Hi everyone,
Firstly, thanks very much for providing this forum. I'm grateful for any help & advice you can offer.
I'm reading a book called Algorithms in a Nutshell and I've come across a sum formula that I'm struggling to understand:
From reading the surrounding material, I think I roughly get what he's trying to express, but it's frustrating to look at the formula a hundred different ways and still not be able to understand it.
What I do understand: from reading Wikipedia I think I get that the large sigma means "sum", the letter above, n, is a variable that represents some kind of "to" value, as in, "from 1 to n", the i represents the "to" value, as in "from 1 to n". As I am a programmer, this kind of looks like a for-loop, where you declare a variable, give it an initial value, the increment it until it reaches an end value, then stop incrementing.
Here are my questions:
A. What does it mean to put a fraction like 1/n immediately before a sum? From my very limited knowledge of math, I assume that when you put two values together without an operator between them, it means multiply. But how on earth does one "multiply" a fraction by a sum? Or is it the result of the sum that's multiplied here? So then if n is 3, then it's the sum of 1 + 2 + 3, which is 6. So are we multiplying 1/n by 6?
B. Now I see i placed after the sum. Again, is that a multiply? Are we multiplying everything to the left of n by n? If not, then how do you distinguish between two values next to eachother meaning multiplication vs. meaning something else? Is it some kind of contextual thing? Or are there definite rules for when something is a multiplication and when it's not?
C. Again, is this multiplication, placing n next to (n + 1)?
D. I understand what 1/2 n means, I think. It means n divided by two, right? Because you're multiplying half (0.5) by the number, which is the same as dividing it by 2. But what on earth does 1/2 on its own mean? Half of what? Is there a general rule in maths for dealing with halves on their own? Am I missing some basic obvious rule?
Since I'm struggling with this simple sum / average operation over a set, I'm guessing my math skills are pretty terrible right now. Can anyone recommend an approach to improving as soon as possible? Are there any mental tricks or tactics to improve my learning speed? Or is it, like learning Chinese, simply a matter of having seen thousands of formulas and tried to solve thousands of math problems until eventually you know them all off by heart? I figure I must be missing a lot of rules and also perhaps my mind isn't disciplined enough to really think hard and solve problems.
I'm thinking of completing basic math courses in Khan Academy, to try and get better. However, looking at the sheer number and depth of the courses, and the fact that I've just spent a solid 1-2 hours trying to grasp a simple sum operation, I'm feeling pretty hopeless. Right now it feels like I could quit my day job and spent 20 years trying to get better at math full-time and at the end of it maybe I'd be as good as a high school graduate in the year 2019. Does anyone have any words of hope or comfort?
If you don't have time to answer the questions, I totally understand, but perhaps you can point me to the general areas of math that I really need to improve on, to get good at solving something like this. Or maybe a good book. Or maybe you think Khan Academy will do the job, if I just work hard and focus for a few years?
Very keen to get your thoughts.
Many thanks ahead of time for any words of advice you can offer!
Firstly, thanks very much for providing this forum. I'm grateful for any help & advice you can offer.
I'm reading a book called Algorithms in a Nutshell and I've come across a sum formula that I'm struggling to understand:
From reading the surrounding material, I think I roughly get what he's trying to express, but it's frustrating to look at the formula a hundred different ways and still not be able to understand it.
What I do understand: from reading Wikipedia I think I get that the large sigma means "sum", the letter above, n, is a variable that represents some kind of "to" value, as in, "from 1 to n", the i represents the "to" value, as in "from 1 to n". As I am a programmer, this kind of looks like a for-loop, where you declare a variable, give it an initial value, the increment it until it reaches an end value, then stop incrementing.
Here are my questions:
A. What does it mean to put a fraction like 1/n immediately before a sum? From my very limited knowledge of math, I assume that when you put two values together without an operator between them, it means multiply. But how on earth does one "multiply" a fraction by a sum? Or is it the result of the sum that's multiplied here? So then if n is 3, then it's the sum of 1 + 2 + 3, which is 6. So are we multiplying 1/n by 6?
B. Now I see i placed after the sum. Again, is that a multiply? Are we multiplying everything to the left of n by n? If not, then how do you distinguish between two values next to eachother meaning multiplication vs. meaning something else? Is it some kind of contextual thing? Or are there definite rules for when something is a multiplication and when it's not?
C. Again, is this multiplication, placing n next to (n + 1)?
D. I understand what 1/2 n means, I think. It means n divided by two, right? Because you're multiplying half (0.5) by the number, which is the same as dividing it by 2. But what on earth does 1/2 on its own mean? Half of what? Is there a general rule in maths for dealing with halves on their own? Am I missing some basic obvious rule?
Since I'm struggling with this simple sum / average operation over a set, I'm guessing my math skills are pretty terrible right now. Can anyone recommend an approach to improving as soon as possible? Are there any mental tricks or tactics to improve my learning speed? Or is it, like learning Chinese, simply a matter of having seen thousands of formulas and tried to solve thousands of math problems until eventually you know them all off by heart? I figure I must be missing a lot of rules and also perhaps my mind isn't disciplined enough to really think hard and solve problems.
I'm thinking of completing basic math courses in Khan Academy, to try and get better. However, looking at the sheer number and depth of the courses, and the fact that I've just spent a solid 1-2 hours trying to grasp a simple sum operation, I'm feeling pretty hopeless. Right now it feels like I could quit my day job and spent 20 years trying to get better at math full-time and at the end of it maybe I'd be as good as a high school graduate in the year 2019. Does anyone have any words of hope or comfort?
If you don't have time to answer the questions, I totally understand, but perhaps you can point me to the general areas of math that I really need to improve on, to get good at solving something like this. Or maybe a good book. Or maybe you think Khan Academy will do the job, if I just work hard and focus for a few years?
Very keen to get your thoughts.
Many thanks ahead of time for any words of advice you can offer!
