1.1 Number systems

1. Know which system you are dealing with.

   In this case, we are dealing with unsigned binary numbers. Our range of possible numbers are between 0 and 2^N-1 .

2. Write out the sum of each digit multiplied by its correct power of two depending on its position
   10101=12^4+02^3+12^2+02^1+12^0=12^4+12^2+1=16+4+1=21

Write out the sum of each digit multiplied by its correct power of two:

1. Find the largest multiple of two less than or equal to the decimal number

   For the decimal number 47, the largest multiple of two is 32 ( $2^5$ ).

2. Subtract the largest multiple from the decimal number.
   $47-32=15$
3. Repeat Step 1 and 2 until there is no remainder
   $15-8=7$
   $7-4=3$
   $3-2=1$
   $1-1=0$
4. Construct the binary number out of the powers of two subtracted from the decimal number
   $47=32+8+4+2+1=2^5+2^3+2^2+2^1+2^0=101111(binary)$

   Note: If necessary, you can check your answer by reversing the steps and converting it back to decimal.

1. Check most significant bit to see if it is negative or positive

   The most significant bit is 1. This means it is negative.

2. Sum the powers of two for the bits after the most significant one
   $1001110=2^6+2^3+2^2+2^1=64+8+4+2=78()$
3. Attach the sign to the decimal number

   Thus the answer is -78.

1. The most significant bit holds the sign value

   In this case, the decimal number is negative so the most significant bit is 1.

2. Subtract the largest multiple of 2 less than or equal to the decimal number
   98-2^6=98-64=34
3. Repeat this step until there is no remainder. Note down the powers of two
   34-2^5=34-32=2
   2-2^1=0
4. Construct the decimal number from the powers of two from the previous steps
   -98=-(2^6+2^5+2^1)=11100010

1. The most significant bit holds the sign value

   In this case, the decimal number is negative so the most significant bit is 0.

   Since we have already calculated the binary representation for 98, we can use the answer from the previous example. The steps are shown again to illustrate this.

2. Subtract the largest multiple of 2 less than or equal to the decimal number
   98-2^6=98-64=34
3. Repeat this step until there is no remainder. Note down the powers of two
   34-2^5=34-32=2
   2-2^1=0
4. Construct the decimal number from the powers of two from the previous steps
   $98=2^6+2^5+2^1=01100010$

1. The most significant bit holds the sign value following the same convention as signed binary

   In this case, the most significant bit is 0. The number is positive.

2. Since this number is positive, we can treat the remaining binary digits as per normal
   $001011=1011=2^3+2^1+2^0=8+2+1=11$

1. Check the most significant bit for sign value

   The first bit is 1 so the number is negative.

2. Invert the remaining bits

   11011 -->00100

3. Add 1 and calculate the decimal value of the number
   $00100+1=00101=2^2+2^0=5$
4. Add the negative sign to decimal value.

   Thus the answer is -5.

1. Use the sign of the decimal value to choose the most significant bit

   The number is negative so the most significant bit will be 1.

2. Convert the decimal value into binary
   $13=8+4+1=2^3+2^2+2^0=1101$
3. Subtract 1 and invert the binary numbers
   $1101-1=1100$

   1100 -->0011

4. Sign extend the number to the appropriate number of digits

   0011 -->11110011

1. Convert the letters to their decimal values using the table

   ABC -->10 , 11 , 12

2. Multiply the decimal values by the correct power of 16 based on their significant position. Add the values
   $1016^2+1116^1+1216^0=2560+176+12=2748$
3. Convert the decimal values to 4-bit binary numbers

   ABC -->10 + 11 + 12 -->1010 1011 1100

4. The binary number can be constructed with these values simply by merging them

   1010 1011 1100 -->101010111100

1. Separate the binary number into blocks of 4 starting from the least significant bit

   1010011110000001 -->1010 0111 1000 0001

2. Convert each 4-bit block into its hexadecimal value to get the answer

   1010 0111 1000 0001 -->A781

3. It is usually easier to convert from hexadecimal to decimal than from binary to decimal

   A781 -->10, 7, 8, 1

   $1016^3+716^2+816^1+116^0=40960+1792+128+1=42881$