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at the point representing 2 If you start with 10 and divide by 5, you cut your distance from the number reflector by a factor of 5 and stay on the same side, so you end up at 2 Dividing by a negative number is trickier In Fig 5-2, suppose you start with 6 and divide it by 3 You jump to the other side of the number reflector and then reduce your distance from it by a factor of 3, finishing at 2 If you start with 10 and divide by 5, you jump to the other side of the number reflector and then cut your distance from it by a factor of 5, finishing at 2 In these situations, the integers divide each other cleanly without remainders Remainders occur when one nonzero integer divides another integer and the result is not an integer Messy division produces fractions, which we ll study in Chap 6 An example is the division of 5 by 3 Another example is the division of 7 by 12 You can add, subtract, or multiply any two integers and always end up with another integer In math jargon, the operations of addition, subtraction, and multiplication are closed over the set of integers When you do a division problem with integers, you don t always get another integer Therefore, the operation of division is not closed over the set of integers When you divide a quantity by another quantity, you get a quotient Sometimes it is called a ratio In a division problem, the first number is sometimes called the dividend, and the second number is sometimes called the divisor
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In the mid-seventies, Gerald Appel was creating all kinds of timing indicators based on price momentum Because the personal computer hadn t yet been invented, he d program his calculations into a calculator Finally, in 1976 he devised the Moving Average Convergence Divergence, or MACD
where z denotes the complex conjugate4 of z This quantity is maximized when the vectors lie in the same direction and is zero when the vectors are orthogonal to each other The fundamental observation we need is the following Lemma The columns of matrix M are orthogonal to each other Proof Take the inner product of any columns j and k of matrix M , 1 + j k + 2(j k) + + (n 1)(j k)
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Division by 0 What happens if you try to divide an integer by 0 You don t get any integer, or any other known type of number Look at this problem inside-out What must you multiply 0 by if
This is a geometric series with rst term 1, last term (n 1)(j k) , and ratio (j k) Therefore it evaluates to (1 n(j k) )/(1 (j k) ), which is 0 except when j = k, in which case all terms are 1 and the sum is n
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you want to get, say, 3 How many times must you add 0 to itself to get anything but 0 No integer can do this trick In fact, no known number solves this problem Most mathematicians will tell you that division by 0 is not defined That s what my 7th-grade math teacher kept saying, and I pestered her about it I would ask, Why not or retort, Let s define it, then! She would repeat herself, Division by 0 is not defined I did not take her seriously, so I started trying to make division by 0 work I never came up with a well-defined way to do it But I came pretty close, and had a lot of fun trying
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Have you heard that dividing a positive integer by 0 gives you infinity If not, you probably will some day Be skeptical! The first thing you must do to figure out if it s really true is to define infinity That s not easy No meaningful, enduring definition of infinity produced by mathematicians has ever had anything to do with division by 0 Again, look at the problem inside-out If you want to multiply 0 by any positive integer n, you must add 0 to itself (n 1) times No matter how large you make n, you always get 0 when you add it to itself (n 1) times Why should adding 0 to itself forever make any difference It s tempting to suppose that it might, but that doesn t prove that it will In mathematics, we need proof before we can claim something is true!
The complex conjugate of a complex number z = rei is z = re i The complex conjugate of a vector (or matrix) is obtained by taking the complex conjugates of all its entries
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