# CAN ANY ONE THEORETICALLY PROVE THAT - * - IS +?????I actually noe the answer . i am just expecting some new methods. many people give logicall proofs. please give me theoritical proofs...

beckden | Certified Educator

We are going to use the following definitons

0 is the additive identity and 0 + a = a + 0 = a

For any number a there exists an additive inverse denoted by -a, which has the property that a + -a = -a + a = 0

(c) Multiplicative identity property
1 is the multiplicative identity and 1 * a = a * 1 = a

(d) Multiplicative property of 0
For any a   0 * a = a * 0 = 0

(d) Distributive property
For any three numbers a, b and c, a(b + c) = ab + ac

For any a, b and c if a = b then a + c = b + c

(f) Transitive property
For any a, b, and c.  If a = c and b = c then a = b.

Lemma : -a = (-1)a

a + -a = 0    Inverse property of addition

a* 0 = 0       Multiplicative property of zero

1 + -1 = 0   Inverse property of addition

a (1 + -1) = 0  because 0 = 1 + -1

a + (-1)a = 0 because of the distributive property

a + -a + (-1)a = -a  because of the additive property of equality

0 + (-1)a = -a    because of the inverse property of equality

(-1)a = -a because of the identity property of addition

By definiton

b + -b = 0   Because of the definiton of additive inverse

-a * 0 = 0   By the multiplicative property of 0

-a (b + -b) = 0   By the transitive property

-a(b) + -a(-b) = 0  By the distributive property.

ab + -ab + (-a)(-b) = ab  By the additive property of equality

0 + (-a)(-b) = ab     By the additive inverse property.

(-a)(-b) = ab    By the additive identity property