#### Decimal TO Binary

#### Comparison with descending powers of two and subtraction

- List the powers of two in a "base 2 table" from right to left. Start at 2
^{0}, evaluating it as "1". Increment the exponent by one for each power. The list, to ten elements, would look like this: 512, 256, 128, 64, 32, 16, 8, 4, 2, 1 - For this example, let's convert the decimal number 156
_{10}to binary. What is the greatest power of two that will fit into 156? Since 128 fits, write a 1 for the leftmost binary digit, and subtract 128 from your decimal number, 156. You now have 28. - Move to the next lower power of two. Can 64 fit into 28? No, so write a 0 for the next binary digit to the right.
- Can 32 fit into 28? No, so write a 0.
- Can 16 fit into 28? Yes, so write a 1, and subtract 16 from 28. You now have 12.
- Can 8 fit into 12? Yes, so write a 1, and subtract 8 from 12. You now have 4.
- Can 4 (power of two) fit into 4 (working decimal)? Yes, so write a 1, and subtract 4 from 4. You have 0.
- Can 2 fit into 0? No, so write a 0.
- Can 1 fit into 0? No, so write a 0.
- Since there are no more powers of two in the list, you are done. You should have 10011100. This is the binary equivalent of the decimal number 156. Or, written with base subscripts: 156
_{10}= 10011100_{2}

Repetition of this method will result in memorization of the powers of two, which will allow you to skip step 1.

#### Short division by two with remainder

This method is much easier to understand when visualized on paper. It relies only on division by two.

- For this example, let's convert the decimal number 156
_{10}to binary. Write the decimal number as the dividend inside an upside-down "long division" symbol. Write the base of the destination system (in our case, "2" for binary) as the divisor outside the curve of the division symbol.`2`__)156__ - Write the integer answer (quotient) under the long division symbol, and write the remainder (0 or 1) to the right of the dividend.
`2`__)156__0

78 - Continue downwards, dividing each new quotient by two and writing the remainders to the right of each dividend. Stop when the quotient is 1.
`2`__)156__0

2__)78__0

2__)39__1

2__)19__1

2__)9__1

2__)4__0

2__)2__0

2__)1__1

2__)0__0 - Starting with the bottom 1, read the sequence of 1's and 0's upwards to the top. You should have 10011100. This is the binary equivalent of the decimal number 156. Or, written with base subscripts: 156
_{10}= 10011100_{2}

Example:

*

So, we have

Conversion for FRACTIONAL PART