Doubling, Powers of Two, and Binary Numbers (12-8 thru 12-14)

<!--StartFragment-->

This week we practiced doubling numbers from 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, and so on.  The children were able to recite the first ten numbers in this sequence in less than 10 seconds. They should practice that sequence this week; and the ambitious Mathletes should try to double 30 times (see the attached worksheet).

We then read One Grain of Rice about a selfish raja in India. He ruled that all the people should give him almost all of their rice for safekeeping, so that in a time of need there would be rice to eat. One year, a famine hit and no one had any rice to eat, but the raja would not give the stored rice to the people because he wanted to save it for himself. One day a clever girl named Rani created a plan to help the starving people of India. She asked the raja for one grain of rice and for each day for thirty days she asked the raja to double the amount of rice given the day before. The raja did not realize how much one grain of rice would amount to if it were doubled every day for one month. The raja learned a valuable lesson about selfishness and Rani saved the people of India from starvation through her cunningness and her understanding of math.

We also posed the question, "Would you rather have $1 million or one penny doubled every day for 30 days?" The children were quickly able to see that taking the penny would amount to 1,073,741,823 pennies or $10,737,418.23 after 30 days.

Some of the older classes learned about the Binary Number system which uses on ("1") and off ("2") switches to represent all numbers in our base ten. The binary system is used internally by all modern computers.

The ancient Indian mathematician Pingala presented the first known description of a binary numeral system in the 3rd century BC.  In November of 1937, George Stibitz, then working at Bell Labs, completed a relaybased computer he dubbed the "Model K" (forK "itchen", where he had assembled it), which calculated using binary addition.

To convert the decimal number 75 to binary, we would find the largest power of 2 less than 75, which is 64. Thus, we would put a 1 in the 2^6 column, and subtract 64 from 75, giving us 11. The largest power of 2 in 11 is 8, or 2^3. Put 1 in the 2^3 column, and 0 in 2^4 and 2^5. Subtract 8 from 11 to get 3. Put 1 in the 2^1 column, 0 in 2^2, and subtract 2 from 3. We're left with 1, which goes in 2^0, and we subtract one to get zero. Thus, our number is 1001011. See the attached worksheet. The first nine binary numbers are:

 Decimal         Binary

          0                0

          1                1

          2               10

          3               11

          4              100

          5              101

          6              110

          7              111

          8             1000

          9             1001

 The Mathletes will receive one Mathlete Dollar for each number they convert to the Binary system.