Home Base, Part A

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Beginning problems about numbers in various bases.

If you haven’t already, take an opportunity now to practice counting in other bases and converting from base ten to other bases and vice versa.

Note: The “free response” answers are not checked for accuracy. To optimize your learning, we recommend you submit your own answer before revealing the hint.

Complete the following sentence:

To my learning, I plan to $\answer [format=string]{submit}$ my own answer revealing the $\answer [format=string]{hint}$ .

Explain why the following “joke” is “funny:” There are $10$ types of people in the world. Those who understand base two and those who don’t.

In base two, 10 is actually two. So people who do not understand base two will not get the joke.

You meet some Tripod aliens, they tally by threes. Thankfully for everyone involved, they use the symbols $0$ , $1$ , and $2$ .

(a): Demonstrate how a Tripod would count, beginning at $11$ .
$11, \answer {12}, \answer {20}, \answer {21}, \answer {22}, \answer {100}, \answer {101}, \answer {102}, \\ \answer {110}, \answer {111}, \answer {112}, \answer {120}, \answer {121}, \answer {122}, \answer {200}, \answer {201}$
(b): What number comes immediately before $10$ ? $\answer {2}$
(c): Before $210$ ? $\answer {202}$
(d): Before $20110$ ? $\answer {20102}$ Explain your reasoning.

Now, suppose that you meet a hermit who tallies by thirteens. Demonstrate the hermit’s counting below. (Note: For bases greater than ten, the convention is to use A for ten, B for eleven, and so on.)

$8, 9, \answer [format=string]{A}, \answer [format=string]{B}, \answer [format=string]{C}, \answer [format=string]{10}, \answer [format=string]{11}, \answer [format=string]{12}, \dots , \\ 18, \answer [format=string]{19}, \answer [format=string]{1A}, \answer [format=string]{1B}, \answer [format=string]{1C}, \answer [format=string]{20}, \answer [format=string]{21}$

Below is an exploding dot machine for an unknown base.

(a): What would the value be if it is a $1\leftarrow 5$ machine? $\answer {2\cdot 5^3+3\cdot 5^2+1\cdot 5+4}$
(b): What would the value be if it is a $1\leftarrow 10$ machine? $\answer {2\cdot 10^3+3\cdot 10^2+1\cdot 10+4}$

Time-saving hint: When a Ximera answer box is expecting a number, you may enter a numerical expression that evaluates to that number. For example, for the $1\leftarrow 5$ machine, if you had entered

2*5^3+3*5^2+1*5+4

Ximera would accept the answer as correct. The idea also works for algebraic expressions, as long as they are algebraically equivalent to the answer.

(a): What would the value be if it is a $1\leftarrow 12$ machine? $\answer {2\cdot 12^3+3\cdot 12^2+1\cdot 12+4}$
(b): What would the value be if it is a $1\leftarrow 8$ machine? $\answer {2\cdot 8^3+3\cdot 8^2+1\cdot 8+4}$
(c): What would the value be if it is a $1\leftarrow x$ machine? $\answer {2\cdot x^3+3\cdot x^2+1\cdot x+4}$

Convert $1134_{ten}$ to the following bases:

(a): base six: $\answer [format=string]{5130}_{six}$
(b): base eight: $\answer [format=string]{2156}_{eight}$
(c): base sixteen: $\answer [format=string]{46E}_{sixteen}$