6 items are filed under this topic.








Multiplication and repeated addition 
20161123 

From Anandmay: Hello
I was looking closely at early arithmetic where I found how we discovered properties of Arithmetic.
Like:2 x 3 = 3 x 2.
This can be proved by considering a 2D figure(actually,quadrilateral) having length consisting of 2 boxes of
1by1 dimensions and breadth of 3 boxes of the same dimensions.
Now,consider it again,but,this time,length of 3,and breadth of 2 of such 1 by 1 boxes.
We now notice that we can fit the 2 types of rectangles formed on each other precisely.
So the multiplicative property of commutativity is true for all natural numbers as we can generalize the result(in our mind,for self satisfaction).
Now,can you find me a nice satisfactory reason of why a fraction times a natural number equals the number times the fraction?
I mean, for example,i can understand the meaning of 3 x 2/3 to be three times 23rd,that is, 2/3+2/3+2/3.Fair enough.
But here is the problem:By definition and actual meaning of multiplication, a x b means the repeated sum of b,done 'a' times.
So what is the meaning of doing 2/3 x 3?The repeated addition of 3 how many times??2/3 times??Not making sense,right?And even
we have not proved yet the commutative property of numbers INCLUDING fractions.So how can we resolve this problem and make these things meaningful?
Thanks Answered by Penny Nom. 





Associative or commutative? 
20070824 

From Terry: 5*(7*2)=(7*5)*2 Is this associative property or commutative ??? Both? Answered by Penny Nom. 





A binary operation 
20070731 

From sofia: Prove that if * is associative and commutative binary operation on a set S, then
(a*b)*(c*d) = [(d*c)*a]*b
for all a,b,c,d element in S. Assume the associative Law only for triples as in the definition that is, assume only
(x*y)*z = x*(y*z)
for all x,y,z element in S. Answered by Penny Nom. 





Binary operations 
20070730 

From jim: prove or disprove:
Every binary operation on a set consisting of a single element is both commutative and associative.
Answered by Penny Nom. 





A commutative subgroup 
20070503 

From moulipriya: If H be a commutative subgroup of a group G. Then can we say that G is also commutative? Answered by Penny Nom. 





Definitions 
19970908 

From SohoGirl13: I am an 8th grader. my email address is SohoGirl13@aol.com. I have a question: what are the associative, communitive, and distributive properties? Answered by Harley Weston. 


