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-------------------- 2518 - MATH - Interesting tricks and shortcuts?
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- Science and Math are special to me. Science is the study of reality. It helps me try to understand what the world is really like. Math is the study of patterns. What is amazing to me is how well math is used to describe nature. Was math discovered in nature, or, was it invented to describe nature. Regardless, it is amazing that it works so perfectly.
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- If you multiply two of the same numbers together you will always get the maximum product for their sum.
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--------------- 5 * 5 = 25 their sum is 10. If you take 4 * 6 you get 24, one less than the maximum. If you take 3 * 7 you get 21 which is 4 less than the maximum, or 2^2 less than the maximum. Try this by squaring any number:
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-------------- 10 * 10 = 100
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-------------- 9 * 11 = 99 --------------- one less than max ----- sum is 20
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-------------- 8 * 12 = 96 --------------- two^2 less than max ----- sum is 20
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-------------- 7 * 13 = 91 --------------- three^2 less than max ----- sum is 20
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- So, the square of any number becomes easier to calculate. Here’s how:
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-------------- 13 * 13 = 169
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-------------- 12 * 14 = 168 --------------- one less than max ----- sum is 26
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-------------- 11 * 15 = 165 --------------- two^2 less than max ----- sum is 26
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-------------- 10* 16 = 160 ---------------
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- So, to square 13, multiple 10*16 = 160 , a much easier calculation to do in your head, and, add 3^2, or 9 to get 169.
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-------------- 13 * 13 = 169
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--------------------------- To square 15, 10*20 = 200 + 5^2 = 225
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--------------------------- To square 21, 22 * 20 = 440 + 1^2 = 441
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- This works for the square of any number.
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- Carl Freidrich Gauss was a math genius at 10 years old. When he was in the 4th grade the teacher tried to keep the class busy by asking them to add up all the numbers between 1 and 100. Gauss immediately blurted out the correct answer 5,050.
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- Gauss did it in his head by taking the long string of numbers 1,2,3, …… 99,100 and folding them back on themselves to create pairs of numbers 1+100, 2+99, 3+98, ….. 50+51. It was easy for him to see that the pair always added up to 101 and this happened just 50 times. So, 101 * 50 can be done in your head, it’s 5,050.
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- The sum of any sequence of patterned numbers is n*(n+1) / 2.
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- Where “n” is the last number in the sequence. Summing the numbers up to 100 is:
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-------------------- 100 (101) / 2 = 10,100 / 2 = 5,050
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- If you did the same problem adding up all the numbers from 1 to 500. Now “n” is 500:
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-------------------- 500 (501) / 2 = 250,500 / 2 = 125,250
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- The formula changes a little if you add up the pattern of all the even numbers to the number “n“, 2+4+6+8+10+……… 100. Where n = 100.
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-------------------- n(n+1)
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----------- 100 ( 101 ) = 10,100 , which is the sum of all the even numbers up to 100.
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- The formula changes to n^2, if you add up the pattern of all odd numbers to the number “n”. Adding up all the odd numbers from 1 to 100 = 100^2 = 10,000.
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- An interesting pattern exists between the squares of numbers and the cubes of numbers. It turns out that the sum of all the cubes of numbers up to “n” is the square of the sum of the numbers up to “n”.
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------------- For example: The sum of all the cubes up to 5 is:
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------------- 1^3 + 2^3 + 3^3 + 4^3 + 5^3 =
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------------- 1 + 8 + 27 + 65 + 125 = 225
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------------- And, (the sum of all numbers up to 5) squared is:
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------------- ( 1 + 2 + 3 + 4 + 5 )^2 = 15^2 = 225
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------------- This holds for all numbers up to any number.
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- Another interesting pattern in squaring numbers is that if number ends in 5 the square always ends in 25.
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----------------- (35)^2 = 1225
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----------------- You can do this calculation in your head because what is also true is that the answer is always preceded by the first number multiplied by one higher than that number. The first number of the number 35 which is being squared is 3. The one higher is 4. 3 * 4 = 12. and, the answer always ends in 25 so the answer is 1225. This works for the square of all numbers ending in 5:
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------------------ (35)^2 = 3*4 + 25 = 1,225
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----------------- (55)^2 = 5*6 + 25 = 3,025
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----------------- (75)^2 = 7*8 + 25 = 5,620
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----------------- (105)^2 = 10*11+ 25 = 11,025
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- If your 4th grade teacher asks you how to add up all the numbers in the multiplication table up to 100 here is how you do it: You start with a 10 by 10 matrix of row and columns that contain the multiplication table. The first row is 1 through 10 and you already know that sum to be n(n+1)/2, remember.
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------------ For the first row the sum is 10*11/2 = 55
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------------ For the second row the sum is 55 * 2 = 110
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------------ For the third row the sum is 55*3 = 165
……………………
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------------ For the tenth row the sum is 55 * 10 = 550
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- So, you can see that the sum is going to be 55*1 + 55*2 + 55*3 + ……..
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- And, you can factor out the 55 and write it as 55*(1 +2 +3 +4 ……..)
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- And, you already know what the sum of the second factor is, it’s 55.
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- Therefore, the sum of all the numbers in the multiplication table up to 100 is 55*55.
55^2 is another thing you already know. How to square any number ending in 5.
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------------------------------- 55^2 = 5*6 ….. 25 = 3,025.
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- If you sum up all the numbers in the multiplication table up to 100 you get 3,025.
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- You learned that squaring any number ending in 5 is always 25 preceded by the first digit times one greater and putting it in front of the 25:
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------------------------ 35^2 = 3 * 4 in front of 25 = 12 …. 25 = 1,225
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------------------------ 55^2 = 5 * 6 ………. 25 = 3,025
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- You can do these multiplications in your head.
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- Actually, the rule being applied here works for squaring any number and, in fact, for multiplying any 2 numbers. To use fancy words for a moment, in algebra it is multiplying polynomials together to get polynomials of the second degree. Algebra uses the quadratic equation to solve polynomial equations of the second degree. None of this algebra do you need to learn in order to perform these tricks in multiplication.
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----------------- 55^2 = (50 + 5)(50+5) = 50( 50+5+5) + 5^2 = 50 * 60 + 25 = 3,025.
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- The algebraic way to say this is (a+b)(a+b) = a^2 +ab +ab + b^2 = a(a+b+b) + b^2.
Now that looks complicated but you will be surprised how easy this algebra is to use after some practice.
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--------------- 95^2 = (90 + 5) (90 + 5) = 90 (100) +25 = 9,025
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- This process does not just work with squaring numbers. It works multiplying any two numbers. The idea is to change the multiplication so it contains easy numbers to multiply in your head.
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------------------ 23 * 28 = (20 + 3)( 20 + 8) = 20 * (20+3+8) + 8*3 = 20 * 31 + 24 = 620 + 24 = 644.
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- The general algebraic formula is (a+b)(c+d) = ac +ad + bc + bd. When you make the first number term the same it becomes (a+b)(a+c) = a^2 + ac + ab + bc = a ( a+b+c) + bc.
Remember these letters in algebra are just substitutes to stand for any number. They call them variables. But, once you choose what a letter stands for than the math is fixed for that result. a,b,c in this case are 100, 7,and 11:
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------------------ 107 * 111 = (100 + 7)( 100 + 11) = 100 ( 100 + 7 + 11) + 7*11 =
100 ( 118) + 77 = 11800 + 77 = 11,877
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-------------------94 * 91 = (100 -6 )(100 - 9) = 100 * (100 - 15) + (-6)(-9) = 100 * 85 + 54 = 8500 + 54 = 8,554.
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- It takes some practice, but when you get the hang of it many complicated multiplications can be made simpler using this algebra trick.
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----------------- 62 * 68 = (60 + 2)( 70 - 2) = 4200 + 140 - 120 - 4 = 4200 +20 - 4 = 4216, or even easier:
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----------------- 62 * 68 = (60 + 2)( 60 + 8) = 60 ( 60+2+8 ) + 16 = 60 ( 70) + 16 = 4200 + 16 = 4,216
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- You can quickly multiply a 2-digit number together by adding them and sticking the sum between the two numbers.
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--------------- 23 * 11 = 253
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- If the sum is greater than 9 remember to carry the one:
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-------------- 96 * 11 = 1,056
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- You can do it with numbers having more than 2 - digits by adding the numbers 2 at a time and sticking their sums between the first and last numbers:
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---------------- 423 * 11 = 4, 653
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- There are many tricks you can play with algebra. Here is another one for you teachers:
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- You ask your students to pick any 4-digit number .
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--------------------------- 1,618
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- Ok, now scramble the number to make a different number.
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--------------------------- 8,611
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- Subtract the smaller number from the larger number.
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--------------------------- 6,993
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- Now add up the digits in that number.
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--------------------------- 27
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- If you have a one digit number you are done. If you have a 2 digit number add those up.
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--------------------------- 9
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- The answer you get is 9. Right? The answer you get will always be 9. Trust me.
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----------------------------------- 4362
----------------------------------- 6234
----------------------------------- 1872
----------------------------------- 18
----------------------------------- 9
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----------------------------------- 9721
----------------------------------- 2791
----------------------------------- 7002
----------------------------------- 9
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Practice these tricks and they will become easy. And, you will have confidence in the results. Let me know if you have any questions. Review that comes next teaches you how to predict the day of the week for any date of the year. Your birthday for instance. This is even more interesting. It is called modular math.
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- December 2, 2019 2518 796 798
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--------------------- Monday, December 2, 2019 --------------------
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