Division of Number Base

Number Base Division 

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Division of number base can be achieved through two methods.

1. The long division method which operates within a specific base. This method is direct and does not require conversion into base 10.

2. Convert both numbers to be divided into base 10; perform the division in base 10, then convert the result back to the original base.

 

Note: When using Long Division, the dividend is the number that is being divided while the divisor is the number you're dividing by. For example, in 4/2, "4" is the dividend and "2", the divisor.

Please read on number base conversion here.

You can read on the base 2 system (binary) here.

 

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Worked Examples on Number Base Division 

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Example 1

• 101101₂ ÷ 101₂

Solution 

Below are steps to follow when using long division of number bases; for example, 101101₂ ÷ 101₂.

Step 1: We begin the long division by comparing the divisor, (101₂), with the first three digits of the dividend, (101101₂). Since (101₂) is equal to the first three digits of the dividend (101₂), our quotient becomes 1. In this regard, we'll write '1' in the quotient, and subtract 101₂ from 101₂ to get a remainder of zero (0)

Step 2: Bring down the next (fourth) digit, (1) from the dividend to the remainder, forming the new number (01₂). Since this remainder is less than the divisor (101), we'll write '0' in the quotient, making it (10₂)

Step 3: Bring down the next (fifth) digit, (0) from the dividend to the remainder, forming the new number (010₂). Since this remainder is less than the divisor, we'll write another '0' in the quotient, making it (100₂). 

Step 4: Bring down the final digit, (1) from the dividend to the remainder, forming the new number (0101₂). Since this value is equal to the divisor (101₂), we'll write '1' in the quotient (which becomes (1001₂) and subtract: (101-101= 0). Once the final remainder is zero, the long division ends.

• Answer: The result of the long division is our quotient 1001₂.

 

Solution through base 10 conversion

Step 1: The first binary number, (101101₂), is converted to base 10 using place values (powers of 2):

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

32 + 0 + 8 + 4 + 0 + 1= 45₁₀

Step 2: The second binary number, (101₂), is converted to base 10:

(1 × 2²) + (0 × 2¹) + (1 × 2⁰)

4 + 0 + 1= 5₁₀

Step 3:  Perform the division in base 10

45/5 = 9₁₀

Step 4: Convert the base 10 result back to base 2. In this case, the base 10 result,(9₁₀), is converted back to binary:

9 ÷ 2 = 4 R 1

4 ÷ 2 = 2 R 0

2 ÷ 2 = 1 R 0

1 ÷ 2 = 0 R 1

Read the remainder from bottom to top

• Answer: 1001₂

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Example 2

• 100011₂ ÷ 101₂

Solution

Using long division method

Step 1: We begin the long division process by comparing the initial digits of the dividend (100011₂) with the divisor (101₂). Since the first three digits (100₂) are less than (101₂), we'll consider the first four digits (1000₂). Since 1000₂ is greater than or equal to 101₂, the first digit of the quotient becomes 1. In this regard, we'll write '1' in the quotient, and subtract (101₂) from (1000₂), which gives us (11₂) as a remainder in our division. 

Step 2: Bring down the next digit, (1), from the dividend to the remainder, forming the new number (111₂). Since the new number (111₂) is greater than or equal to the divisor (101₂), the next digit of the quotient is (1), making it (11₂). Next, we'll subtract (101₂) from (111₂), leaving a remainder of (10₂). 

Step 3: Bring down the final digit, (1), from the dividend to the remainder, forming the number (101₂). Since this value is equal to the divisor (101₂), the final digit of the quotient becomes (1), making it (111₂). Subtracting (101₂) from (101₂) yields a final remainder of (0₂), and there are no more digits to bring down. 

• Answer:111₂.

 

Solution through base 10 conversion

Step 1: The first binary number, (100011₂), is converted to base 10 using place values (powers of 2):

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

32 + 0 + 0 + 0 + 2 + 1= 35₁₀

Step 2: The second binary number, (101₂), is converted to base 10:

(1 × 2²) + (0 × 2¹) + (1 × 2⁰)

4 + 0 + 1= 5₁₀

Step 3:  Perform the division in base 10

35/5 = 7₁₀

Step 4: Convert the base 10 result back to base 2. In this case, the base 10 result,(7₁₀), is converted back to binary:

7 ÷ 2 = 3 R 1

3 ÷ 2 = 1 R 1

1 ÷ 2 = 0 R 1

Read the remainder from bottom to top

• Answer: 111₂

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Example 3

• 111100₂ ÷ 110₂

Solution

Step 1: We begin the long division process by comparing the initial digits of the dividend (111100₂) with the divisor (110₂). Since 111 is greater than or equal to 110, the first digit of the quotient becomes '1'. We'll then subtract (110₂) from (111₂), which gives us (1₂) as a remainder in our division. 

Step 2: We'll bring down the next digit (1) to our remainder, which gives (11₂). Since the divisor (110₂) is less than (11₂), the next digit of the quotient is (0), making it (10₂), 

Step 3: We'll bring down the next digit (0) to our remainder to give (110₂). Since the divisor (110₂) is greater than or equal to 110₂, the next digit of the quotient becomes (1), making it (101₂), leaving a remainder of (000₂).

Step 4: We'll bring down the final digit (0) to our remainder to give (0000₂). Since the divisor (110₂) is less than (0000), the next digit of the quotient is (0), making it (1010₂), leaving a remainder of (0000₂). At this point, there aren't any digits to bring down.

• Answer: The result of the long division is our quotient 1010₂.

 

Solution through base 10 conversion

Step 1: The first binary number, (111100₂), is converted to base 10 using place values (powers of 2):

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

32 + 16 + 8 + 4 + 0 + 0= 60₁₀

Step 2: The second binary number, (110₂), is converted to base 10:

(1 × 2²) + (1 × 2¹) + (0 × 2⁰)

4 + 2 + 0 = 6₁₀

Step 3:  Perform the division in base 10

60/6 = 10₁₀

Step 4: Convert the base 10 result back to base 2. In this case, the base 10 result,(10₁₀), is converted back to binary:

10 ÷ 2 = 5 R 0

5 ÷ 2 = 2 R 1

2 ÷ 2 = 1 R 0

1 ÷ 2 = 0 R 1

Read the remainder from bottom to top

• Answer: 1010₂

Please read on the multiplication of number bases here.

Please read on the addition of number bases here.

Please read on the subtraction of number bases here.

 

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Assignment on Number Base Division

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Simplify the following 

• 1001011₂ ÷ 1001₂

• 321₄ + 21₄

• 111111₂ ÷ 10112

• 65432₇ ÷ 2345₇

• 1011101₂ ÷ 100101₂

• 4674₈ ÷ 1333₈

• 201001₃ ÷ 100211₃

• 321010₄ ÷ 13220₄

• 44231₅ ÷ 1234₅

Please read on word problems leading to algebraic expressions here.

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