Addition of Number Bases

Number Base Addition 

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The addition of numbers in different bases is quite straightforward. It follows a similar procedure as in the addition of base 10. The principle of addition of number base is stated below:

In number base addition, when an addition in a column equals or exceeds the base number, we'll divide the sum by the base; writing down the remainder, and carrying the whole number quotient to the next column. 

This above principle is shown in binary or base 2 addition:

• 0 + 0 = 0

• 0 + 1 = 1

• 1 + 1= 2/2 (Carry 1 and write down remainder 0)

• 1 + 1 + 1 = 3/2 (Carry 1 and write down remainder 1)

Note: This same principle applies to other bases. That is: "when an addition in a column equals or exceeds the base number, we'll divide the sum by the base, while writing down the remainder, and carrying the whole number quotient to the next column". 

Please read on base 2 system (binary) here.

 

It is noteworthy to state that number base addition can easily be carried out through base 10 conversion. The steps below are followed using this method:

• Step 1: Convert the bases to be added into base 10.

• Step 2: Add the base 10 numbers (derived from step 1).

• Step 3: Convert the base 10 addition result (in step 2) to the required base in the question.

Please read on number base conversion here.

You can read on the division of number bases here.

 

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Worked Examples on Number Base Addition 

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Example 1

• 10₂ + 11₂

Solution 

Align the two binary numbers by their place values. This is shown below:

 

Step 1: Add the rightmost digits.

0 + 1= 1

Step 2: Add the leftmost digits We add the leftmost digits:

1 + 1 = 2/2 (Carry 1 and write down remainder 0)

Step 3: 1 carried = 1

• Answer: Combine the direct answers and remainder = 101₂

 

Solution Through Base 10 Conversion

• 10₂ + 11₂

Step 1: Convert 10₂ to base 10

(1 × 2¹) + (0 × 2⁰)

2 + 0 = 2₁₀

Step 2: Convert 11₂ to base 10

(1 × 2¹) + (1 × 2⁰)

2 + 1 = 3₁₀

Step 3: Add both base 10 results

2 + 3 = 5₁₀

Step 4: Covert the base 10 result to base 2

5 ÷ 2 = 2 R 1

2 ÷ 2 = 1 R 0

0 ÷ 2 = 0 R 1

Read remainder from bottom to top.

• Answer: 101₂

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Example 2

• 11011₂ + 1101₂

Solution 

Align the two base numbers by their place values. If possible, the shorter number can be padded with leading zeros for clarity. This is shown below:

 

Step 1: Perform base addition with carries. Add the digits from right to left, following the base addition rule.

1 + 1 = 2/2 (Carry 1 and write down remainder 0)

Step 2: Second rightmost column

1 carried + 1 + 0 = 2/2 (Carry 1 and write down remainder 0)

Step 3: Third rightmost column

1 carried + 0 + 1 = 2/2 (Carry 1 and write down remainder 0)

Step 4: Fourth rightmost column

1 carried + 1 + 1 = 3/2 (Carry 1 and write down remainder 1)

Step 5: Left column

1 carried + 1 + 0 = 2/2 (Carry 1 and write down remainder 0)

Step 6: 1 + 0 = 1 

• Answer: Combine the direct answers and remainders = 101000₂

 

Solution Through Base 10 Conversion

• 11011₂ + 1101₂

Step 1: Convert 11011₂ to base 10

(1 × 2⁴) + (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)

16 + 8 + 0 + 2 + 1 = 27₁₀

Step 2: Convert 1101₂ to base 10

(1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)

8 + 4 + 0 + 1 = 13₁₀

Step 3: Subtract both base 10 results

27 + 13 = 40₁₀

Step 4: Covert the base 10 result to base 2

40 ÷ 2 = 20 R 0

20 ÷ 2 = 10 R 0

10 ÷ 2 = 5 R 0

5 ÷ 2 = 2 R 1

2 ÷ 2 = 1 R 0

1 ÷ 2 = 0 R 1

Read remainder from bottom to top.

• Answer: 101000₂

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Example 3

• 2033₄ + 30012₄

Solution 

Align the two base 4 numbers by their place values. If possible, the shorter number can be padded with leading zeros for clarity. This is shown below:

 

Step 1: Perform base 4 addition with carries. Add the digits from right to left, following the base addition rule.

3 + 2 = 5/4 (Carry 1 and write down remainder 1)

Step 2: Second rightmost column

1 carried + 3 + 1 = 5/4 (Carry 1 and write down remainder 1)

Step 3: Third rightmost column

1 carried + 0 + 0 = 1

Step 4: Fourth rightmost column

2 + 0 = 2

Step 5: Left column

0 + 3 = 3

• Answer: Combine the direct answers and remainders = 32111₄

 

Solution Through Base 10 Conversion

• 2033₄ - 30012₄

Step 1: Convert 2033₄ to base 10

(2 × 4³) + (0 × 4²) + (3 × 4¹) + (3 × 4⁰)

128 + 0 + 12 + 3 = 143₁₀

Step 2: Convert 30012₄ to base 10

(3 × 4⁴) + (0 × 4³) + (0 × 4²) + (1 × 4¹) + (2 × 4⁰)

768 + 0 + 0 + 4 + 2 = 774₁₀

Step 3: Add both base 10 results

143 + 774 = 917₁₀

Step 4: Covert the base 10 result to base 5

917 ÷ 4 = 229 R 1

229 ÷ 4 = 57 R 1

57 ÷ 4 = 14 R 1

14 ÷ 4 = 3 R 2

3 ÷ 4 = 0 R 3

Read remainder from bottom to top.

• Answer: 32111₄

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Example 4

• 50442₆ + 335₆

Solution 

Align the two base 6 numbers by their place values. If possible, the shorter number can be padded with leading zeros for clarity. This is shown below:

 

Step 1: Perform base 6 addition with carries. Add the digits from right to left, following the base addition rule.

5 + 2 = 7/6 (Carry 1 and write down remainder 1)

Step 2: Second rightmost column

1 carried + 4 + 3 = 8/6 (Carry 1 and write down remainder 2)

Step 3: Third rightmost column

1 carried + 4 + 3 = 8/6 (Carry 1 and write down remainder 2)

Step 4: Fourth rightmost column

1 carried + 0 + 0 = 1

Step 5: Left column

5 + 0 = 5

• Answer: Combine the direct answers and remainders = 51221₆

 

Solution Through Base 10 Conversion

• 50442₆ + 335₆

Step 1: Convert 50442₆ to base 10

(5 × 6⁴) + (0 × 6³) + (4 × 6²) + (4 × 6¹) + (2 × 6⁰)

6480 + 0 + 144 + 24 + 2 = 6650₁₀

Step 2: Convert 335₆ to base 10

(3 × 6²) + (3 × 6¹) + (5 × 6⁰)

108 + 18 + 5 = 131₁₀

Step 3: Add both base 10 results

6650 + 131 = 6781₁₀

Step 4: Covert the base 10 result to base 5

6781 ÷ 6 = 1130 R 1

1130 ÷ 6 = 188 R 2

188 ÷ 6 = 31 R 2

31 ÷ 6 = 5 R 1

5 ÷ 6 = 0 R 5

Read remainder from bottom to top.

• Answer: 51221₆

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Example 5

• 101₂ + 111₂

Solution 

Align the base numbers Align the two binary numbers by their place values.

 

Step 1: Perform binary addition with carries. Add the digits from right to left, following base addition rules:

1 + 1 = 2/2 (Carry 1 and write down remainder 0)

Step 2: Second rightmost column

1 carried + 0 + 1 = 2/2 (Carry 1 and write down remainder 0)

Step 3: Left column

1 carried + 1 + 1 = 3/2 (Carry 1 and write down remainder 1)

Step 4: 1 carried = 1

Answer: Combine the direct answers and remainders = 1100₂

 

Solution Through Base 10 Conversion

• 101₂ + 111₂

Step 1: Convert 43002₅ to base 10

(1 × 2²) + (0 × 2¹) + (1 × 2⁰)

4 + 0 + 1 = 5₁₀

Step 2: Convert 111₂ to base 10

(1 × 2²) + (1 × 2¹) + (1 × 2⁰)

4 + 2 + 1 = 7₁₀

Step 3: Add both base 10 results

5 + 7 = 12₁₀

Step 4: Covert the base 10 result to base 5

12 ÷ 2 = 6 R 0

6 ÷ 2 = 3 R 0

3 ÷ 2 = 1 R 1

1 ÷ 2 = 0 R 1

Read remainder from bottom to top.

• Answer: 1100₂

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Example 6

• 10432₅ + 4321₅

Solution

Align the two base 5 numbers by their place values. If possible, the shorter number can be padded with leading zeros for clarity. This is shown below:

Step 1: Perform base 5 addition with carries. Add the digits from right to left, following the base addition rule.

2 + 1 = 3

Step 2: Second rightmost column

3 + 2 = 5/5 (Carry 1 and write down remainder 0)

Step 3: Third rightmost column

1 carried + 4 + 3 = 8/5 (Carry 1 and write down remainder 3)

Step 4: Fourth rightmost column

1 carried + 0 + 4 = 5/5 (Carry 1 and write down remainder 0)

Step 5: Left column

1 carried + 1 + 0 = 2 

• Answer: Combine the direct answers and remainders = 20303₅

 

Solution Through Base 10 Conversion

• 10432₅ + 4321₅

Step 1: Convert 10432₅ to base 10

(1 × 5⁴) + (0 × 5³) + (4 × 5²) + (3 × 5¹) + (2 × 5⁰)

625 + 0 + 100 + 15 + 2 = 742₁₀

Step 2: Convert 4321₅ to base 10

(4 × 5³) + (3 × 5²) + (2 × 5¹) + (1 × 5⁰)

500 + 75 + 10 + 1 = 586₁₀

Step 3: Add both base 10 results

742 + 586 = 1328₁₀

Step 4: Covert the base 10 result to base 5

1328 ÷ 5 = 265 R 3

283 ÷ 5 = 53 R 0

53 ÷ 5 = 10 R 3

10 ÷ 5 = 2 R 0

2 ÷ 5 = 0 R 2

Read remainder from bottom to top.

• Answer: 20303₅

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Example 7

• 22222₃ + 11111₃

Solution 

Align the two base 3 numbers by their place values. This is shown below:

 

Step 1: Perform base 3 addition with carries. Add the digits from right to left, following the base addition rule.

2 + 1 = 3/3 (Carry 1 and write down remainder 0)

Step 2: Second rightmost column

1 carried + 2 + 1 = 4/3 (Carry 1 and write down remainder 1)

Step 3: Third rightmost column

1 carried + 2 + 1 = 4/3 (Carry 1 and write down remainder 1)

Step 4: Fourth rightmost column

1 carried + 2 + 1 = 4/3 (Carry 1 and write down remainder 1)

Step 5: Left column

1 carried + 2 + 1 = 4/3 (Carry 1 and write down remainder 1)

Step 6: 1 carried = 1

• Answer: Combine the direct answers and remainders = 111110₃

 

Solution Through Base 10 Conversion

• 22222₃ + 11111₃

Step 1: Convert 22222₃ to base 10

(2 × 3⁴) + (2 × 3³) + (2 × 3²) + (2 × 3¹) + (2 × 3⁰)

162 + 54 + 18 + 6 + 2 = 242₁₀

Step 2: Convert 11111₃ to base 10

(1 × 3⁴) + (1 × 3³) + (1 × 3²) + (1 × 3¹) + (1 × 3⁰)

81 + 27 + 9 + 3 + 1 = 121₁₀

Step 3: Add both base 10 results

242 + 121 = 363₁₀

Step 4: Covert the base 10 result to base 5

363 ÷ 3 = 121 R 0

121 ÷ 3 = 40 R 1

40 ÷ 3 = 13 R 1

13 ÷ 3 = 4 R 1

4 ÷ 3 = 1 R 1

1 ÷ 3 = 0 R 1

Read remainder from bottom to top.

• Answer: 111110₃

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Example 8

• 1110₂ + 1001₂

Solution 

Align the two binary numbers by their place values. This is shown below:

 

Step 1: Perform binary addition with carries. Add the digits from right to left, following the base addition rule:

0 + 1 = 1

Step 2: Second rightmost column

1 + 0 = 1

Step 3: Third rightmost column

1 + 0 = 1

Step 4: Left column

1 + 1 = 2/2 (Carry 1 and write down remainder 0) 

Step 5: 1 carried = 1

• Answer: Combine the direct answers and remainders = 10111₂

 

Solution Through Base 10 Conversion

• 1110₂ + 1001₂

Step 1: Convert 1110₂ to base 10

(1 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰)

8 + 4 + 2 + 0 = 14₁₀

Step 2: Convert 1001₂ to base 10

(1 × 2³) + (0 × 2²) + (0 × 2¹) + (1 × 2⁰)

8 + 0 + 0 + 1 = 9₁₀

Step 3: Add both base 10 results

14 + 9 = 23₁₀

Step 4: Covert the base 10 result to base 5

23 ÷ 2 = 11 R 1

11 ÷ 2 = 5 R 1

5 ÷ 2 = 2 R 1

2 ÷ 2 = 1 R 0

1 ÷ 2 = 0 R 1

Read remainder from bottom to top.

• Answer: 10111₂

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Example 9

100100₂ + 11₂

Solution 

Align the two binary numbers by their place values. If possible, the shorter number can be padded with leading zeros for clarity. This is shown below:

 

Step 1: Perform binary addition with carries. Add the digits from right to left, following the base addition rules

0 + 1 = 1

Step 2: Second rightmost column

0 + 1 = 1

Step 3: Third rightmost column

1 + 0 = 1

Step 4: Fourth rightmost column

0 + 0 = 0

Step 5: Fifth rightmost column

0 + 0 = 0

Step 6: Left column

1 + 0 = 1

Answer: Combine the direct answers and remainders = 100111₂

 

Solution Through Base 10 Conversion

• 100100₂ - 11₂

Step 1: Convert 100100₂ to base 10

(1 × 2⁵) + (0 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (0 × 2⁰)

32 + 0 + 0 + 4 + 0 + 0 = 36₁₀

Step 2: Convert 11₂ to base 10

(1 × 2¹) + (1 × 2⁰)

2 + 1 = 3₁₀

Step 3: Add both base 10 results

36 + 3 = 39₁₀

Step 4: Covert the base 10 result to base 5

39 ÷ 2 = 19 R 1

19 ÷ 2 = 9 R 1

9 ÷ 2 = 4 R 1

4 ÷ 2 = 2 R 0

2 ÷ 2 = 1 R 0

1 ÷ 2 = 0 R 1

Read remainder from bottom to top.

• Answer: 100111₂

Please read on subtraction of number bases here.

Please read on multiplication and division in algebra here.

Please read on the use of BODMAS in mathematics here.

 

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Assignment on Addition of Number Bases

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Add the following bases:

• 321₄ + 213₄

• 4231₅ + 1234₅

• 3871₉ + 3887₉

• 10110111₂ +.110111₂

• 700001₈ +.166667₈

• 654321₇ + 123456₇

• 43211₅ + 22143₅

• 64443₇ + 16334₇

• 787649₁₀ + 99181₁₀

• 88123₉ + 23568₉

Please read on word problems leading to algebraic expressions here.

Please read on mathematics week 9 topic for jss1 first term here.

Please read on mathematics scheme of work for jss1 first term here.