base2

Consider the following chart:

On the left we have the column of sequential integers. Let this column represent, say, time, or, say, initial members of an organisation. The column on the right represents the rate of expansion over time, or per each different original group member. The left column reads up and down, and the right column reads side to side. Here is how to read them: begin with square one, expansion rate two. Next in sequence would be to start with two, and measure the expansion rate four, etc. The chart goes up to integer five, rate of expansion thirty two, but of course you can see how applying this method works.

The three middle columns show some relationships we can draw across the board between sequential integers and expansion rates. 1+1=2, 2+2=4 and 3+3=6 represent the initial integers and expansion rates. The difference between each initial integer and the next across is the corresponding row on the left column. The difference between each integer sum and the next across corresponds to the expansion rate column on the right.

Basically to read across a row, you count the difference between the sums by the factor to the right. Thus: (2,4,6)=2; (32,64,96)=32. The expansion rate in the right column is the same as that for the 1+1=2 column because this entire table represents only multiples of two. 1+1=2 and 2+2=4 establish the base two system as a root function. The rest of the chart is based on multiplying each sum by two.

So, in short, this chart represents the corner-stone of the Base-Two system. There are similar charts that can be constructed for all the multiples, and it will show that, for each multiplicative step by its base number system (3,4,5,6... etc.) it will increase in the same fashion: As the sequence of integers expands "arithmetically," the column on the right, the expanson rate, will increase "exponentially."