Question Click to View Answer What is a base-ten number. Base-ten numbers are the numbers we're used to seeing in our everyday life. Base-ten math seems natural to us because we've been doing using base-ten for our entire lives. Humans use base-ten because we have 10 fingers and it is easy for us to count to ten. If humans had 8 fingers, we would probably use base-8 (aka octal) numbers. Convert 101 base-2 to the corresponding base-ten number (i.e. 101 is a binary number and your objective is to compute the equivalent base-ten number). The following lines show the digits and numbering of the base-2 number: ```digits: 1 0 1 numbering: 2 1 0 ``` Every digit must be multipled by 2 raised to the numbering. All of these products are summed to compute the result. `1*(2**2) + 0*(2**1) + 1*(2**0) = 5` This link has an awesome description on how to convert number bases. Convert 1110 base-2 to the corresponding base-ten number. We will again map out the digits and numbering of the digits to aid with the conversion. ```digits: 1 1 1 0 numbering: 3 2 1 0 1*(2**3) + 1*(2**2) + 1*(2**1) + 0*(2**0) = 14 ``` Convert 8 base-ten to the corresponding base-2 number. The divide by two algorithm can be used to convert a base-ten number into a base-2 number. The algorithm starts by dividing the base-ten number by 2. The remainder is the last digit of the binary number and the remainder is divided by 2 again. The algorithm keeps repeating until the result of the division is 0 or 1 - this final result is prepended to the binary number. ```1 => 1 2 / 2 => remainder 0 => 0 4 / 2 => remainder 0 => 0 8 / 2 => remainder 0 => 0 ``` 8 base-ten is equivalent to 1000 base-2. This page has a great description of the divide by two algorithm. This video describes the algorithm as well. Convert 12 base-ten to the corresponding base-2 number. We will use the divide by two algorithm again to solve this problem. We are using integer division when running the algorithm. Integer division is when the fractional part of the division is discarded, so 3 / 2 is 1 with integer division (not 1.5). ``` 1 => 1 3 / 2 => remainder 1 => 1 6 / 2 => remainder 0 => 0 12 / 2 => remainder 0 => 0 ``` The result is `1100`. Convert 545 base-8 (an octal number) to the corresponding base-ten number. ```digits: 5 4 5 numbering: 2 1 0 5*(8**2) + 4*(8**1) + 5*(8**0) = 357 ``` Convert D base-16 to the corresponding base-ten number. Base-16 is also referred to as hexadecimal. Base-16 numbers use the following digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. A is 10, B is 11, C is 12, and so forth. D base-16 is equivalent to 13 base-ten. Convert 357 base-ten to the corresponding hexadecimal number. We will use a generalized version of the divide by two algorithm again to solve this problem (we want to convert to a hexadecimal number, so we'll divide by 16). We start by dividing 357 by 16 to get 22 with a reminder of 5. The remainder is prepended to the result and then 22 is divided by 16 to get 1 with a remainder of 6. The divisor (1) is now less than 16, so it is prepended to the result to get 165. ``` 1 => 1 22 / 16 => remainder 6 => 6 357 / 16 => remainder 5 => 5 ``` The result is 165. Convert 63933 base-ten to the corresponding hexadecimal number. ```15 => F 249 / 16 => remainder 9 => 9 3995 / 16 => remainder 11 => B 63933 / 16 => remainder 13 => D ``` The result is `F9BD`. How many different numbers can be represented by a two digit base-16 number? How many different numbers can be represented by a two digit base-ten number? A two digit base-16 number can represent 256 different numbers `16*16`. A two digit base-ten number can only represent 100 different numbers `10*10`. Convert BD4 base-16 to the corresponding base-ten number. ```digits: B D 4 numbering: 2 1 0 11*(16**2) + 13*(16**1) + 4*(8**0) = 3028 ```