imaginary cube roots is 1 or, the product of three cube roots of unity is 1.
Let us assume that, Ο = β1ββ3i2; then, Ο2 = β1+β3i2
Therefore, the product of the two imaginary or complex cube roots = Ο β Ο2 = β1ββ3i2 Γ β1+β3i2
Or, Ο3 = 14[(-1)2 - (β3i)2] = 14[1 - 3i2] = 14[1 + 3] = 14 Γ 4 = 1.
Again, the cube roots of unity are 1, Ο, Ο2. So, product of cube roots of unity = 1 β Ο β Ο2 = Ο3 = 1.
Therefore, product of the three cube roots of unity is 1.
Property IV: Ο3 = 1
We know that Ο is a root of the equation z3 - 1 = 0. Therefore, Ο satisfies the equation z3 - 1 = 0.
Consequently, Ο3 - 1 = 0
or, Ο = 1.
Note: Since Ο3 = 1, hence, Οn = Οm, where m is the least non-negative remainder obtained by dividing n by 3.
Property V: The sum of the three cube roots of unity is zero i.e., 1 + Ο + Ο2 = 0.
We know that, the sum of the three cube roots of unity = 1 + β1ββ3i2 + β1+β3i2
Or, 1 + Ο + Ο2 = 1 - 12 + β32i - 12 - β32i = 0.
Notes:
(i) The cube roots of 1 are 1, Ο, Ο2 where, Ο = β1ββ3i2 or, β1+β3i2
(ii) 1 + Ο + Ο2 = 0 β 1 + Ο = - Ο2, 1 + Ο2 = - Ο and Ο + Ο2 = -1
(iii) Ο4 = Ο3 β Ο = 1 β Ο = Ο;
Ο5 = Ο3 β Ο2 = 1 β Ο2 = Ο2;
Ο6 = (Ο3)2 = (1)2 = 1.
In general, if n be a positive integer then,
Ο3n = (Ο3)n = 1n = 1;
Ο3n+1 = Ο3n β Ο = 1 β Ο = Ο;
Ο3n+2 = Ο3n β Ο2 = 1 β Ο2 = Ο2.
Property VI: The reciprocal of each imaginary cube roots of unity is the other.
The imaginary cube roots of unity are Ο and Ο2, where Ο = β1+β3i2.
Therefore, Ο β Ο2 = Ο3 = 1
β Ο = 1Ο2 and Ο2 = 1Ο
Hence, we conclude that the reciprocal of each imaginary cube roots of unity is the other.
Property VII: If Ο and Ο2 are the roots of the equation z2 + z + 1 = 0 then - Ο and - Ο2 are the roots of the equation z2 - z + 1 = 0.
Property VIII: Cube roots of -1 are -1, - Ο and - Ο2.
11 and 12 Grade Math
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