By Martin Gardner

The next books are a hundred% IDENTICAL:

"Aha! perception through Gardner, Martin". ISBN: 071671017X

&

"Aha! Aha! perception via Gardner, Martin". ISBN: 0894540017

Don't be fooled via the various ISBN's - the covers are exact and flipping via either books, the pages are identical.

As for the e-book itself, i have never had an opportunity but to learn it - i am nonetheless attempting to straighten out the entire duplicates i've got.

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**Additional info for Aha! Aha! Insight**

**Example text**

Then (1) (2) (3) (4) (5) (6) (7) (8) xϩy ϭ y ϩx x ϩ (y ϩ z) ϭ (x ϩ y) ϩ z 0ϩx ϭ xϩ0 ϭ x x ϩ (Ϫx) ϭ (Ϫx) ϩ x ϭ 0 c(x ϩ y) ϭ cx ϩ cy (c ϩ d)x ϭ cx ϩ dx (cd)x ϭ c(dx) 1x ϭ x Commutative Law of Addition Associative Law of Addition Existence of Identity Element for Addition Existence of Inverse Elements for Addition Distributive Laws of Scalar Multiplication over Addition Associativity of Scalar Multiplication Identity Property for Scalar Multiplication In part (3), the vector 0 is called an identity element for addition because 0 does not change the identity of any vector to which it is added.

16. In R3 , the vectors i, j, and k are mutually orthogonal; that is, the dot product of any pair of these vectors equals zero. In general, in Rn the standard unit vectors e1 ϭ [1, 0, 0, . . , 0], e2 ϭ [0, 1, 0, . . , 0], . . , en ϭ [0, 0, 0, . . , 1] form a mutually orthogonal set of vectors. The next theorem gives an alternative way of describing parallel vectors in terms of the angle between them. 3 (see Result 4), and the proof of the other case is similar. 9 Let x and y be nonzero vectors in Rn .

Then cx ϭ |c| x . That is, the length of cx is the absolute value of c times the length of x. 1 is left as Exercise 23 at the end of this section. We have noted that in R2 , the vector cx is in the same direction as x when c is positive and in the direction opposite to x when c is negative, but have not yet discussed “direction” in higher-dimensional coordinate systems. We use scalar multiplication to give a precise deﬁnition for vectors having the same or opposite directions. Deﬁnition Two nonzero vectors x and y in Rn are in the same direction if and only if there is a positive real number c such that y ϭ cx.