Four Four's, 2-10
Date: 11/24/97 at 11:48:42 From: Zachary Burns Subject: Numerical Expression The numerical expression 4 + 4/4 - 4 uses four 4s and has a value of 1. Find a numerical expression using only four 4s for each integer from 2 to 10. You may use any operations and grouping symbols. I do not know where to start on this problem. Can you please help me out? Thanks, Zachary
Date: 11/24/97 at 13:52:11 From: Doctor Rob Subject: Re: Numerical Expression I would start out by figuring out what operations I felt comfortable using. Certainly +, -, *, and / should be allowed, and powers. What about the square root operation? What about decimal points (like 4.4)? What about adjunction (like 444)? And so on .... Some of the operations require only one 4, like Sqrt. These are called "unary operations" (pronounced "YEW-nair-y"). Others require two 4's, like 4/4. These are called "binary operations" (pronounced "BYE-nair-y"). I would take one four and apply all the unary operations to make a list of values that result: 4, .4, Sqrt = 2, -4, and so on. Then I would apply all the unary operators again, to expand the list, then again, and again. If we continued this indefinitely, we would have a list of all numbers expressible with one 4, such as -Sqrt[-Sqrt[.4]]. Duplicates can be deleted, such as -(-4) = 4. Then I would combine these using all the binary operations and all numbers on the list to include numbers representable with two 4's, such as Sqrt^(-4). Then I would apply all the unary operators, perhaps several times, to these results. This produces "all" numbers representable with two 4's. Part of the list for two 4's would contain: 4 + 4 = 8, 4 - 4 = 0, 4 * 4 = 16, 4 / 4 = 1, 4^4 = 256, 4^(-4) = 1/256 44, 4.4, .44, .4 / 4 = 1/10, Sqrt^4 = 16 (duplicate of 4*4 = 16), Sqrt^(-Sqrt) = 1/4, and so on. If you continue this process until you have four 4's, you will have a list of "all" numbers representable with four 4's. Most of them will not be integers from 0 to 10, so you can discard them. If you are clever, you can start with the number you want to write and combine it with two 4's to create a number on your two 4's list. For example, let's do 11. Now 11 and 44 are easily combined: 11/44 = 1/4 = Sqrt^(-Sqrt), the last from the list above, so 11 = 44*Sqrt^(-Sqrt). Understand? -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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