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### 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.

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
4.4)? What about adjunction (like 444)?  And so on ....  Some of the
operations require only one 4, like Sqrt[4]. 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[4] = 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]^(-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]^4 = 16 (duplicate of 4*4 = 16),
Sqrt[4]^(-Sqrt[4]) = 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[4]^(-Sqrt[4]),

the last from the list above, so

11 = 44*Sqrt[4]^(-Sqrt[4]).

Understand?

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
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