BC - Mental Mathematics
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Amazing Mathematics

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The present purpose is to discuss a fun ,practical, and very  useful subject:  “Amazing Mathematics”.  This subject is also known as “Mental Mathematics”,  “Math Gymnastics”, or “Math Tricks”.  Soon, you will not only be impressing yourself but also your friends.


 This is a free brief course.  It  summarizes the associated fee based short course which is conducted  face-to-face in person or online.  To enroll in the short course, phone Lee at 702-945-1294.


 I’m Lee Hayden and I commit  to making this a course of high quality and exceptionally beneficial to you.






We shall start slow and build a firm foundation.  Quite frequently, it is more important to grasp a principle of operation and not the end result.  For example, consider the following products:


 









Note that the product consists of two single digit numbers that sum to 9.  Thus, 9*3 = 27 and 2 + 7 = 9.  Take 3 and lower it by -1 and get 2.  We need another number such that 2 plus this other number equals 9.  In this case, the other number is 7, i.e., 2 + 7 = 9.  Remove the + sign to get 27.  Then, the product 9*3=27.


The principle shown in Table 1:  The product of the number 9 and a single digit number X  is a  result consisting if two numbers m and n such that the sum of m and n is 9  and m as the first number is  m=X-1.


Can you guess the principle used in Table 2?
















In Table 2, consider the multiplication of a two digit number by 11:   Thus, 11*43=473.  To perform this multiplication mentally, image the 4 and the 3 at the ends of the product.  Add the 4 and 3 to obtain 7; then place the 7 in the middle between the 4 and 3 to obtain 473.  These operations are shown in yellow.


Consider the multiplication 11*64=704.  Mentally place the 6 and 4 at the ends.  Then, add 6 and 4 to obtain 10.  The number 10 consists of two digits instead of one digit and thus can not be placed between the 6 and 4.  Instead, place the 0 between the 6 and 4 and carry the one  onto the 6 to obtain 704.  These operations are shown in green.

 


In Table 3 below, consider the multiplication of 3, 4, and 6 digit numbers by 11.  Thus, consider 11* 123=1353. Mentally place the 1 and 3 at opposite ends and then, from right to left, add 2 and 3 to obtain 5, and add 2 and 1 to obtain 3. Produce the answer from left to right as 1353.  The multiplications in yellow to not require the carry of one to the next digit to the left. The multiplications in green require the carry of one to the next digit to the left.

























Consider Table 4. Squaring two numbers ending in 5.  Note that each result ends in the two digit number 25.  Take the first number m and increase  it by  +1 to get n = m +1.  Compute m^n and tack on the number 25 at the end..  For 45^2 = 2025, m =4 and n=4+1=5.  Thus, 45^2 =4*5 = 20 with 25 placed at the end. Finally, 20 with the 25 placed at the end is 2025.


Mathematically, Table 4 results can be obtained as follows:


The equation (a + b)(a -b) = a^2 - b^2 can be written as


a^2 = b^2  + (a + b)(a -b).  If we are squaring two digit number ending in 5 as in Table 4, then let b =5.  Thus,,


a^2  = 25 + (a + 5)(a - 5).


For example, assume we want 45^2, where a = 45.  Making the calculation:


a^2 = 25 + (45 + 5)(45 - 5) = 25 + 50*40 = 25 + 2000 = 2025





Consider Table 5. Multiplying pairs of  two digit numbers where each of the two numbers has the same firstl digit and the sum of the second digits is 10. If m is the first digit, n is the second digit of the first number, p is the second digit of the second number. Compute m*(m+1) and n*p.  The product of the two numbers is m*(m+1) with n*p appended.


For example, on our calculator, 88*82= 7216.  For our mental calculation, m=8 and m*(m+1)= 8*9=72.  Additionally, n*p=8*2=16.  Thus, 88*82= 72 with 16 appended or 7216.


Interesting Number Facts


The number 1 has a unique personality. It I the only number which remains the same when raised to any power or when any root is taken.  No matter how many time you multiply 1 by itself, the result will always be 1 and that can be said of no other number.


The number 2 is the only even number that is also a prime number.  A prime number, by definition, is any number without factors; it is divisible only by itself and 1, so no even number exceapt 2 can ever be prime.


Also note that 2 and 6 are the only numbers whose factors, when multiplied, equal their sum.

2 + 2 = 4 and 2 x 2=4

1 + 2 + 3 = 6 and 1 X 2  X 3  = 6




Some interesting number arrangements,


1 + 2 = 3

4 + 5 + 6 = 7+ 8

9 + 10 + 11 + 12 = 13 + 14 + 15

16 + 17 + 18 + 19 + 20 = 21  + 22 + 23 + 24

25 + 26 + 27 + 28 + 29 + 30 = 31 + 32 + 33 + 34 + 35

36 + 37 + 38 + 39 + 40 + 41 + 42 = 43 + 44 + 45 + 46 + 47 + 48

49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 57 + 58 + 59 + 60 +61 + 62 + 63

Notice that row n  begins with 2^n.



Fast Multiplication .


Multiplication, a*b, where a and b are single digit numbers.


Multiplication of two numbers where each number is a single digit is given by  the basic multiplication table, learned in elementary school.







Multiplication, a*b, where a is a single digit and b is two digits.



Multiplication, a*b, where a and b are two digit numbers.



Multiplying two numbers is very easy  when each of the two numbers are both two digits. Mathematically, the product  x*y is to be computed where .x and y are both two digits. and such multiplication can be formed in the head without resorting to paper.  


The learning process begins by learning to multiple 11* ab.


Multiplying two digits , a and b by 11, I.e,  11*ab.

If a + b <10, then

11*ab = a(a+b)b

11*ab = axb, where x = a + b

If a + b > 9,  then

11*ab = (a +1)(a +b)b

11*ab = (a+1)xb, where x = last digit of a + b



Table 1. Multiplication of single digit by 9




9*1=09


9*6=54

9*2=18


9*7=63

9*3=27


9*8=72

9*4=36


9*9=81

9*5=45



Table 2. Multiplication of Two Digits By 11




11*11=121

11*41=451

11*61=671

11*12=132

11*42=462

11*62=682

11*13=143

11*43=473

11*63=693

11*14=154

11*44=484

11*64=704

11*15=165

11*45=495


11*16=176

11*46=506


11*17=187



11*18=198



11*19=209



Table 4. Squaring Two Digit Numbers Ending in 5


15^2 = 225

25^2 = 625

35^2 =1225

45^2 = 2025

55^2 = 3025

65^2 =4225

75^2=5625

85^2=7225

95^2=9025

Table 3. Multiplication Of 2, 4, and 6 Digit Numbers By 11




11*123=1353

11*1234=13574

11*123456=1358016

11*333=3663

11*3333=36663

11*333333=3666663

11*444=4884

11*4444=48884

11*444444=4888884

11*555=6105

11*5555=61105

11*555555=6111105

11*999=10989

11*9999=109989

11*999999=10999989













Table 5.





81*89=7209

21*29=609

41*49=2009

71*79=5609

82*88=7216

22*28=616

42*48=2016

72*78=5616

83*87=7221

23*27=621

43*47=2021

73*77=5621

84*86=7224

24*26=624

44*46=2024

74*76=5624

85*85=7225

25*25=625

45*45=2025

75*75=5625

86*84=7224

26*24=624

46*44=2024

76*74=5624

87*83=7221

27*23=621

47*43=2021

77*73=5621

88*82=7216

28*22=616

48*42=2016

78*72=5616

89*81=7209

29*21=609

49*41=2009

79*71=5609