Exploring Whole Numbers: Fill & Download for Free

How many ways can you give 8 apples to 4 people? Not everyone has to receive the same number of apples, and someone could even receive no apples.Sounds pretty simple... there couldn't be that many different ways, right? Let's give the people names: Alice, Bob, Carol, and Dave. So you could give Alice 8 apples, or you could give her 7 apples and give someone else 1 apple, or you could give Alice 6 apples and then... OK, maybe this is more complicated than we thought.Let's just start listing examples at random and see if we can find a pattern.Apples given to (Alice, Bob, Carol, Dave):(8, 0, 0, 0)(7, 1, 0, 0)(7, 0, 1, 0)(6, 2, 0, 0)(2, 2, 2, 2)(5, 1, 0, 2)So what's going on? We really just need to pick four whole numbers that add up to 8. But how many ways can we do that?Hmm, let's try representing the apples with stars instead. Same examples as before:Alice | Bob | Carol | Dave  ******** | | | ******* | * | | ******* | | * | ****** | ** | | ** | ** | ** | ** ***** | * | | ** But wait. What if we squished down each of the rows? It just becomes a sequence of stars and dividers:Alice|Bob|Carol|Dave ********||| *******|*|| *******||*| ******|**|| **|**|**|** *****|*||** So now these are sequences of 11 symbols that always contain 8 *'s and 3 |'s. But can we get any sequence like this? ... Yes, any sequence with 8 *'s and 3 |'s can be turned uniquely into a way to divide up the apples. Just count the number of stars between dividers.What about the other way around? Does every way to divide the apples give a different sequence? Yes, that's true too. So we have a 1-to-1 correspondence, also called a bijection. That means the number of ways to divide up the apples is the exact same as the number of 11-symbol sequences that have 8 *'s and 3 |'s.But that number is simple. It's exactly 11 choose 3, or the number of ways to get from the top of Pascal's triangle to row 11, entry 3. So we know the answer is [math]\binom{11}{3} = \frac{11 \cdot 10 \cdot 9}{3 \cdot 2 \cdot 1} = 165.[/math]When math is no longer just a homework assignment telling you exactly what formula to use, it becomes a puzzle, an exploration.Sometimes seemingly unrelated concepts can link together. Sometimes we can put together things that we know in order to solve a problem that looks like something we could never know. And sometimes we can learn something new along the way. That's what math has taught me.

It’s used in two rather different ways.As a condition, a “weak” condition is one that doesn’t require much. For example, “Let [math] N[/math] be a whole number.” That’s a weak condition: all you require is that you have a whole number. In contrast, a strong condition requires a lot: “Let [math] N[/math] be a perfect cube that can be expressed as the product of two distinct nonsquare numbers.”As a result, a “weak” result is one that doesn’t give you much. “Then [math] N[/math] is greater than 0.” Conversely, a strong result gives you a lot: “Then [math] N[/math] can be expressed uniquely as a product of primes.”Now there’s four possible combinations of conditions and results:A strong condition giving a weak resultA strong condition giving a strong result,A weak condition giving a weak result,A weak condition giving a strong resultRoughly speaking, mathematical research proceeds from the first to the last: We start with a whole bunch of assumptions and find something that’s not too interesting. Then we explore a bit more and find something that’s a lot more interesting. Then we try to reduce the starting conditions.

Sure, if you can come up with some good reason to call those things 1 and 2 and +; some analogy to the familiar concepts, despite the change.For example, perhaps you will take + to be bitwise XOR (which is a natural kind of combination, still; a kind of addition, if you like), while taking whole numbers to have their usual meaning. Then 1 + 1 = 0 instead of 2 (and 10 + 9 = 3, and all sorts of things). This what you get if you do addition in base 2 but without any “carrying”; in many programming languages, it’s a standard and useful operation.Of course, none of this is in contradiction to the usual meaning of the statement “1 + 1 = 2”; it’s just another thing we can talk about, with analogies that sometimes make it useful to use similar terminology when talking about it, but also notable differences.Perhaps you will find some other system where it is natural to talk about things this way. It’s up to you to explore and play around. That’s how math works! You can explore and play around with whatever you like, and see if you find anything interesting in doing so.