At 0:48 the answer x = 3/2 is _not_ wrong. You failed to specify the set over which you want to solve the equation. The sole answer to "solve 16x^4 = 81 over the positive reals" is x = 3/2. The two answers to "solve 16x^4 = 81 over the reals" are x = ±3/2. The four answers to "solve 16x^4 = 81 over the complex numbers" are x = ±3/2, ±3i/2. The "fundamental theorem of algebra" only applies over the field of complex numbers. Over the complex numbers, x^4 = (3/2)^4 has four solutions given by x = 3/2 times the four roots of unity. Over the reals, unity has only two roots; over the positive reals, unity has only one square root. It's as simple as that, as long as you get the _question_ right.
Thanks a lot for your feedback. I appreciate it. If you watched till the end, though, you would have heard when I said that the number of roots you need depends on the particular application for which you solved the equation. You might need all four roots, or just the two real roots, or even just the positive real root. The aim of the video is to shed light on the fact that the solution to that equation is NOT ONLY 3/2.
@@NonsoMaths Oddly enough, I did watch to the end, and saw that you realised that the context of the question matters when deciding on the solutions. It's almost always better to have an understanding of what you expect the answers to be and work within that framework, than to blindly follow an algebraic algorithm, only to eventually discard some possible solutions. If you're trying to find a distance, you know that you only want a positive solution. If you're trying to find a velocity in a given direction, you'll need to include both positive and negative solutions. Your shouldn't be telling your viewers that a correct answer is wrong, when your reason is because you didn't explain the context. So do you now understand why the question you posed initially was deficient?
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x_j=(3/2)[cos(jπ/2)+i*sin(jπ/2)],j=0,1,2,3 & i=√(-1)
At 0:48 the answer x = 3/2 is _not_ wrong. You failed to specify the set over which you want to solve the equation. The sole answer to "solve 16x^4 = 81 over the positive reals" is x = 3/2.
The two answers to "solve 16x^4 = 81 over the reals" are x = ±3/2. The four answers to "solve 16x^4 = 81 over the complex numbers" are x = ±3/2, ±3i/2.
The "fundamental theorem of algebra" only applies over the field of complex numbers. Over the complex numbers, x^4 = (3/2)^4 has four solutions given by x = 3/2 times the four roots of unity.
Over the reals, unity has only two roots; over the positive reals, unity has only one square root. It's as simple as that, as long as you get the _question_ right.
Thanks a lot for your feedback. I appreciate it.
If you watched till the end, though, you would have heard when I said that the number of roots you need depends on the particular application for which you solved the equation. You might need all four roots, or just the two real roots, or even just the positive real root. The aim of the video is to shed light on the fact that the solution to that equation is NOT ONLY 3/2.
@@NonsoMaths Oddly enough, I did watch to the end, and saw that you realised that the context of the question matters when deciding on the solutions.
It's almost always better to have an understanding of what you expect the answers to be and work within that framework, than to blindly follow an algebraic algorithm, only to eventually discard some possible solutions. If you're trying to find a distance, you know that you only want a positive solution. If you're trying to find a velocity in a given direction, you'll need to include both positive and negative solutions.
Your shouldn't be telling your viewers that a correct answer is wrong, when your reason is because you didn't explain the context. So do you now understand why the question you posed initially was deficient?