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EXAMPLE 54 The equivalence class [(4, 12)] contains all of the pairs (4, 12), (1, 3), ( 2, 6) (Of course it contains in nitely many other pairs as well) This equivalence class represents the fraction 4/12 which we sometimes also write as 1/3 or ( 2)/( 6) If [(a, b)] and [(c, d)] are rational numbers then we de ne their product to be the rational number [(a c, b d)] This is well de ned (unambiguous), for the following reason Suppose that (a, b) is related to (a, b) and (c, d) is related to (c, d) We would like to know that [(a, b)] [(c, d)] = [(a c, b b)] is the same equivalence class as [(a, b)] [(c, d)] = [(a c, b d)] In other words we need to know that (a c) (b d) = (a c) (b d) But our hypothesis is that a b =a b and c d =c d (52)
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Multiplying together the left sides and the right sides we obtain (a b) (c d) = (a b) (c d) Rearranging, we have (a c) (b d) = (a c) (b d) But this is just Eq 52 So multiplication is unambiguous EXAMPLE 55 The product of the two rational numbers [(3, 8)] and [( 2, 5)] is [(3 ( 2), 8 5)] = [( 6, 40)] = [( 3, 20)] This is what we expect: the product of 3/8 and 2/5 is 3/20 If q = [(a, b)] and r = [(c, d)] are rational numbers and if r is not zero (that is, [(c, d)] is not the equivalence class zero in other words, c = 0) then we de ne
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the quotient q/r to be the equivalence class [(ad, bc)] We leave it to you to check that this operation is well de ned EXAMPLE 56 The quotient of the rational number [(4, 7)] by the rational number [(3, 2)] is, by de nition, the rational number [(4 ( 2), 7 3)] = [( 8, 21)] This is what we expect: the quotient of 4/7 by 3/2 is 8/21 How should we add two rational numbers We could try declaring [(a, b)] + [(c, d)] to be [(a + c, b + d)], but this will not work (think about the way that we usually add fractions) Instead we de ne [(a, b)] + [(c, d)] = [(a d + b c, b d)] That this de nition is well de ned (unambiguous) is left for the exercises We turn instead to an example EXAMPLE 57 The sum of the rational numbers [(3, 14)] and [(9, 4)] is given by [(3 4 + ( 14) 9, ( 14) 4)] = [( 114, 56)] = [(57, 28)] This coincides with the usual way that we add fractions: 9 57 3 + = 14 4 28
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Notice that the equivalence class [(0, 1)] is the rational number that we usually denote by 0 It is the additive identity, for if [(a, b)] is another rational number then [(0, 1)] + [(a, b)] = [(0 b + 1 a, 1 b)] = [(a, b)] A similar argument shows that [(0, 1)] times any rational number gives [(0, 1)] or 0 By the same token the rational number [(1, 1)] is the multiplicative identity We leave the details for you Of course the concept of subtraction is really just a special case of addition [that is, a b is the same as a + ( b)] So we shall say nothing further about subtraction
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In practice we will write rational numbers in the traditional fashion: 2 19 22 24 , , , , 5 3 2 4 In mathematics it is generally not wise to write rational numbers in mixed form, such as 2 3 , because the juxtaposition of two numbers could easily be mistaken for 5 multiplication Instead we would write this quantity as the improper fraction 13/5 De nition 51 A set S is called a eld if it is equipped with a binary operation (usually called addition and denoted + ) and a second binary operation (usually called multiplication and denoted ) such that the following axioms are satis ed: A1 S is closed under addition: if x, y S then x + y S A2 Addition is commutative: if x, y S then x + y = y + x A3 Addition is associative: if x, y, z S then x + (y + z) = (x + y) + z A4 There exists an element, called 0, in S which is an additive identity: if x S then 0 + x = x A5 Each element of S has an additive inverse: if x S then there is an element x S such that x + ( x) = 0 M1 S is closed under multiplication: if x, y S then x y S M2 Multiplication is commutative: if x, y S then x y = y x M3 Multiplication is associative: if x, y, z S then x (y z) = (x y) z M4 There exists an element, called 1, which is a multiplicative identity: if x S then 1 x = x M5 Each nonzero element of S has a multiplicative inverse: if 0 = x S then there is an element x 1 S such that x (x 1 ) = 1 The element x 1 is sometimes denoted by 1/x D1 Multiplication distributes over addition: if x, y, z S then x (y + z) = x y + x z Eleven axioms is a lot to digest all at once, but in fact these are all familiar properties of addition and multiplication of rational numbers that we use every day: the set Q, with the usual notions of addition and multiplication (and with the usual additive identity 0 and multiplicative identity 1), forms a eld The integers, by contrast, do not: nonzero elements of Z (except 1 and 1) do not have multiplicative inverses in the integers Let us now consider some consequences of the eld axioms
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