ALGEBRA Sec. 5 IDENTITY AXIOMS. MathHands.com. IDENTITY AXIOMS: Identities

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1 IDENTITY AXIOMS IDENTITY AXIOMS: Identities It is helpful to recall the definition of a binary operation. As we have stated it, a binary operation is a mixing recipe for mixing two items. We used the color analogy, and we return to it now to establish a meaningful notion of an identity. Identities have nothing to do with identities, in the usual, sense. This is one case where the name of an axiom is not very descriptive. Identities are what they are because of what they do when they are mixed with other numbers. More specifically, they are identities because they do nothing to other numbers when operated with or mixed with. Imagine the color red mixed with a few drops of water. The resulting color will be exactly the same, red. In the world of adding, we will assume the number 0 is the additive identity. We will assume it does nothing when mixed (by addition) with any number or quantity. It should be noted that there are many types of identities and many worlds where identities live, and simply because 0 is the identity in the world of addition, does not mean that it its also the identity in the world of multiplication. ADDITIVE IDENTITY [AId] Up to this section, zero was not officially in our vocabulary, only natural numbers or variables. We take a moment here to meet zero formally and to learn the Additive Identity Axiom. It says that for zero is like water in the world of addition. That is, all numbers remain unchanged when added with zero. Formally, for any quantity, A, A + 0 = A < AId > MULTIPLICATIVE IDENTITY [MId] 0 + A = A < AId > We follow our pattern of delivering axioms in pairs, one for addition, one for multiplication. In the world of multiplication the harmeless, do-nothing element is 1. That is, all numbers remain unchanged when multiplied with 1.Thus we call 1 the multiplicative identity. We accept this as an axiom. Formally, for any quantity, A, A 1 = A < MId > 1 A = A < MId >

2 Some questions to think about 1. (True, False, WDKY (we don t know yet)) Do you know why? explain. 1x = x TRUE by MId 2. (True, False, WDKY (we don t know yet)) Do you know why? explain. 1x = 1x + 0 TRUE by AId 3. (True, False, WDKY (we don t know yet)) Do you know why? explain. 1x = 1(x + 0) TRUE by AId 4. (True, False, WDKY (we don t know yet)) Do you know why? explain. x + 0 = 1(x + 0)

3 TRUE by MId 5. (True, False, WDKY (we don t know yet)) Do you know why? explain. 2 3 = TRUE by MId 6. (True, False, WDKY (we don t know yet)) Do you know why? explain = 2 3 TRUE by MId 7. (True, False, WDKY (we don t know yet)) Do you know why? explain. x 2 = x TRUE by AId 8. (True, False, WDKY (we don t know yet)) Do you know why? explain. 0 = 0 + 0

4 TRUE by AId 9. (True, False, WDKY (we don t know yet)) Do you know why? explain. 0 = 0 + t false (True, False, WDKY (we don t know yet)) Do you know why? explain. 1 = 1 1 TRUE by MId 11. (True, False, WDKY (we don t know yet)) Do you know why? explain. 1 1 = 1 TRUE by MId 12. (True, False, WDKY (we don t know yet)) Do you know why? explain. 1 1 = 1

5 who knows... not by anything we have learned thus far (True, False, WDKY (we don t know yet)) Do you know why? explain. 1 0 = 0 who knows... not by anything we have learned thus far (True, False, WDKY (we don t know yet)) Do you know why? explain. 2 3 = TRUE by Aid 15. (True, False, WDKY (we don t know yet)) Do you know why? explain. (3 + 0) = (3) TRUE by Aid 16. (True, False, WDKY (we don t know yet)) Do you know why? explain. (3 + 0) = 1 (3 + 0)

6 TRUE by Mid 17. (True, False, WDKY (we don t know yet)) Do you know why? explain. 3 1 = 3 TRUE by Mid OR also TRUE by TT 18. (True, False, WDKY (we don t know yet)) Do you know why? explain. x 1 = x TRUE by Mid