5 ms·

You can start explaining fractional powers with roots, e.g. x^0.5

robomartin 5y ago
Right, the problem is that you quickly run into the "miracle occurs" territory. The square root of a number takes us from an area to the length of the side of the square corresponding to that area. The cube root is the same for a cube. What is the 10th root of x? It's a number that, when multiplied by itself ten times equals x. OK. How do you compute this number? The best I can offer at this point is, for simplicity, a brute force search or, for faster results, a bisection search algorithm. In other words, the "and then a miracle occurs" moment is right there. The fact that I can key these numbers into a calculator and get the answer isn't the kind of explanation I want to use for my kid. I don't want to say "once you get here you pick-up your calculator", because the legitimate question then might be "If it's magic, why don't I just pick it up at the start of the problem?" To be clear, I don't mean "miracle" as anything other than "this shit is hard-to-impossible to explain or calculate by hand". That said, you could probably run through a quick bisection search by hand and likely converge on a low error answer in 2 to 5 cycles. The meaning of of the e root of b explained with exponentiation and the exponentiation is explained with the root.
- hdctambien 5y ago
  I think the magic/miracle of math is that you can go from "real world" into "math world" then back into "real world". If a rule is true for c and n and n+1, and you can physically represent the idea when n=2 and n=3, then you can apply that representation theoretically to n>3 to understand ideas that are not easily understandable. The 10th root of x takes you from a measurement of an 10 dimensional object to the measurement of a 9 dimensional object. That's crazy, right? Without needing to "understand" what an 10 dimensional object is, you know something about it because you understand what roots mean with lower values... Of course, that doesn't help you actually calculate the 10th root of x. Is there a better way than basically guess, check, and refine? The calculator is just really fast at doing that (and only needs to calculate a relatively small number of significant digits). Sometimes that's just how math is. The only magic there is that computers are very fast at computation compared to people.
  - hexane360 5y ago
    >The 10th root of x takes you from a measurement of an 10 dimensional object to the measurement of a 9 dimensional object. Doesn't it take you from 10d to 1d? For instance, 10^10 is the hypervolume of a 10-cube with all side lengths = 10.
    - robomartin 5y ago
      Imagine you are trying to explain this to a 15 year old. If math is going to make sense to kids we can't resort to explanations that sound like "and then a miracle occurs". BTW, I am not being critical of your answer. What I am saying is that there are these corners in seemingly simple math that have me scratching my head when it comes to explaining the concepts to a kid in a manner that makes sense and isn't circular. I have yet to find good answers to these questions. Kid: What does the 10th. root of n mean? Dad: It's the number, let's call it x, that, when raise to the 10th power is equal to n Kid: So: n = x * x * x * x * x * x * x * x * x * x? Dad: Yes! You got it! Kid: How do you calculate it? Dad: Well... Kid: What if it is the 10.1 root of n? Dad: Well, that's a little different... Kid: How? Dad: It's the number than when raised to the p-1 power times the base raised to the fractional portion of the power is equal to n Kid: What's the fractional portion? Dad: For the case of p = 10.1, it's 0.1 Kid: x * x * x * x * x * x * x * x * x * x^(p - int(p)) then? Dad: Yeah. Kid: How do I calculate x to the 0.1 power? Dad: Well, you could use your calculator...(now starting to sweat) Kid: How does the calculator do the math. You know, like when the math teacher says "Show your work" Dad: Well, you could use logarithms... Kid: What are logarithms? Dad: A better method could be to use Newton's method. Here: https://en.wikipedia.org/wiki/Newton%27s_method https://en.wikipedia.org/wiki/Newton%27s_method Kid: It says: "start with an initial guess which is reasonably close to the true root, then to approximate the function by its tangent line using calculus, and finally to compute the x-intercept of this tangent line by elementary algebra" Dad: Yes... Kid: I don't know calculus. Is that the only way? I just wanted to understand how to calculate the 10th root of a number? Dad: OK, let's try this. I just threw it together: # Calculate the exp root of n using a binary search # def root_binary_search(n, exp): # Return b, which is the exp root of n # b**exp should be equal to n # min = 0 # For exponents < 1 the max needs to be sufficiently large max = n if exp < 1: while max**exp < n: max *= 2 max_error = 0.00001 while True: b = (max + min) / 2 b_exp = b**exp error = abs(n - b_exp) # print(f"min: {min:15.4f} max: {max:15.4f} b: {b:15.4f} b_exp: {b_exp:15.4f} n: {n:15.4f} error: {error:5.8f}") if error <= max_error: return b else: if b_exp > n: max = b else: min = b # Tests print(root_binary_search(4, 2), f" result should be: {4**(1/2)}") print(root_binary_search(16, 2), f" result should be: {16**(1/2)}") print(root_binary_search(5, 0.1), f" result should be: {5**(1/0.1)}") print(root_binary_search(2, 10), f" result should be: {2**(1/10)}") print(root_binary_search(4, 0.25), f" result should be: {4**(1/0.25)}") Kid: So...you are telling me to guess? Dad: Yeah...? (looking embarrassed) Kid: And to accept an error? 4-squared is 256, not 255.998046875? Dad: Well, you have to understand that with a binary search... Kid: And, did you see what happens if I run this case? print(root_binary_search(4, 1), f" result should be: {4**1}") Kid: Dad? Dad: I have to get back to work. Why don't you ask your math teacher tomorrow?

Right, the problem is that you quickly run into the "miracle occurs" territory. The square root of a number takes us from an area to the length of the side of the square corresponding to that area. The cube root is the same for a cube. What is the 10th root of x? It's a number that, when multiplied by itself ten times equals x. OK. How do you compute this number? The best I can offer at this point is, for simplicity, a brute force search or, for faster results, a bisection search algorithm. In other words, the "and then a miracle occurs" moment is right there. The fact that I can key these numbers into a calculator and get the answer isn't the kind of explanation I want to use for my kid. I don't want to say "once you get here you pick-up your calculator", because the legitimate question then might be "If it's magic, why don't I just pick it up at the start of the problem?" To be clear, I don't mean "miracle" as anything other than "this shit is hard-to-impossible to explain or calculate by hand". That said, you could probably run through a quick bisection search by hand and likely converge on a low error answer in 2 to 5 cycles. The meaning of of the e root of b explained with exponentiation and the exponentiation is explained with the root.

hdctambien 5y ago
I think the magic/miracle of math is that you can go from "real world" into "math world" then back into "real world". If a rule is true for c and n and n+1, and you can physically represent the idea when n=2 and n=3, then you can apply that representation theoretically to n>3 to understand ideas that are not easily understandable. The 10th root of x takes you from a measurement of an 10 dimensional object to the measurement of a 9 dimensional object. That's crazy, right? Without needing to "understand" what an 10 dimensional object is, you know something about it because you understand what roots mean with lower values... Of course, that doesn't help you actually calculate the 10th root of x. Is there a better way than basically guess, check, and refine? The calculator is just really fast at doing that (and only needs to calculate a relatively small number of significant digits). Sometimes that's just how math is. The only magic there is that computers are very fast at computation compared to people.
- hexane360 5y ago
  >The 10th root of x takes you from a measurement of an 10 dimensional object to the measurement of a 9 dimensional object. Doesn't it take you from 10d to 1d? For instance, 10^10 is the hypervolume of a 10-cube with all side lengths = 10.
  - robomartin 5y ago
    Imagine you are trying to explain this to a 15 year old. If math is going to make sense to kids we can't resort to explanations that sound like "and then a miracle occurs". BTW, I am not being critical of your answer. What I am saying is that there are these corners in seemingly simple math that have me scratching my head when it comes to explaining the concepts to a kid in a manner that makes sense and isn't circular. I have yet to find good answers to these questions. Kid: What does the 10th. root of n mean? Dad: It's the number, let's call it x, that, when raise to the 10th power is equal to n Kid: So: n = x * x * x * x * x * x * x * x * x * x? Dad: Yes! You got it! Kid: How do you calculate it? Dad: Well... Kid: What if it is the 10.1 root of n? Dad: Well, that's a little different... Kid: How? Dad: It's the number than when raised to the p-1 power times the base raised to the fractional portion of the power is equal to n Kid: What's the fractional portion? Dad: For the case of p = 10.1, it's 0.1 Kid: x * x * x * x * x * x * x * x * x * x^(p - int(p)) then? Dad: Yeah. Kid: How do I calculate x to the 0.1 power? Dad: Well, you could use your calculator...(now starting to sweat) Kid: How does the calculator do the math. You know, like when the math teacher says "Show your work" Dad: Well, you could use logarithms... Kid: What are logarithms? Dad: A better method could be to use Newton's method. Here: https://en.wikipedia.org/wiki/Newton%27s_method https://en.wikipedia.org/wiki/Newton%27s_method Kid: It says: "start with an initial guess which is reasonably close to the true root, then to approximate the function by its tangent line using calculus, and finally to compute the x-intercept of this tangent line by elementary algebra" Dad: Yes... Kid: I don't know calculus. Is that the only way? I just wanted to understand how to calculate the 10th root of a number? Dad: OK, let's try this. I just threw it together: # Calculate the exp root of n using a binary search # def root_binary_search(n, exp): # Return b, which is the exp root of n # b**exp should be equal to n # min = 0 # For exponents < 1 the max needs to be sufficiently large max = n if exp < 1: while max**exp < n: max *= 2 max_error = 0.00001 while True: b = (max + min) / 2 b_exp = b**exp error = abs(n - b_exp) # print(f"min: {min:15.4f} max: {max:15.4f} b: {b:15.4f} b_exp: {b_exp:15.4f} n: {n:15.4f} error: {error:5.8f}") if error <= max_error: return b else: if b_exp > n: max = b else: min = b # Tests print(root_binary_search(4, 2), f" result should be: {4**(1/2)}") print(root_binary_search(16, 2), f" result should be: {16**(1/2)}") print(root_binary_search(5, 0.1), f" result should be: {5**(1/0.1)}") print(root_binary_search(2, 10), f" result should be: {2**(1/10)}") print(root_binary_search(4, 0.25), f" result should be: {4**(1/0.25)}") Kid: So...you are telling me to guess? Dad: Yeah...? (looking embarrassed) Kid: And to accept an error? 4-squared is 256, not 255.998046875? Dad: Well, you have to understand that with a binary search... Kid: And, did you see what happens if I run this case? print(root_binary_search(4, 1), f" result should be: {4**1}") Kid: Dad? Dad: I have to get back to work. Why don't you ask your math teacher tomorrow?