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number base 4 addition and conversion

by Trish Jacobs
(North Haverhill, New Hampshire, United States)












































Adding with number base 4

What is 33, base four, plus 21, base four?

Convert the total to number base 10 (the corresponding decimal system).

Comments for number base 4 addition and conversion

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May 17, 2014
base 4 addition
by: Staff


Answer

Part I

There are a couple of ways to solve this problem:

(A) add the two numbers as they are (leave the two numbers in base 4)

or

(B) convert both numbers to base 10, add the base 10 numbers, and then convert the answer back into base 4.


Both methods are shown below:


(A) add the two numbers as they are (leave the two numbers in base 4)


This approach may be easier to understand if you begin by writing the numbers 33₄ and 21₄ in expanded exponential notation.

Base 4 numbers use a place-value notation system (also called positional notation), just like numbers written in the Base 10.

The place-value notation system allows the same digits to be used over and over again.

The ability to use the same digits over and over makes the place-value notation system superior to other numbering systems.

Other numbering systems require the user to memorize different symbols for different values (eg: roman numerals, Egyptian Number System , Greek Number System , etc.).

Regardless of what number is being written, the base 4 system only uses the four single digits 0, 1, 2, 3 over and over again.

Regardless of what number is being written, the base 10 system only uses the ten single digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 over and over again.

The value of the digit depends on the value of its position within the number.

For example, the number 222, written in the base 10 system is:

200 + 20 + 2

The digit 2 has a different value depending on whether it is written in the 1’s column, the 10’s column, or the 100’s column.

222₁₀ can also be written:

222₁₀ = (2 * 100) + (2 * 10) + (2 * 1)

or, in expanded exponential notation:

222₁₀ = (2 * 10²) + (2 * 10¹) + (2 * 10⁰)

If the same digits, 222, are written in the base 4 system:

222₄ = (2 * 4²) + (2 * 4¹) + (2 * 4⁰)

The digit 2 has a different value depending on whether it is written in the 4⁰ column, the 4¹ column, or the 4² column.



For this problem, both 33₄ and 021₄ can be written in expanded exponential notation as follows:


033₄ = 0×4² + 3×4¹ + 3×4⁰ (number base 4) = 0 + 12 + 3 = 15₁₀ (number base 10)

021₄ = 0×4² + 2×4¹ + 1×4⁰ (number base 4) = 0 + 8 + 1 = 9₁₀ (number base 10)



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May 17, 2014
base 4 addition
by: Staff

------------------------------------


Part II


 base 4 numbers:  expanded exponential notation 

*** Click to enlarge image ***





The arithmetic problem is: 033₄ + 021₄


0 3 3₄

+ 0 2 1₄
------------



adding base 4 numbers in vertical format 

*** Click to enlarge image ***







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May 17, 2014
base 4 addition
by: Staff

------------------------------------


Part III



Step 1: Add the far right column (the 4⁰ column, also called the ones’ place digits since 4⁰ = 1)


3 + 1 = 4


4 does not use the digits 0, 1, 2, or 3. Base 4 numbers can only use one of the four digits which are less than 4, either 0, 1, 2, or 3.

(In the base four system, the number 4₁₀ is represented as 10₄, as shown below.

10₄ = 1×4¹ + 0×4⁰ = 4 + 0 = 4₁₀)

To proceed with the arithmetic, simply subtract 4 from the column total: 4 – 4 = 0.

0 is one of the digits which can be used. Write 0 in the far right column, like this


0 3 3₄

+ 0 2 1₄
------------

0₄



add the far right column (the ones' place digits column) 

*** Click to enlarge image ***





But the total was 4, not 0. The 4 must be accounted for.

Add 1 to the next column (the "4¹" column, also called the fours’ column since 4¹ = 4), as shown below.


0 3⁺¹3₄

+ 0 2  1₄
------------

0₄





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May 17, 2014
base 4 addition
by: Staff


------------------------------------


Part IV


add +1 to the fours’ column  (the fours' place digits column) 

*** Click to enlarge image ***





Step 2: Add the second column (the 4¹ column, also called the fours’ place digits since 4¹ = 4)


1 + 3 + 2 = 6

6 does not use the digits 0, 1, 2, or 3. Base 4 numbers can only use one of the four digits which are less than 4, either 0, 1, 2, or 3.

(In the base four system, the number 6₁₀ is represented as 12₄, as shown below.

12₄ = 1×4¹ + 2×4⁰ = 4 + 2 = 6₁₀)


add the digits in the fours’ column 

*** Click to enlarge image ***






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May 17, 2014
base 4 addition
by: Staff

------------------------------------


Part V



6 is not less than 4

Therefore, subtract 4: 6 – 4 = 2.

2 is less than 4, so write 2 in the far 2nd column, like this


0 3⁺¹3₄

+ 0 2  1₄
------------

2  0₄



But the total was 6, not 2. The extra 4 must be accounted for.

Add 1 to the next column (the "4²" column).



0⁺¹ 3⁺¹ 3₄

+ 0   2   1₄
-------------

2    0₄




 add +1 to the sixteens’ column  (the 4² place digits column) 

*** Click to enlarge image ***






Step 3: Add the third column (the 4² column)


1 + 0 + 0 = 1


1 is less than 4, so write 1 in the 3rd column (the 4² column)


0⁺¹ 3⁺¹ 3₄

+ 0   2   1₄
-------------

1    2    0₄




------------------------------------

May 17, 2014
base 4 addition
by: Staff


------------------------------------


Part VI


add the digits in the 4² column 

*** Click to enlarge image ***




The final answer is:

33₄ + 21₄ = 1 2 0₄




addition of base 4 numbers  33₄ + 21₄ - final answer 

*** Click to enlarge image ***








------------------------------------

May 17, 2014
base 4 addition
by: Staff


------------------------------------


Part VII


(B) convert both numbers to base 10, add the base 10 numbers, and then convert the answer back into base 4.

Base 10 equivalent of the base 4 addition

033₄ = 0×4² + 3×4¹ + 3×4⁰ (number base 4) = 0 + 12 + 3 = 15₁₀ (number base 10)

021₄ = 0×4² + 2×4¹ + 1×4⁰ (number base 4) = 0 + 8 + 1 = 9₁₀ (number base 10)

033₄ + 021₄ = 15₁₀ + 9₁₀ = 24₁₀


Convert 24₁₀ back into base 4

033₄ + 021₄ (number base 4) = 24₁₀(number base 10) = ?×4² + ?×4¹ + ?×4⁰ (number base 4)



To accomplish this, use the REMAINDER METHOD:

Since we are converting to the number base 4, divide by 4.

24₁₀ ÷ 4 = 6 R 0 (this is the far right column, the 4⁰ column)

6 ÷ 4 = 1 R 2 (this is the next column, the 4¹ column)

1 ÷ 4 = there is no remainder, the 1 is the last digit (this is the far left column, the 4² column)



033₄ + 021₄ (number base 4) = 24₁₀(number base 10) = 1×4² + 2×4¹ + 0×4⁰ (number base 4)


033₄ + 021₄ (number base 4) = 1 2 0₄ (number base 4)



As before, the final answer is:

33₄ + 21₄ = 1 2 0₄







Thanks for writing.

Staff
www.solving-math-problems.com



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