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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 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.

Do the same division that you did before, keeping track of the remainders. (You may want 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.

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

Now remember that each digit stands for the number of copies we 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 2006-2008 All Rights Reserved

• Convert 357₁₀ to the corresponding base-seven number.

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.

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 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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