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Number Bases: Base 4 and Base 7

Purplemath

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 (that is, how many four-times-fours) you have; the fourth tells you how many sixty-fours (that is, how many four-times-four-times-fours) you have; and so on.

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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 for binaries, keeping track of the remainders. (You may want to use scratch paper for this.)

Naturally, since this is base-4, I'll be dividing by 4s.

357 ÷ 4 = 89, R1; 89 ÷ 4 = 22, R1; 22 ÷ 4 = 5, R2; 5 ÷ 4 = 1, R1

Then 357₁₀ converts to 11211₄.

Convert 807₁₀ to the corresponding base-four number.

I'll divide again by 4s.

807 ÷ 4 = 201, R3; 201 ÷ 4 = 50, R1; 50 ÷ 4 = 12, R2; 12 ÷ 4 = 3, R0

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

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

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.

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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.

Convert 357₁₀ to the corresponding base-seven number.

I do the division, this time by 7s:

357 ÷ 7 = 51, R0; 51 ÷ 7 = 7, R3; 7 ÷ 7 = 1, R0

Then 357₁₀ = 1020₇.

Convert 13346₁₀ to the corresponding base-seven number.

13346 ÷ 7 = 1906, R4; 1906 ÷ 7 = 272, R2; 272 ÷ 7 = 38, R6; 38 ÷ 7 = 5, R3

When I got to that "5" on top, I had to stop, because 7 can't divide into 5.

Then 13346₁₀ = 53624₇.

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.

URL: https://www.purplemath.com/modules/numbbase2.htm

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Number Bases: Base 4 and Base 7

Purplemath

Base 4

MathHelp.com

Convert 35710 to the corresponding base-four number.

Convert 80710 to the corresponding base-four number.

Convert 302134 to the corresponding decimal number.

Base Seven

Convert 35710 to the corresponding base-seven number.

Convert 1334610 to the corresponding base-seven number.

Convert 536247 to the corresponding decimal number.

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K12 Math

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Share This Page

Convert 357₁₀ to the corresponding base-four number.

Convert 807₁₀ to the corresponding base-four number.

Convert 30213₄ to the corresponding decimal number.

Convert 357₁₀ to the corresponding base-seven number.

Convert 13346₁₀ to the corresponding base-seven number.

Convert 53624₇ to the corresponding decimal number.