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Integer Properties Strategies
Now that we’ve finished talking about all the important properties of integers, we’ll review the strategies on this very important question type. The first and most important strategy when dealing with integers is: make sure you’re dealing with integers. Now this sounds a bit paradoxical, but you see, the test loves to give problems with variables and set straps for students who automatically assume that the variables represent integers.
So yes, you know a lot about integer properties, but don’t go automatically applying all these properties if you’re not guaranteed that the variables of the numbers of the problem are in fact integers. If the question identifies the variables as integers. If they say explicitly x and y are integers or if they use the words even, odd, prime then yes, then we do know that the variables are integers but there has to be some kind of implicit statement.
The second most important strategy when dealing with integers is don’t forget about zeros and negative. People, sometimes they just get positive happy, they think, they hear the word number, they think oh, number, has to be 1, 2, 3, 4, 5, numbers like that, they forget about negatives, they forget about 0, fractions, decimals, 0, et cetera.
You always have to think about all possible categories of numbers. So even if you’re told that it’s an integer, it could be 0 or could be negative. Even an odd numbers could certainly be negative and 0 is an even number. Prime numbers do have to be positive. So if you’re told the number is prime number you do know it is a positive integer.
Also the question only talks about remainders, if all the numbers involved are positive integers. So, that, that’s another case where you know your dealing with positive integers. Remember the many interchangeable ways to talk about factors. 13 is a factor 78, 13 is a divisor of 78, 78 is divisible by 13, 78 is a multiple of 13, 13 is part of the prime factorization of 78.
All of those represent the same fundamental mathematical fact, and the test uses these ideas interchangeably, and expects you to use them in interchangeably as well. For the prime numbers, I recommend memorizing the all the primes below 60. So those are the prime numbers below 60, again it will save you a ton of time if you don’t have to figure out, wait, is 59 a prime number or not?
It’s better just to know that. Of course remember that one is not a prime number it is conspicuously absent from this list. 1 is not a prime number and remember that all prime numbers are odd except for 2. 2 is the lowest prime number and the only even prime number. The test loves to ask about that.
Remember that prime factorizations unlock all kinds of secrets about numbers. Prime factorization is the single most important strategy for unlocking a number of problems dealing with integer properties. If a number larger than 100 appears in any question, chances are very good that finding the prime factorization of the number will help you answer the question.
So, especially if it’s a question asking something about factors or multiples or division or something like that and there’s a big number in the question somewhere, chances are pretty good that if you do the prime factorization it will give you some kind of insight. It essential to memorize the perfect squares of the first 10 integers. And it’s helpful to memorize the perfect squares of 11 through 15 as well.
If you know that you’ll save yourself a lot of time. Remember that of course you can square 0, 0 squared is 0. And remember that a negative squared is positive. So just as positive 9 squared is 81, negative 9 squared is also positive 81. For questions on which you need to find the Greatest Common Factor or Least Common Multiple, remember the procedures we have discussed, as well as that magic formula and of course with that formula remember the dangers associated with it.
You’d never want to perform the multiplication and the numerator first, and then divide. That would be the absolute worst way, the most unintelligent way, to use this formula. You always want to cancel, because of course, the greatest common factor, because it’s a common factor, it will cancel both with p and with q.
You actually have a choice about how you cancel. But definitely cancel before you multiply. Also remember to practice so that you build your intuition. If you’re given small numbers, it’s much more efficient simply to use your intuition than apply a formal procedure. It’s really one of the traps of studying to try and get procedure happy.
Trying to memorize a recipe or a procedure for everything, that is not the path to success on the test, you really have to develop your number sense and your mathematical intuition. For questions about even and odd, remember that you can always use the substitution, 1 equals odd and 2 equals even. If two variables are given remember you have to test, all four cases.
Both even, both odd, and then even and odd either way. And again this is another case where you can use the procedure, but it’s also good to use logic and intuition. That deepens your mathematical thinking. For consecutive numbers, remember that questions about these often appear in variable forms, and you have to recognize that the algebraic expressions represent consecutive numbers.
So for example, if we have an expression like this, we factor out the t, we rearrange the factors and sure enough t-2, t-1 times t, that is the product of three consecutive numbers. But we know, very important this is three consecutive integers, only if we know that t is an integer.
We have to have that guarantee. If t is not an integer, all bets are off and this expression doesn’t have any particular meaning. Finally, for all questions involving a remainder, remember two powerful strategies are, listing possible dividends. For example, listing all the dividends that when divided by 6 could have a remainder of 3, something like that.
And the other one is using the rebuilding the dividend formula which is given here in verbal form. So I’ll say, math is not a spectator sport. You learn math only by doing math, by solving problems. So if your only exposure to integer properties so far has been what has happened in these videos, yes there’s a lot of information in these videos, but you really don’t own it until you solve problems.
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