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Number Bases: Base 4 and Base 7 (page 2 of 3)

Sections: Introduction & binary numbers, Base 4 & base 7, Octal & hexadecimal

Base 4

In base four, each digit in a number represents the number of copies of that power of four. That is, the first digit tells you how many ones you have; the second tells you how many fours you have; the third tells you how many sixteens (four-times-fours) you have; the fourth tells you how many sixty-fours (four-times-four-times-fours) you have; and so on. The methodology for conversion between decimal and base-four numbers is just like that for converting between decimals and binaries, except that binary digits can be only "0" or "1", while the digits for base-four numbers can be "0", "1", "2", or "3". (As you might expect, there is no single solitary digit in base-four math that represents the quantity "four".)

• Convert 35710 to the corresponding base-four number.

I will do the same division that I did before, keeping track of the remainders. (You may want to use scratch paper for this.)

Then 35710 converts to 112114.

• Convert 80710 to the corresponding base-four number.

Note: Once I got "3" on top, I had to stop, because four cannot divide into 3.

Reading the numbers off the division, I get that 80710 converts to 302134.

Now YOU try it!

• Convert 302134 to the corresponding decimal number.

I will list out the digits, and then number them from the RIGHT, starting at zero:

 digits: 3  0   2  1  3 numbering: 4  3   2  1  0

Each digit stands for the number of copies I need for that power of four:

3×44 + 0×43 + 2×42 + 1×41 + 3×40
= 3×256 + 0×64 + 2×16 + 1×4 + 3×1
= 768 + 32 + 4 + 3
= 807

As expected, 302134 converts to 80710.

Now YOU try it!

Base Seven

I can't think of any particular use for base-seven numbers, but they will serve us by providing some more practice with conversions.   Copyright © Elizabeth Stapel 2001-2011 All Rights Reserved

• Convert 35710 to the corresponding base-seven number.

I do the division:

Then 35710 = 10207.

• Convert 1334610 to the corresponding base-seven number.

Then 1334610 = 536247.

Now YOU try it!

• Convert 536247 to the corresponding decimal number.

I will list the digits, and count them off from the RIGHT, starting at zero:

 digits: 5  3   6  2  4 numbering: 4  3   2  1  0

Then I'll do the multiplication and addition:

5×74 + 3×73 + 6×72 + 2×71 + 4×70
= 5×2401 + 3×343 + 6×49 + 2×7 + 4×1

= 12005 + 1029 + 294 + 14 + 4

= 13346

Then 536247 = 1334610.

Now YOU try it!

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 Cite this article as: Stapel, Elizabeth. "Number Bases: Base 4 and Base 7." Purplemath. Available from     http://www.purplemath.com/modules/numbbase2.htm. Accessed [Date] [Month] 2016

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