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Expression CxDxT7 2CxDxT6 + QxT4 CxDxT5 + 2QxT6 + 3BxCxT7 2CxDxT4 + QxT5 + 6(BxCx + AxDx)T6 + 2PxT7 CxDxT3 + 2QxT4 + 3BxCxT5 + 4PxT6 + 5AxBxT7 2CxDxT2 + QxT3 + 6BxCxT4 + 2PxT5 + 10AxBxT6 + 3A2T7 x CxDxT1 + 2QxT2 + 3BxCxT3 + 4PxT4 + 5AxBxT5 + 6A2T6 x QxT1 + 6(BxCx + AxDx)T2 + 2PxT3 + 10AxBxT4 + 3A2T5 x 3BxCxT1 + 4PxT2 + 5AxBxT3 + 6A2T4 x 2PxT1 + 10AxBxT2 + 3A2T3 x 5AxBxT1 + 6A2T2 x A2T1 x

The number systems that we use in calculus are the natural numbers, the integers, the rational numbers, and the real numbers Let us describe each of these:

(1) h10 (1) 11 (1) 12

(2) 6 (2) 7 (2) 8 (2) 9

6- 3 Two ships are traveling at right angles The first ship, traveling at 8m/s , crosses the path of the second ship when it is 1OOOm away (from the point where the paths cross) and traveling at 6 m/s What are their positions, separation, and rate of separation 300 s after their paths cross

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The natural numbers are the system of positive counting numbers 1, 2, 3, We denote the set of all natural numbers by N The integers are the positive and negative whole numbers: , 3, 2, 1, 0, 1, 2, 3, We denote the set of all integers by Z The rational numbers are quotients of integers Any number of the form p/q, with p, q Z and q = 0, is a rational number We say that p/q and r/s represent the same rational number precisely when ps = qr Of course, you know that in displayed mathematics we write fractions in this way:

Expression CyDyT7 2CyDyT6 + QyT4 CyDyT5 + 2QyT6 + 3ByCyT7 2CyDyT4 + QyT5 + 6(ByCy + AyDy)T6 + 2PyT7 CyDyT3 + 2QyT4 + 3ByCyT5 + 4PyT6 + 5AyByT7 2CyDyT2 + QyT3 + 6ByCyT4 + 2PyT5 + 10AyByT6 + 3A2T7 y CyDyT1 + 2QyT2 + 3ByCyT3 + 4PyT4 + 5AyByT5 + 6A2T6 y QyT1 + 6(ByCy + AyDy)T2 + 2PyT3 + 10AyByT4 + 3A2T5 y 3ByCyT1 + 4PyT2 + 5AyByT3 + 6A2T4 y 2PyT1 + 10AyByT2 + 3A2T3 y 5AyByT1 + 6A2T2 y A2T1 y

7 1 2 + = 2 3 6

(2) h10 (2) 11 (2) 12

TABLE C6 h(3) i h(3) 1 h(3) 2 h(3) 3 h(3) 4 h(3) 5 h(3) 6 h(3) 7 h(3) 8 h(3) 9

The real numbers are the set of all decimals, both terminating and nonterminating This set is rather sophisticated, and bears a little discussion A decimal number of the form

Expression CxDyT7 2CxDyT6 + ZxT7 CxDyT5 + 2ZxT6 + WxT7 2CxDyT4 + ZxT5 + 2WxT6 + VxT7 CxDyT3 + 2ZxT4 + WxT5 + 2VxT6 + UxT7 2CxDyT2 + ZxT3 + 2WxT4 + VxT5 + 2UxT6 + 3AxAyT7 CxDyT1 + 2ZxT2 + WxT3 + 2VxT4 + UxT5 + 6AxAyT6 ZxT1 + 2WxT2 + VxT3 + 2UxT4 + 3AxAyT5 WxT1 + 2VxT2 + UxT3 + 6AxAyT4 VxT1 + 2UxT2 + 3AxAyT3 UxT1 + 6AxAyT2 AxAyT1

on an x-y coordinate system with the first ship going in the y direction and the second ship going in the x-direction Figure 6-3 is for t = 0 , the time when the ships cross paths The drawing helps to visualize the problem The position of the first ship at any time t is y = (8m/s)t d y - 8 y t s dt

(3) 11

x = 316792

(4) 1

is actually a rational number, for it represents x = 316792 = A decimal number of the form m = 427519191919 , with a group of digits that repeats itself interminably, is also a rational number To see this, notice that 100 m = 427519191919 and therefore we may subtract: 100m = 427519191919 m= Subtracting, we see that 99m = 423244 or m= 423244 99000 4275191919 316792 100000

Expression CyDxT7 2CyDxT6 + ZyT7 CyDxT5 + 2ZyT6 + WyT7 2CyDxT4 + ZyT5 + 2WyT6 + VyT7 CyDxT3 + 2ZyT4 + WyT5 + 2VyT6 + UyT7 2CyDxT2 + ZyT3 + 2WyT4 + VyT5 + 2UyT6 + 3AyAxT7 CyDxT1 + 2ZyT2 + WyT3 + 2VyT4 + UyT5 + 6AyAxT6 ZyT1 + 2WyT2 + VyT3 + 2UyT4 + 3AyAxT5 WyT1 + 2VyT2 + UyT3 + 6AyAxT4 VyT1 + 2UyT2 + 3AyAxT3 UyT1 + 6AyAxT2 AyAxT1 TABLE C8 Coefx Px Qx Ux Vx Wx Zx Expression B2 + 2AxCx x C2 + 2BxDx x 2AyBx + 3AxBy AyCx + 2BxBy + 3AxCy ByCx + 2BxCy CxCy + 2BxDy

+ ,

(4) 4

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