| Math Mojo - Making Math Meaningful |
This was the question:
Could you please tell me what the place values are for 5 or 6 values, in each base beginning with 3 - 9?
Example: In base 2, the columns go like this:
| The
16s column |
The
8s column |
The
4s column |
The
2s column |
The
1s column |
Professor Homunculus' answer:
Nice question.
I assume that you know how base systems work.
Just in case, here is the skinny
on the subject:
The units place (also called the units column) is the base to the zero power.
That always means the number of the base itself, because any natural number
to the zero power is the number itself. So the first column in any base is always
the 1's column. If you find it weird that this is so (and most people do, until
it is explained to them) check out the lesson about Anything
raised to the power of Zero.
The column after that is base to the first power. That always means the number of the base itself, because any whole number to the first power is the number itself. So the second column in any base is always the (number of the base)'s column. For example, the second column in base 4 is the 3's column, and the second column in base 11 is the 11's column.
The third column would be the base to the second power, the fourth column would
be the base to the third power, the fifth column would be the base to the fourth
power, and so on.
That would make it look like this in exponential notation and standard notation:
(By the way, the symbol ^ means "to the power of...")
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in base 3 :
which in standard notation would be:
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in base 4 :
which in standard notation would be:
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in base 5 :
which in standard notation would be:
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in base 6 :
which in standard notation would be:
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in base7 :
which in standard notation would be:
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in base 8 :
which in standard notation would be:
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in base 9 :
which in standard notation would be:
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Base 10, as you know, is our normal number system. Of course bases can go above 10, too, so I will give an example of base 11, and the rest is up to you!
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in base 11 :
which in standard notation would be:
(Just out of curiousity, have you noticed a pattern in how the bases of 11 grow in standard notation? Do you think the pattern continues after 11^4? Why or why not?)
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