A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which eighth-grade students have to locate the square root of 10 on a number line.<br /><br /><br /><br />The square root of 10 is an irrational number, meaning it cannot be represented as a fraction: its decimals do not repeat.<br /><br />On a calculator, it comes out to 3.1622776601683793319988935444327…..<br /><br />That is just a little greater than 3 but less than 4, and since only one of the choices on the number line is higher than 3 (to the right of 3) and lower than 4 (to the left of 4), that one has to be the answer.<br /><br />The other dots are at about 0.9, which is less than 1 and therefore can’t be the answer; 2.6, which is less than 3, not greater than 3; and at 5, which is the square root of 25, not of 10.<br /><br />Use our online resource library at VoxLearn.org, select one of the math collections and enter the search terms “estimate square roots” if you are having trouble understanding why the number used as the square root of 10 is the correct number to use.<br /><br />Although many of these lessons will give you sets of practice problems, we haven’t found one that takes you through the preparation steps:<br /><br />First, memorize all the perfect squares from 1 to 15, and then memorize those for the 5’s after that. Here they are:<br /><br /><br />Number Perfect Square<br />1 1<br />2 4<br />3 9<br />4 16<br />5 25<br />6 36<br />7 49<br />8 64<br />9 81<br />10 100<br />11 121<br />12 144<br />13 169<br />14 196<br />15 225<br />20 400<br />25 625<br
Since 3 is a factor of 18 and 18/3=6, then 18=3x6. We need only test the smaller number, 3, to see if it is a factor of 18 and not 6. Similarly, 5 is a factor 20 and 20=5x4 so we need only test the smaller number 4 as a factor. But what if we don't know the factors of a number? So, testing a number to see if it is prime means we need only test the "smaller" factors. But where do smaller factors stop and larger factors start? The principle here is:
Suppose one number is a factor of N and that it is smaller than the square-root of the number N. Then the second factor must be larger than the square-root.
We can see this with the examples above: For 18, we need only test numbers up to the square-root of 18 which is 4.243, ie up to 4! This is much quicker than testing all the numbers up to 17!!
For 25, we need to test numbers up to and including the square-root of 25, which is 5.
And for 37, we need only go up to 6 (since 6x6=36 so the square-root of 37 will be only a little bit larger).
Which numbers to test as factors
So, putting these two shortcuts together, we need only test those prime numbers up to 6 to see if they are factors of 37. If any are, the number is not prime (it is composite) and if none of them are, then the only factors would be 1 and 37 and 37 would be prime.
The numbers to test are therefore:
and 37 is not even, so 2 is not a factor of 37
and 37/3 is 12.3333 so 3 does not divide exactly into 37 either
and 37 does not end with 0 or 5 so 5 is not a factor of 37
7 is the next prime,
but it is bigger than the square-root of 36, so we can stop now.
So 37 has no factors (except 1 and 37 of course) and therefore
37 is a prime number.
Can you... ...extend the list of prime numbers above up to 100?
[Hint: since 10x10=100, you only need to know the primes up to 10: 2,3,5 and 7 and only use these as test divisors in ord