Norway | A Nice Algebra Problem | Math Olympiad
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Sqr(x+2)=t, t+i*t=4, t=4/(1+i), ... X+2=+-(8/i) ...
For those who want to check the result (it _is_ surprising),
x + 2 = ±8i ,
so you effectively have
√(8i) + √(-8i) = 4, when x=-2+8i,
and
√(-8i) + √(8i) = 4, when x=-2-8i,
which means you just need to prove one to prove the other value of x works too.
8i
= 8e^(i(π/2))
= 8(cos(π/2) + i*sin(π/2))
√(8i)
= √(8)√(e^(i(π/2)))
= √(8)e^(i(π/2))^(1/2)
= √(8)e^(i(π/4))
= (2√2)(cos(π/4) + i*sin(π/4))
= (2√2)(1/√2 + i/√2)
= 2 + 2i
-8i - r>0 in re^(iθ) always,
so θ must be adjusted.
= -8e^(i(π/2)) = 8e^(i(π/2 - π))
= 8(cos(-π/2) + i*sin(-π/2))
√(-8i)
= √(8)e^(i(-π/2))^(1/2)
= √(8)e^(i(-π/4))
= (2√2)(cos(-π/4) + i*sin(-π/4))
= (2√2)(1/√2 - i/√2)
= 2 - 2i
(2+2i) + (2-2i) = 4
2+2 + 2i-2i = 4
4 + 0 = 4
4 = 4 ✅
Warum nicht alles vorher mit 17,5x multiplizieren?
Gostei muito!
Complex numbers is fake invented math because
(1) the definition of a complex number contradicts to the laws of formal logic, because this definition is the union of two contradictory concepts: the concept of a real number and the concept of a non-real (imaginary) number-an image. The concepts of a real number and a non-real (imaginary) number are in logical relation of contradiction: the essential feature of one concept completely negates the essential feature of another concept. These concepts have no common feature (i.e. these concepts have nothing in common with each other), therefore one cannot compare these concepts with each other. Consequently, the concepts of a real number and a non-real (imaginary) number cannot be united and contained in the definition of a complex number. The concept of a complex number is a gross formal-logical error;
(2) the real part of a complex number is the result of a measurement. But the non-real (imaginary) part of a complex number is not the result of a measurement. The non-real (imaginary) part is a meaningless symbol, because the mathematical (quantitative) operation of multiplication of a real number by a meaningless symbol is a meaningless operation. This means that the theory of complex number is not a correct method of calculation. Consequently, mathematical (quantitative) operations on meaningless symbols are a gross formal-logical error;
(3) a complex number cannot be represented (interpreted) in the Cartesian geometric coordinate system, because the Cartesian coordinate system is a system of two identical scales (rulers). The standard geometric representation (interpretation) of a complex number leads to the logical contradictions if the scales (rulers) are not identical. This means that the scale of non-real (imaginary) numbers cannot exist in the Cartesian geometric coordinate system.
LOL. Just wait until you get to quaternions and octonions!
Hint: math doesn't care what is "logical". You define objects however you want, and then you derive their properties. Some objects turn out to be useful.
@@AlexBesogonov Can your limited iq provide a counter argument to 1,2,3? If not, shushhhhh dork.