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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.)

     

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      converting 357_10 to base four

    Then 35710 converts to 112114.

  • Convert 80710 to the corresponding base-four number.
    • convert 807_10 to base four

    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:

      344 + 043 + 242 + 141 + 340
          = 3256 + 064 + 216 + 14 + 31
          = 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:

      convert 357_10 to base seven

    Then 35710 = 10207.

  • Convert 1334610 to the corresponding base-seven number.
    • convert 13346_10 to base seven

    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:

      574 + 373 + 672 + 271 + 470
          = 52401 + 3343 + 649 + 27 + 41

          = 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
 

 



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