1.2 What these Machines are Doing
Lesson materials located below the video overview.
Let’s go back to the \(1 \leftarrow 2\) rule for a moment.
THE \(1 \leftarrow 2\) RULE: Whenever two dots sit in any one box they “explode,” that is, disappear, and become one dot in the next box to their left
Two dots in the right-most box are worth one dot in the next box to the left.
And since each dot in the rightmost box is worth \(1\) (that is how I have set up the machine) we see that a dot one place to the left is worth \(2\).
But also, two dots in the box of value \(2\) is equivalent to one dot in the box just to their left:
This next box must be worth two 2s, that is, worth \(4\) .
And two of these fours makes \(8\) .
|Question 14: What are the values of the boxes a few more places to the left?|
We said earlier that the \(1 \leftarrow 2\) code for \(13\) was \(1101\). Let’s check:
Yep! We do indeed have that \(13\) is one \(8\), one \(4\), no \(2\)s and one \(1\):
What number has code \(10110\)? It’s now easy.
|Question 15: What number has \(1 \leftarrow 2\) code \(100101\) ?|
|Question 16: What is the \(1 \leftarrow 2\) code for the number two hundred?|
FANCY LANGUAGE: People call numbers written in code binary numbers. (The prefix bi means “two.”)
Recall that all the solutions to these questions appear in the COMPANION GUIDE to this EXPLODING DOTS course.
|Question 17: In the \(1 \leftarrow 3\) system we have three dots in one box worth one dot one place to the left. This gives the numbers:
a) What is the next number?
We said that the \(1 \leftarrow 3\) code for fifteen is \(120\). We see that this is correct because …
\( 9 + 3 + 3 = 15\).
b) Could we say that the \(1 \leftarrow 3\) code for fifteen was \(0120\) ? That is, is it okay to put zeros in the front of these codes?
What about zeros at the ends of codes? Are they optional? Is it okay to leave off the last zero of the code\(120\) for fifteen and just write instead \(12\) ?
c) What number has \(1 \leftarrow 3\) code \(21002\)?
d) What is the \(1 \leftarrow 3\) code for two hundred?
|Question 18: In the \(1 \leftarrow 4\) system four dots in one box are worth one dot one place to the left. What is the value of each box?
a) What is the \(1 \leftarrow 4\) code for \(29\)?
b) What number has \(1 \leftarrow 4\) code \(132\)?
|Question 19: a)
b) What is the code for the number \(98723\) in the \(1 \leftarrow 10\) system?
c) When we write the number \(7842\) the “\(7\)” is represents what quantity? The “\(4\)” is four groups of what value? The “\(8\)” is eight groups of what value? The “\(2\)”is two groups of what value?
d) Why do human beings like the \(1 \leftarrow 10\) system for writing numbers? Why the number \(10\) ? What do we like to use on the human body for counting? Would Martians likely use the \(1 \leftarrow 10\) system for their mathematics? Why or why not?
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