For instance: You're remodeling your house; you have 13 "4x8" panels of sheetrock on hand, and you discover that you have 650 square feet of wall to cover. Question: How many additional panels of sheetrock will you have to go out and buy?
That's not the most difficult problem in the world, true, but let's run through it once in binary arithmetic, using our fingers as computers. First we need to convert to binary numbers-but only because we chose to start out with decimal ones; it isn't fair to include conversion time as part of the time required for solving the problem.
In binary numbers you have 1101 "100 X 1000" panels on hand, and 10100,01010 square feet of wall to cover.
1101x100x1000, obviously, is merely a matter of pointing off places; you represent 01101 on your left hand, and 00000 on your right hand; that's how many square feet of sheetrock you have-uh, on hand, so to speak. Then the subtraction is merely a matter of considering the successive digits, reading from the right, subtracting the digit shown on your finger from the corresponding digit in the written number you are subtracting from, and carrying "borrowed" numbers. (Are you able to remember how much trouble you had with "carrying" when you first learned the principles of decimal subtraction? Then don't give up on binary subtraction if it takes you a few minutes to get the hang of "carrying" here.)
The result you "write," one digit at a time, on your fingers. That is, by the time you are subtracting your right-thumb digit from the written figure, the remaining fingers on your right hand are already indicating the last four digits of the answer. When you're done, you read the answer off.
As already shown, the number of square feet of sheetrock you need to buy is 111,01010 (we padded the left-hand group out with zeroes to indicate all five finger positions in writing the subtraction). There are 1,00000 square feet in a panel; 111,01010 divided by 1,00000
10100,01010 sq/ft wall to cover
-01101,00000 sq/ft sheetrock on hand
============
00111,01010 sq/ft needed is obviously 111 and a fraction. But you can't buy part of a panel so you add 1 to the 111 and get 1000. Answer: You need to buy 1000 panels. (Or, in decimal numbers, 8.)
Look hard? Once again, consider it from the perspective of relative difficulty. After all, this is probably your first binary problem. Make up a few more; by the time you've done six, it won't be hard at all; by the time you've done a hundred, it will be semi-automatic; by the time you've done a thousand- Well, hold on for a moment before you do your thousand; perhaps it will cheer you up to know that there are some special cases of binary arithmetic which aren't ever hard, not even the first time.
For example: Multiplication (or division) by powers of 2 is an obvious case; you simply point off and add zeroes. True, the decimal system has a similar situation in regard to powers of 10. But you still have to give the verdict to binary on this point, simply because in any finite series there are more powers of 2 than powers of 10.
But if you want to see something really easy, consider the strange case of the problem 1023-n. Let's arbitrarily take n as 626 (because we happen to have a binary equivalent conveniently to hand-any other number less than 1023 would do as well, of course). Do this one on your fingers. First show yourself the binary representation of 1023:
11111,11111
Then cancel that and represent on your fingers the binary equivalent of 626:
10011,10010
Don't bother about subtracting; you've already done it! Just reverse your convention for reading finger representations; read an extended finger as "0," a retracted finger as "1," and you get:
11111,11111
10011,10010
01100,01101
In other words, any number n in binary notation is always the "reverse" of the number 1023-n. Not only that, but the same sort of rule can be made for the cases 511-n, 255-n, 127-n, etc.-for any number whose binary representation is "all ones," as you may already have realized. Try it and see.
It may be objected that such special cases are comparatively rare. This is true enough, but in the decimal system they are not only rare; they do not exist at all. And we have not, by any means, exhausted binary's bag of tricks. It is, in fact, hardly possible that any reader can spend as much as a single evening trying out experiments in binary arithmetic without discovering additional shortcuts to this one.
Decimal system?
That clumsy, sprawly, quaint old thing!
Grandy Devil
MAHLON begat Timothy, and Timothy begat Nathan, and Nathan begat Roger, and the days of their years were long on the Earth. But then Roger begat Orville, and Orville was a heller. He begat Augustus, Wayne, Walter, Benjamin and Carl, who was my father, and I guess that was going too far, because that was when Gideon Upshur stepped in to take a hand.
I was kissing Lucille in the parlor when the doorbell rang and she didn't take kindly to the interruption. He was a big old man with a burned-brown face. He stamped the snow off his feet and stared at me out of crackling blue eyes and demanded, "Orvie?"
I said, "My name is George."
"Wipe the lipstick off your face, George," he said, and walked right in.
Lucille sat up in a hurry and began tucking the ends of her hair in place. He looked at her once and calmly took off his coat and hung it over the back of a chair by the fire and sat down.
"My name is Upshur," he said. "Gideon Upshur. Where's Orville Dexter?"
I had been thinking about throwing him out up until then, but that made me stop thinking about it. It was the first time anybody had come around looking for Orville Dexter in almost a year and we had just begun breathing easily again.
I said, "That's my grandfather, Mr. Upshur. What's he done now?"
He looked at me. "You're his grandson? And you ask me what he's done?" He shook his head. "Where is he?"
I told him the truth: "We haven't seen Grandy Orville in five years."
"And you don't know where he is?"
"No, I don't, Mr. Upshur. He never tells anybody where he's going. Sometimes he doesn't even tell us after he comes back."
The old man pursed his lips. He leaned forward, across Lucille, and poured himself a drink from the Scotch on the side table.
"I swear," he said, in a high, shrill, old voice, "these Dexters are a caution. Go home."
He was talking to Lucille. She looked at him sulkily and opened her mouth, but I cut in.
"This is my fiancee," I said.
"Hah," he said. "No doubt. Well, there's nothing to do but have it out with Orvie. Is the bed made up in the guest room?"
I protested, "Mr. Upshur, it isn't that we aren't glad to see any friend of Grandy's, but Lord knows when he'll be home. It might be tomorrow, it might be six months from now or years."
"I'll wait," he said over his shoulder, climbing the stairs.
Having him there wasn't so bad after the first couple of weeks. I phoned Uncle Wayne about it, and he sounded quite excited.
"Tall, heavy-set old man?" he asked. "Very dark complexion?"
"That's the one," I said. "He seemed to know his way around the house pretty well, too."
"Well, why wouldn't he?" Uncle Wayne didn't say anything for a second. "Tell you what, George. You get your brothers together and--"
"I can't, Uncle Wayne," I said. "Harold's in the Army. I don't know where William's got to."
He didn't say anything for another second. "Well, don't worry. I'll give you a call as soon as I get back."
"Are you going somewhere, Uncle Wayne?" I wanted to know.
"I certainly am, George," he said, and hung up.
So there I was, alone in the house with Mr. Upshur. That's the trouble with being the youngest.