## 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
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
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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