By Herbert S. Wilf
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If we were to allow too few different digits, then we would find that some numbers have no representation at all. For instance, if we were to use the decimal system with only the digits 0, 1, . . , 8, then infinitely many numbers would not be able to be represented, so we had better keep the 9s. 2. 9. Let b > 1 be a positive integer (the ‘base’). Then every positive integer n can be written in one and only one way in the form n = d0 + d1 b + d2 b2 + d3 b3 + · · · if the digits d0 , d1 , . . lie in the range 0 ≤ di ≤ b − 1, for all i.
In general, if we write out the string of digits that represents a number in the decimal system, as dm dm−1 · · · d1 d0 , then the number that is being represented by that string of digits is: n= m X di 10i . i=0 Now let’s try the binary system. Instead of using 10s we’re going to use 2s. So we imagine that the powers of 2 are displayed before us as: . . , 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. To represent a number, we will now specify how many copies of each power of 2 we would like to have.
Vk } of vertices of G such that ∀i = 1, k − 1 : (vi , vi+1 ) ∈ E(G). 2 Did you realize that the number of people who shook hands an odd number of times yesterday is an even number of people? 6. Graphs 41 A graph is connected if there is a path between every pair of its vertices. A path P is simple if its vertices are all distinct, Hamiltonian if it is simple and visits every vertex of G exactly once, and Eulerian if it uses every edge of G exactly once. A subgraph of a graph G is a subset S of its vertices together with a subset of just those edges of G both of whose endpoints lie in S.