Memorizing the first 100 perfect squares (2022)
The article outlines techniques for memorizing the first 100 perfect squares, emphasizing patterns and tricks over brute force memorization to aid mental calculations and recognize square numbers efficiently.
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The article discusses techniques for memorizing the first 100 perfect squares, which are numbers of the form \( x^2 \) where \( x \) is a positive integer. The author shares personal experiences and insights on the benefits of memorizing these squares, such as aiding in mental multiplication and recognizing square numbers quickly. The article emphasizes that brute force memorization is not necessary; instead, it introduces various tricks and patterns to facilitate learning. Key strategies include recognizing the last digits of squares, using algebraic rules for numbers ending in 5, and applying specific formulas for squares near 50 and 100. The author also highlights the importance of understanding the structural patterns in the decimal representation of squares, which can simplify the memorization process. By leveraging these techniques, one can efficiently reconstruct the table of perfect squares without memorizing each one individually. The article concludes with a systematic approach to filling in the remaining squares based on previously established patterns.
- Memorizing perfect squares aids in mental calculations and recognizing square numbers.
- Various tricks and patterns can simplify the memorization process.
- Understanding the last digits and algebraic rules can help calculate squares efficiently.
- The article emphasizes that brute force memorization is not necessary for learning perfect squares.
- A systematic approach can be used to fill in squares based on established patterns.
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12 commentsFor e.g. 23 x 23, subtract three from the first number, add three to the second, and then add 3² to the product. So 20 x 26 + 9, the idea being that multiplying by a multiple of 10 is easier to do mentally.
But this doesn't always work and you still need to be good at adding/subtracting.
So... 73^2 is 4900 + 9 + 420 = 5329. The really nice part is getting estimates for square roots of numbers.
So, sqrt(3895)? 60^2 + 120n = 3600 + 120n => n=2; that's 3844 (from above); the difference is 51; the residual estimate is then: 62 51/(62*2).
For 27^2, one can just memorize, or: 27-25=2 => 2x100=200, 25-2=23, 23^2=529. 27^2=200+529=729.
As long as one knows square of numbers up to 25, it is done for all up to 100 and more … :-).
There are a several little triplet "patterns" in this first batch that make it easy to this point:
3.1415 926 535 8 979 323 84 626 433.
(10a+5)² = [a * (a+1)][25]
(a+1)² = a² + a + (a+1)
(a-1)² = a² - a - (a-1)
(a+2)² = a² + 4(a+1)
(a-2)² = a² - 4(a-1)
the problem is that I don't know why do this.
nonetheless I can report that I've memorized 3 instances of two consecutive twin primes
11,13,17,19, then 101,103,107,109 (which just raises questions that I can only aspire to ask, nevermined answering, about the what, why, and how of decimal system),
and then 191,193,197,199. the next prime is 211. but the cool thing is how 210 = 2*3*5*7, which are all primes before the first double twin prime
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