advanced numerical factoring
summary
the content focuses on leveraging algebraic factoring techniques, particularly the difference of two squares, to elegantly solve problems involving the prime factorization of large numbers and the simplification of decimals.
- prime factorization is crucial for understanding the number of factors a number has and for solving various mathematical problems.
- the difference of two squares is a powerful technique that simplifies the factorization of large numbers that would otherwise be difficult to factor.
- this method not only aids in finding prime factorizations of large numbers but also in simplifying decimals just less than one.
- practical examples include factoring numbers like 1599, 2491, and 9975 using the difference of squares to reveal their prime factors.
- the technique is also applicable in simplifying decimal expressions, demonstrating its versatility and utility in solving a range of problems.
chapters
00:00
understanding prime factorization
00:35
the elegance of difference of squares
04:08
simplifying decimals using difference of squares
q: for the practice problem around 5:03, how does mike come up with (1-000049)/(1-0.007) as a substitute for 0.999951/0.993? what's the best way to rewrite a number as "1 minus something?"
a: to be honest, he probably just used a calculator. :)
i know we discourage using the calculator too much, but there is definitely a time and place where it's appropriate to use. this is one of those cases! any time you see a number with lots and lots of decimals -- and no easy way to convert it to a fraction -- then the calculator should be your go-to option!
so, to get the 1 minus a number, you just subtract the original numbers from 1, like so:
1-0.999951 = 0.000049
so, 0.999951 = 1 -0.000049
