Divisibility:

If any number ends in zero or five, then that number itself is evenly divisible by five with no remainder which is pretty much obvious.

What is somewhat less obvious however is that If the sum of the digits of any number is evenly divisible by three with no remainder, then the number itself is evenly divisible by three with no remainder.

For example, in the number 9638184, the sum of the digits is 39 which is evenly divisible by three with no remainder which means that the number 9638184 is itself is evenly divisible by three with no remainder.

Thus, 9638184 / 3 = 3212728 exactly with no remainder.

Square root rationality:

If you take the square root of any number that has a finite and odd number of significant digits to the right of the decimal point, that square root will be an irrational number.

Why?

It's because any rational number with a either none or a finite number of significant digits to the right of the decimal point that you square will yield either an integer result with no digits at all to the right of the decimal point or will yield a non-integer result which will have exactly twice the number of significant digits to the right of the decimal point as did the original number. "Twice" is always even, never odd.

Conversely, the square root of a number having an odd number of significant digits to the right its own decimal point cannot have a square root with half of an odd number of significant digits to the right of its own decimal point. Reductio ad absurdum.

Thus:

26.55² = 704.9025 - and - sqrt ( 704.9025 ) = 26.55

The number 26.55 = 531 / 20 which is rational because it is expressible as one integer (531) divided by another integer (20).

Similarly:

96.371² = 9287.369641 - and - sqrt ( 9287.369641 ) = 96.371

In each case, there are twice as many significant digits to the right of the decimal point of the number itself as there are in the square root of that number.

However:

sqrt ( 704.902 ) = 26.5499905838.... which is irrational.

The number itself with three, an odd number, of significant digits to the right of the decimal point yields a square root which is an irrational number whose significant digits go on forever, which is not a repeating decimal and which therefore cannot be expressed as a ratio of two integers.

That does not mean that every number with an even number of significant digits to the right of its decimal point will yield a rational number for the square root, but absolutely no number with an odd number of significant digits to the right of the decimal point will yield a rational number for the square root.

Good stuff for basic understanding of numbers.

Does it have applications in day-to-day electrical/electronics engineering?

BTW, for a good selection of methods for checking integer divisibility:

see http://en.wikipedia.org/wiki/Divisibility_rule

Posted by: george storm | February 04, 2012 at 09:21 AM