The Mathematics Thread

[Max]

Is anybody else interested in mathematics? In this thread, we can discuss any math related topic.

I will begin by covering some topics in number theory (the study of integers). Here is a list of divisibility rules for every number from 1 to 10 (other than 7):

1 - Every number is divisible by 1

2 - A number is divisible by 2 if it is even (i.e., if it ends in a 0, 2, 4, 6, or 8)

3 - A number is divisible by 3 if the sum of that number's digits is divisible by 3 (e.g., 789 is divisible by 3 because 7 + 8 + 9 = 24, which is divisible by 3)

4 - A number is divisible by 4 if its last two digits are divisible by four (e.g., 752 is divisible by 4, but 206 is not)

5 - A number is divisible by 5 if its last digit ends in 0 or 5

6 - A number is divisible by 6 if it is divisible by both 2 and 3

8 - This is not a terribly helpful rule, but a number is divisible by 8 if its last 3 digits are divisible by 8 (e.g., 4,064 is divisible by 8 but 4,564 is not)

9 - A number is divisible by 9 if the sum of that number's digits is divisible by 9 (e.g., 198 is divisible by 9 because 1 + 9 + 8 = 18, which is divisible by 9)

10 - A number is divisible by 10 if it ends in a 0

There is no quick divisibility rule for 7, so dividing by 7 is the fastest way to test for divisibility.

Closely related to the concept of divisibility are prime numbers. A common (but erroneous) definition of a prime number is that it is a (positive) number that is only divisible by 1 and itself. The prime numbers between 1 and 100 (inclusive) are the following: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

1 itself is not a prime number, even though it is divisible only by 1 and itself. That's because the real definition of a prime number is a (positive) number that is divisible by exactly two numbers. While 1 is divisible by itself, itself is 1, so 1 is only divisible by one number.

[allmc2008]

1+1=4

[quartermainefan]

I was a math whiz in school, but not complicated math theories with geometry and calculus. I was able to multiply 3 digits by 3 digits in my head at one point. That is an ability you lose because it has no use and therefore never gets used, but I can still do addition with multiple digits in my head. Division was always a weak spot for me but I can get a rough division in my head for multiple digits via rounding. Prime numbers I never got why people found them so interesting. They are prime, but now that I know they are prime I cannot think of any way to make use of that knowledge.

[Max]

That's really impressive how you can do all those calculations in your head, Qfan.

I honestly am having trouble thinking of a reason why one would need to use prime numbers in the real world.

Here's some more useless stuff:

Behavior of Odd and Even Integers:

Odd + (or -) Odd = Even

Even + (or -) Even = Even

Odd + (or -) Even = Odd

When adding (or subtracting) multiple integers, the sum (or difference) will be odd if an odd number of numbers is odd; it will be even if an even number of numbers is odd.

Odd * Odd = Odd

Even * Even = Even

Odd * Even = Even

When multiplying multiple integers, the product will be even if just one of the numbers is even; otherwise, it will be odd.

Odd/Odd: Could result in a non-integer or an odd integer

Even/Even: Could result in a non-integer, an even integer, or an odd integer.

Odd/Even: Will always result in a non-integer.

Even/Odd: Could result in a non-integer or an even integer.

Behavior of Positive and Negative Numbers (including positive and negative non-integers):

Positive + Positive = Positive

Negative + Negative = Negative

Positive - Negative = Positive

Negative - Positive = Negative

(The other result of the other combinations, Positive + Negative, Positive - Positive, Negative + Positive, and Negative - Negative, depend upon the absolute value of what is being added or subtracted.)

Positive * Positive = Positive (true for division as well)

Negative * Negative = Positive (true for division as well)

Positive * Negative = Negative (true for division as well)

When multiplying multiple numbers, the result will be positive if there are an even number of negative numbers in the operation, and will be negative if there are and odd number of negative numbers in the operation. (The exception to this rule is if zero is among the many numbers in the multiplication operation; then the product will be zero.)