## Exploding Dots

### 3.7 Wild Explorations

Here are some “big question” investigations you might want to explore, or just think about. Have fun!

 EXPLORATION 1: THERE IS NOTHING SPECIAL ABOUT BASE TEN FOR ADDITION  Here is an addition problem in a $$1 \leftarrow 5$$ machine. (That is, it is a problem in base five.) This is not a $$1 \leftarrow 10$$ machine addition. a) What is the $$1 \leftarrow 5$$ machine answer? b) What number has code $$20413$$ in a $$1 \leftarrow 5$$ machine? What number has code $$13244$$ in a $$1 \leftarrow 5$$ machine? What is the sum of those two numbers and what is the code for that sum in a $$1 \leftarrow 5$$ machine?   [Here are the answers so that you can check your clever thinking. The sum, as a $$1 \leftarrow 5$$ machine problem, is $$20413+13244 = 3|3|6|5|7 = 3|4|1|5|7 = 3|4|2|0|7 = 3|4|2|1|2 = 34212$$. In a $$1 \leftarrow 5$$ machine, $$20413$$ is two $$625$$s, four $$25$$s, one $$5$$, and three $$1$$s, and so is the number $$1358$$ in base ten; $$13244$$ is the number $$1074$$ in base ten; and $$34212$$ is the number $$2432$$ in base ten.  We have just worked out $$1358 + 1074 = 2432$$.]

 EXPLORATION 2: THERE IS NOTHING SPECIAL ABOUT BASE TEN FOR MULTIPLICATION  Let’s work with a $$1 \leftarrow 3$$ machine.   a) Find $$111 \times 3$$ as a base three problem. Also, what are $$1202 \times 3$$ and $$2002 \times 3$$?      Can you explain what you notice?   Let’s now work with a $$1 \leftarrow 4$$ machine.   b) What is $$133 \times 4$$ as a base four problem? What is $$2011 \times 4$$? What is $$22 \times 4$$?      Can you explain what you notice?   In general, if we are working with a $$1 \leftarrow b$$  machine, can you explain why multiplying a number in base $$b$$ by $$b$$ returns the original number with a zero tacked on to its right?

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