I just need a description of the following:

Associative property of Multiplication
Associative Property of Addition
Commutative properties
Distributive Property
Identity property(I don't need to know much about this)
Zero property(i don't need to know much about this)

ADDITION:

associative a+(b+c) = (a+b) + c
commutative a+b = b+a
identity : there is a number 0 with
a+0=0+a = a
inverse a + -a = -a + a = 0

MULTIPLICATION:
associative a(bc) = (ab)c
commutative ab=ba
identity there is a number 1 with
a*1 = 1*a = a
inverse If a not zero then there is an 1/a such that a(1/a) = (1/a)a = 1

BOTH (the biggie!!)
distributive a(b+c) = ab + ac

Associative property of Multiplication:

(x*y)*z = x*(y*z)

Associative property of Addition:

(x + y) + z = x + (y + z)

Commutative properties:

x*y = y*x

x + y = y + x

Distributive Property :

x*(y + z) = x*y + x*z

Identity property: There exists a number 1 such that:

1*x = x for all x.

Zero property: There exists a number 0 such that:

0 + x = x for all x

Inverse properties:

For every x there exists a number -x, such that:

x + (-x) = 0

For every x not equal to zero there exists a number x^(-1), such that:

x*x^(-1) = 1

Examples:

There can only be one 0. Proof suppose there were two numbers 0 and 0' that both satisfy the property that the zero element has to satsify, then:

0 + 0' = 0'

because 0 is a zero element.

But because 0' is also a zero element, you also have:

0 + 0' = 0

This means that 0 = 0'

Example:

(-1)*x = -x

Proof:

Let's check of (-1)*x satisfies the criterium of being the inverse (relative to addition) of x:

x + (-1)*x =

1*x + (-1)*x =

(1 + (-1))*x = (use that -1 is the inverse relative to addition of 1)

0*x = 0

So, we can conclude that (-1)*x is the inverse of x relative to addition, which means that (-1)*x = -x

thank u soooooo much!

thank u soooooooooooooooo much, i really needed to understand this.

Sure! I'd be happy to explain each of these properties to you.

1. Associative property of multiplication:
The associative property of multiplication states that, when multiplying three or more numbers, the grouping of the numbers does not affect the result. In other words, it doesn't matter how you group the numbers when performing the multiplication - the result will be the same. Mathematically, it can be written as: (a * b) * c = a * (b * c). This property is helpful when dealing with long multiplication problems, as it allows you to group the numbers in a way that is more convenient to calculate.

2. Associative property of addition:
Similar to the associative property of multiplication, the associative property of addition states that the grouping of numbers does not affect the sum. This means that when adding three or more numbers, you can change the grouping without changing the result. Mathematically, it can be written as: (a + b) + c = a + (b + c). This property is useful when adding multiple numbers, as you can add them in any order or group them based on preference without affecting the final sum.

3. Commutative properties:
The commutative properties apply to both addition and multiplication and state that the order of the numbers does not affect the result.

- Commutative property of addition: This property states that the order in which you add two numbers doesn't matter. Mathematically, it can be written as: a + b = b + a. For example, 2 + 3 is the same as 3 + 2.

- Commutative property of multiplication: This property states that the order in which you multiply two numbers doesn't matter. Mathematically, it can be written as: a * b = b * a. For example, 4 * 5 is the same as 5 * 4.

These properties are helpful when rearranging terms in a mathematical expression or when simplifying and solving equations.

4. Distributive property:
The distributive property is used when you need to multiply a number by a sum or a difference. It states that you can distribute the multiplication across each term in the sum or difference. Mathematically, it can be written as: a * (b + c) = (a * b) + (a * c). For example, 2 * (3 + 4) is the same as (2 * 3) + (2 * 4), which simplifies to 14.

The distributive property is a fundamental property in algebra and is often used when simplifying expressions or solving equations.

I hope this helps! Let me know if you have any further questions.