Algebra, ring question with even integers

• jasonfarley89
So it would be a * (b + c) = (a(b + c))/2 = (ab + ac)/2. Does that make sense? In summary, the conversation is about defining a new multiplication operation on the set of even integers, where the operation is defined as a*b = ab/2 using normal multiplication. The discussion also covers the associative and distributive properties of this operation, with the correct equations being (a*b)*c = abc/4 and a*(b+c) = (ab + ac)/2 respectively.
jasonfarley89
We have E the set of even integers with ordinary addition Define new multiplication * on E defined as

a*b = ab/2 where on the right hand side of the equation is just normal multiplication.

I am just a bit confused i am trying to show Associative multiplication meaning i have to show

(a*b)*c = a*(b*c)

when i do (a*b)*c am i supposed to get (ab/2)(c/2) ?? i feel like i am just doing something really stupid here

Can someone explain? Its binary operation so it can only take two elements at a time right?

also i am bit confused with distributive law,

so a*(b+c) = a(b+c)/2? i want to be equal to (ab)*(ac) = ab(ac)/2 right?

Last edited:
should it be that (a*b)*c = (ab/2) *c = ab/2(c/2) so i have ab(c/2)/2 = (abc/2)/2 = abc/4? that correct?

Yes, that's correct, although I might parenthesize it a little differently: $$(a * b) * c = (ab/2) * c = ((ab/2)c)/2 = abc/4$$.

For the distributive law, you want $$a * (b + c) = a * b + a * c$$, not $$a * (b + c) = ab * ac$$.

1. What is algebra?

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations. It involves the use of letters and symbols to represent numbers and unknown quantities.

2. What is a ring in algebra?

In algebra, a ring is a mathematical structure that consists of a set of elements and two operations, addition and multiplication. The operations must follow certain rules, such as closure, associativity, and distributivity, for the set to be considered a ring.

3. What are even integers?

Even integers are whole numbers that are divisible by 2. They can be represented as 2n, where n is any integer. Examples of even integers include 2, 6, -10, and 100.

4. How do you solve a ring question with even integers?

To solve a ring question with even integers, you can use the properties of rings, such as closure, associativity, and distributivity. For example, to solve an equation involving even integers, you can use the distributive property to simplify the expression and then solve for the unknown variable.

5. What are some real-life applications of algebra and rings with even integers?

Algebra and rings with even integers are used in various fields, such as engineering, physics, and computer science. They can be used to model and solve real-world problems, such as calculating forces in a mechanical system or encrypting data in computer networks.

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