We do apply math of numbers(Arithmetic) on regular basis in our day to day activities where we are buying any goods, scheduling any even in the day, counting the remaining days for a party etc. This has become integral part of our thinking. I also do the same. Today I took a pause and started thinking about some interesting facts about this Arithmetic and here is what came to my mind.

**1.**Multiply any number by 3 and sum all individual digits of the result to make it single digit, it's always 3 or 9

Example

- 79*3= 237 Sum->2+3+7=12 -> 1+2=3
- 8547*3=25641 Sum->2+5+6+4+1=18 ->1+8=9
- 896512*3=2689536 Sum->2+6+8+9+5+3+6=39 ->3+9=12 ->1+2=3

**2.**Multiply any number with 9 and sum all individual digits of the result to make it single digit, it's always 9.

Example

- 15*9 = 135 Sum-> 1+3+5=9
- 1764*9= 15876 Sum-> 1+5+8+7+6=27 ->2+7=9
- 96862*9=871758 Sum-> 8+7+1+7+5+8=36 ->3+6=9

**3.**Square of any number having last digit as 5 is always ending with 25(square of 5) and the value preceding 25 in result is multiplication of value preceding 5 by its own increment by 1.

Example

- 25*25 =625 -> It's ending with 25 and preceded by 6 which is nothing but 2*(2+1).
- 925*925=855625 -> It's ending with 25 and preceded by 8556 which is nothing but 92*(92+1)
- 4565*4565=20839225 -> It's ending with 25 and preceded by 20839 which is nothing but 456*(456+1)

**4.**Square of a number having all digits as 1 is always starts and end with 1 and the it has always less than one digits of the starting number digits. Also it's an ascending number sequence till middle and then descending number sequence.

Example

- 11*11 =121 ->Its 3 (2*2-1) digits, starts and ends with 1 and 12 is ascending order digits group and 21 is descending digit group.
- 111*111=12321 -> Its 5 (3*2-1) digits, starts and ends with 1 and 123 is ascending order digits group and 321 is descending digit group.
- 11111*11111=123454321 -> Its 9 (5*2-1) digits, starts and ends with 1 and 12345 is ascending order digits group and 54321 is descending digit group.

**5.**Pi is number which is both irrational and transcendental. Out of its known decimal digits, there are no repeating patterns. There is no zero its first 31 decimal digits. Notice yourself

from the below:

- Value of the Pi up to its 31 decimal digits is 3.1415926535897932384626433832795 (no zeros)
- Value of Pi up to its 100 decimal digits is 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679

**6.**Most interesting thing I find in math is about its numeral system. No matter which numeral system we pick e.g. binary, ternary, octal, decimal, hexadecimal, we can apply same principle of mathematical operations just keeping the numeral system base in mind.

Example

Let's try adding two set of numbers (8,9 and 436,753) through different numeral systems.

**Binary**

1000 | 110110100 | ||

+1001 | +1011110001 | ||

------ | ----------- | ||

10001->17(Decimal) | 10010100101 -->1189(Decimal) |

**Ternary**

22 | 121011 | ||

+100 | +1000220 | ||

------ | ----------- | ||

122->17(Decimal) | 1122001-->1189(Decimal) |

**Octal**

10 | 664 | ||

+11 | +1361 | ||

------ | ----------- | ||

21->17(Decimal) | 2245 -->1189(Decimal) |

**Decimal**

8 | 436 | ||

+9 | +753 | ||

------ | ----------- | ||

17 | 1189 |

**Hexadecimal**

8 | 1B4 | ||

+9 | +2F1 | ||

------ | ----------- | ||

11->17(Decimal) | 4A5 -->1189(Decimal) |

The fact I am trying to illustrate here is that I am applying the same mechanism to add numbers one digit at a time starting from right to left and taking the carry overs if applicable. We can perform all other operations such as subtraction, multiplication and division as well. Not to worry about the numeral system. Just need to take care of the basic principle.

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ReplyDeleteJust one small contribution from my side :).

ReplyDeletewhen you multiply numbers ending with 5 and having difference of 10,the result always ends with 75 and the value preceding 75 in result is multiplication of smaller value preceding 5 by its own increment by 2.

eg: 35*45 =1575-> It's ending with 75 and preceded by 15 which is nothing but3*(3+2).

125*135 = 16875-> It's ending with 75 and preceded by 168 which is nothing but 12*(12+2).

Good collection....

ReplyDeleteHere goes few more:

1.A number is divisible by 3 if the sum of its digits is divisible by 3.

For example, 135 is divisible by 3 because the sum of its digits (1 + 3 + 5 = 9) is divisible by 3.

2. To compare two fractions, cross-multiply. The larger product will be on the same side as

the larger fraction.

e.g.

5/6 VS 6/7

Cross-multiplying gives 5 ⋅ 7 vs. 6 ⋅ 6, or 35 vs. 36. Now 36 is larger than 35, so

6/7 is greater then 5/6