# Parts and powers

How much is the power "n raised to zero"? This question ended the previous column. Today we will try to clarify the issue of powers of natural numbers. You will have no difficulty in conceiving natural numbers as sets Æ = 0, {Æ} = 1, {Æ, {Æ}} = 2, {Æ, {Æ}, {Æ, {Æ}}} = 3 ,…, {0, 1, 2, 3,…, no -1} = noBe sure to write down natural numbers up to 10 as an exercise to make sure you really understand what natural numbers are. The count of the parts of an assembly that has no sets has to do with the power . We really need to ask what a power means . An easy way to conceive of it is by thinking that it means m*m*… *m, no times But what does it mean to "multiply" the number m so many times for himself?

Multiply a number m on the other hand, say Pmeans "add" m with himself P times But what does it mean to add a number m "P times "? We've found that we still don't know what it means to add two natural numbers. But that's easy. First, let's see how we can add m + 1. We simply wrote m + 1 = m È {m}. This is why we can say that 0 + 1 = 1, 2 + 1 = 3, etc. For example, note that 2 + 1 = 2 is {2} = {0, 1} is {2} = {0, 1 } È {{0, 1}} = {0, 1, {0, 1}} = {0, 1, 2} = 3. To make sure you really understand the definition of "adding of m with 1 ", demonstrate for yourself that: 0 + 1 = 1, 1 + 1 = 2, 3 + 1 = 4, and a few more truths you want to demonstrate. We're in a fascinating moment in our mathematical adventure: so we've just To do so, we immediately realize that there are infinite truths in mathematics. Just imagine the truths 0 + 2 = 2, 2 + 2 = 4, 3 + 2 = 5,…, etc…

You may be asking, but what does it mean? m + 2? You can answer this question yourself: means (m + 1) + 1. Since you already know what m + 1 means, so easily you also know what it means m + 2. If you keep thinking like that by analogy, you'll easily know what it means. m + no.

Getting back to our problem of how to multiply m per Pwe simply agree that m*P = m + m +… + m, P times Now we are ready to explain what power means . Simply she means m*m*… *m, no times Again, it's not worth pursuing if you're not sure you understand what a natural power is. Let's take a simple example: .

If you do not understand this example, read this text again from the beginning. Calculate it for yourself right now: , ,, and other very simple powers to check your understanding.

Our problem is, finally, to explain very clearly and simply how you can to interpret a power . For this, let's imagine a set transformation. This idea is nothing more than matching sets from one set, other sets from a second set. Example: {1, 2} to {2, 4}. The transformation here is one that "doubles" numbers. A set transformation can have the most arbitrary join rule. For example: take a set, consider a part of yourself, and form a "package"; with the remaining part form "another package", the remaining package. In other words, think of the set {0, 1, 2, 3}. Consider your part {0, 1, 2} and the remaining part {3}. So we say {0, 1, 2} is a package and {3} is the remaining package. Thus, we transform part {0, 1, 2} into set 0 (forming a package) and part {3} into set 1 (forming the remaining package). That is, the numbers 0, 1, and 2 became 0, and 3 became 1. So when we ask how many parts have a set {0, 1,…, no - 1}, we are wondering how many times we can form packages. In other words, we are wondering how many transformations there are of the set {0, 1,…, no - 1} in the set {0, 1}. For example, how many "bundles" of {0, 1, 2} are there? That is, how many transformations are there from {0, 1, 2} to {0, 1}? Show that there are 8. Do not forget to form the empty package. Then answer "for packages": how much is "n raised to zero"?

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