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Exercise - Base 10 conversion to base 2

Here's an exercise to practice converting a base 10 (decimal) number to base 2 (binary):

Exercise

Convert the decimal number 156 to its binary equivalent.

Solution

To convert the decimal number 156 to binary, follow these steps:

  1. Divide the Number by 2:
  2. Write down the decimal number.

  3. Divide it by 2, and record the quotient and the remainder.

  4. Repeat the Process:

  5. Continue dividing the quotient by 2, recording each remainder, until the quotient is 0.

  6. Read the Remainders Backwards:

  7. The binary equivalent is obtained by reading the remainders from bottom to top.

Let's go through the process step by step:

  1. Divide 156 by 2: [ 156 \div 2 = 78 \text{ with a remainder of } 0 ]

  2. Divide 78 by 2: [ 78 \div 2 = 39 \text{ with a remainder of } 0 ]

  3. Divide 39 by 2: [ 39 \div 2 = 19 \text{ with a remainder of } 1 ]

  4. Divide 19 by 2: [ 19 \div 2 = 9 \text{ with a remainder of } 1 ]

  5. Divide 9 by 2: [ 9 \div 2 = 4 \text{ with a remainder of } 1 ]

  6. Divide 4 by 2: [ 4 \div 2 = 2 \text{ with a remainder of } 0 ]

  7. Divide 2 by 2: [ 2 \div 2 = 1 \text{ with a remainder of } 0 ]

  8. Divide 1 by 2: [ 1 \div 2 = 0 \text{ with a remainder of } 1 ]

Now, list the remainders from bottom to top: [ 1 \text{ (from the last division)} \ 0 \ 0 \ 1 \ 1 \ 1 \ 0 \ 0 \text{ (from the first division)} ]

So, the binary representation of 156 is \( 10011100_2 \).

Answer

The binary equivalent of the decimal number 156 is \( 10011100_2 \).

Algebraic Method

To convert a decimal number to binary using an algebraic method, you can express the decimal number as a sum of powers of 2. Here's how we can use this method to convert the decimal number 156 to binary:

Algebraic Method:

  1. Step 1: Find the largest power of 2 less than or equal to 156.
  2. The powers of 2 are: \( 2^0 = 1 \), \( 2^1 = 2 \), \( 2^2 = 4 \), \( 2^3 = 8 \), \( 2^4 = 16 \), \( 2^5 = 32 \), \( 2^6 = 64 \), \( 2^7 = 128 \), \( 2^8 = 256 \) (which is greater than 156).

  3. The largest power of 2 less than or equal to 156 is \( 2^7 = 128 \).

  4. Step 2: Subtract \( 128 \) (i.e., \( 2^7 \)) from 156 and repeat for the remainder. [ 156 - 128 = 28 ]

  5. Step 3: Find the largest power of 2 less than or equal to 28.

  6. The largest power of 2 less than or equal to 28 is \( 2^4 = 16 \).

  7. Step 4: Subtract \( 16 \) (i.e., \( 2^4 \)) from 28 and repeat for the remainder. [ 28 - 16 = 12 ]

  8. Step 5: Find the largest power of 2 less than or equal to 12.

  9. The largest power of 2 less than or equal to 12 is \( 2^3 = 8 \).

  10. Step 6: Subtract \( 8 \) (i.e., \( 2^3 \)) from 12 and repeat for the remainder. [ 12 - 8 = 4 ]

  11. Step 7: Find the largest power of 2 less than or equal to 4.

  12. The largest power of 2 less than or equal to 4 is \( 2^2 = 4 \).

  13. Step 8: Subtract \( 4 \) (i.e., \( 2^2 \)) from 4 and repeat for the remainder. [ 4 - 4 = 0 ] Now the remainder is 0, so we are done.

Final Expression:

Now, we have expressed 156 as a sum of powers of 2: [ 156 = 2^7 + 2^4 + 2^3 + 2^2 ]

Step 9: Write the binary representation.

Each power of 2 corresponds to a digit in the binary representation. A 1 is placed in the positions of the powers of 2 we used, and a 0 is placed in the positions we skipped.

  • \( 2^7 = 128 \)
  • \( 2^6 = 0 \)
  • \( 2^5 = 0 \)
  • \( 2^4 = 16 \)
  • \( 2^3 = 8 \)
  • \( 2^2 = 4 \)
  • \( 2^1 = 0 \)
  • \( 2^0 = 0 \)

Thus, the binary representation of 156 is: [ 10011100_2 ]

This matches the result we obtained earlier using the division method.