Conversions of Number Systems and the Importance of Number Positioning

To convert 110010101 to binary, first we begin with the number on the far right and we move left until the last digit. So (1*(2^0) =1). The next number to the left is 0; (0*(2^1) = 0). The next number to the left is 1; (1*(2^2) = 4). The next number to the left is 0; (0*(2^3) =0). The next number to the left is 1; (1*(2^4) =16). The next number to the left is 0; (0*(2^5) =0). The next number to the left is 0; (0*(2^6) =0). The next number to the left is 1; (1*(2^7) =128). The next number to the left is 1; (1*(2^8) =256). Now we take the sum of all the answers and we should be done with the conversion. 1+0+4+0+16+0+0+128+256=405. The binary number 110010101 is equal to 405 in decimals.

To convert decimal to binary, we check to see if the decimal number is divisible by 2, if it is we write 0 and if it is not then we write 1 and subtract 1 from the decimal number and divide the number by 2. We keep doing this until the decimal number becomes 0 always writing a 1 or 0 to the left of the previous number. To convert the decimal number 529 to binary, we check to see if 529 is divisible by 2. It is not so we right 1 and subtract 1 making it 528 and divide it by 2. Again we check if 264 is divisible by two. It is so we write a 0 to the left of the previous 1 making it 01. We divide 264 by two and we get 132. We check if 132 is divisible by 2 and it is so we write another 0 to the left of 01 making it 001 and divide 132 by 2. We check if 66 is divisible by 2 and it is so we write another 0 to the left of 001 making it 0001 and divide 66 by 2. We check if 33 is divisible by 2 and it is not so we write 1 to the left of 0001 making it 10001, subtract 1 from 33 making it 32 and divide by 2. We check if 16 is divisible by 2 and it is so we write 0 to the left of 10001 making it 010001 and divide 16 by 2. We check if 8 is divisible by 2 and it is so we write 0 to the left of 010001 making it 0010001 and divide 8 by 2. We check if 4 is divisible by 2 and it is so we write 0 to the left of 0010001 making it 00010001 and divide 4 by 2. We check if 2 is divisible by 2 and it is so we write 0 to the left of 00010001 making it 000010001 and divide 2 by 2. We check if 1 is divisible by 2 and it is not so we write 1 to the left of 000010001 making it 1000010001, subtract 1 from 1 making the decimal number 0. So, 529 in decimal converted to binary is 1000010001.

The difference between a positional and non-positional numbers are the order or "position" of the numbers. For example, binary numbers are positional numbers. 101 in binary has a different value than 110 because the position of the 1s provides a different value. 101 is equal to 5 in decimal numbers and 110 is equal to 6. On the other hand, in non-positional numbers, the order of symbols does not matter and does not change the value of number. For example, in the Egyptian numerals a frog is equal to 100,000 and a vertical line is equal to 1. Regardless of how the frog and the vertical line are ordered (frog then vertical line or vertical line then frog) they will amount to the same value, 100,001.