Mathematics
You love Maths?. Here are some (random) ideas that Math Lovers will find interesting. Hope you enjoy.
Fibonacci sequence and Geometric sequence
You may have heard of Fibonacci sequence. It is a sequence in which one member is defined as the sum of previous two members. There is an interesting relation between Fibonacci sequence and Geometric sequence.
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Study of Unit Length Circle
While using smartphones you may have noticed an icon indicating ‘system busy’. This is a small circle in which a rotation of a point is animated. If closely watched you may notice that actually there are two points rotating through the border of the circle, one in clockwise direction and other in anticlockwise direction. Naturally while rotating these two points meet several times. While experimenting I reached at some interesting mathematical features of this.
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The period of 1/k for integer k is always < k
The meaning of the term ‘period’ is as follows. Take the case of 1/11 = 0.090909. Here the two digits 0 and 9 are repeated forever. So the period of 1/11 is 2. There is a Theorem about the period which state that the period of 1/k for integer k is always < k. I have a tricky proof for this.
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Single precision float value to string
When doing microcontroller programming one may have come across a task of converting a float value to a string. For example when a float value is to be displayed on LCD or it is to be passed out through a UART port. The simple method is to use sprint() function. This function is exactly same as commonly used printf() function. The only difference is that sprint() pass the output to an array instead of some kind of display.
sprintf() is an extremely simple way to convert float to string. But the main disadvantages are the memory consumption and time for execution. We have developed a very compact and furiously fast code for this conversion.
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Sum of digits of a number
Some of you may be aware that sum of digits of a number always give the remainder produced when the number is divided by 9.
For example sum of digits of the number 29307634 = 2 + 9 + 3 + 0 + 7 + 6 + 3 + 4 = 34 and again the sum of digits 3 + 4 = 7. So when 2307634 is divided by 9 the remainder will be 7. But what about the proof?.
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Sum of first N Odd numbers
There is an interesting property for the sum of first N odd numbers. For example sum of first 3 odd numbers, 1 + 3 + 5 = 9 which is 3^{2}. Take another example: sum of first 5 odd numbers, 1 + 3 + 5 + 7 + 9 = 25 which is 5^{2}.
This means that sum of first N odd numbers is N^{2}.
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