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 facetoface in person or online. To enroll in the short course, phone Lee at 7029451294.
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=X1.
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 
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