Math Review

Positional Number Systems

Decimal

3210 column  position
1493₁₀ = 1*10³ + 4*10² + 9*10¹ + 3*10⁰

3210-1-2 column  position
1493.76₁₀ = 1*10³ + 4*10² + 9*10¹ + 3*10⁰ + 7*10^-1 + 3*10^-2

Binary

43210 column  position
10110₂ = 1*2⁴ + 0*2³ + 1*2² + 1*2¹ + 0*2⁰
       = 16 + 4 + 2
       = 22₁₀

43210-1-2 column  position
10110.01₂ = 1*2⁴ + 0*2³ + 1*2² + 1*2¹ + 0*2⁰ + 0*2^-1 + 1*2^-2
           = 16 + 4 + 2 + 1/4
          = 22 1/4₁₀

Hexadecimal

3210 column  position
12B4₁₆ = 1*16³ + 2*16² + 11*16¹ + 4*16⁰
       = 4096 + 512 + 176 + 4
       = 4788₁₀

3210-1-2 column  position
12B4.0A₁₆ = 1*16³ + 2*16² + 11*16¹ + 4*16⁰ + 0*16^-1 + 10*16^-2
          = 4096 + 512 + 176 + 4 + 10/256
          = 4788 10/256₁₀

Memorize 0-15 Decimal/Binary/Hexadecimal Conversions

Decimal/Binary/Hexadecimal

Decimal	Binary	Hexadecimal		Decimal	Binary	Hexadecimal
0	0000	0		8	1000	8
1	0001	1		9	1001	9
2	0010	2		10	1010	A
3	0011	3		11	1011	B
4	0100	4		12	1100	C
5	0101	5		13	1101	D
6	0110	6		14	1110	E
7	0111	7		15	1111	F

Question

Expand from positional representation and write as decimal.

1. 123.4₁₀
2. 1AB₁₆
3. 11001.1₂

Conversions

From any Base to Decimal - Write in positional form Base 10. See Positional Number Systems above.

From Decimal to any Base - Algorithm can be roughly stated as:

1. Divide Number by base.
2. Write Remainder
3. Set Number = Quotient
4. Go to 1 if Quotient not equal 0
5. Result is remainder of each division, written in reverse order produced.

• Decimal to Decimal

Convert 245₁₀ to decimal.

     24 Quotient              2               0           
10/ 245                   10/24           10/ 2         
   -20                      -20              -0          
     45                       4               2          
    -40
      5  Remainder

Result 245₁₀ = 245₁₀


 

• Decimal to Binary

Convert 25₁₀ to binary.

   12  Quotient              6              3           1          0 
2/ 25                    2/ 12           2/ 6         2/3        2/1
   24                       12              6           2          0
    1  Remainder             0              0           1          1

Result 25₁₀ = 11001₂


 

• Decimal to Hexadecimal

Convert 4788₁₀ to hexadecimal.
 

    299 Quotient             18                1             0
16/4788                 16/ 299           16/ 18          16/1
     :                       :                16             0
     4 Remainder             1110 = B16         2             1

Result 4788₁₀ = 12B4₁₆

 

Binary to Hexadecimal - Group right to left by 4's and use memorized table on each group.

• 1011011010₂
• grouped by 4's is: 0010  1101  1010
• Converted is:        2     D    A₁₆
   
• Result is: 1011011010₂ = 2D4₁₆

 

Hexadecimal to Binary - Use memorized table to convert one hexadecimal digit into corresponding 4 binary digits.

• 12B4₁₆
• Hexadecimal:    1    2    B    4
• Binary:        0001 0010 1011 0100
  
   
• Result is:    12B4₁₆ = 0001001010110100₂

Question

Convert 83₁₀ to:

4. Binary
5. Hexadecimal
    
6. Convert 11001101₂ to hexadecimal.
7. Convert 1B42₁₆ to binary.

Arithmetic

You know these!

Binary Definitions

+        0       0        1       1
        +0      +1       +0      +1
         0       1        1      10

-        0        0        1       1
        -0       -1       -0      -1
         0       -1        1       0

*        0        0        1        1
        *0       *1       *0       *1
         0        0        0        1

/        0/0 undefined    1/0 undefined
         0/1 = 0          1/1 = 1

       10110        10110        1010              1010
       +1011        -1011        *101        101/110010
      100001         1011        1010           -101
                                0000              010
                              +1010              -000  
                               110010              101
                                                  -101
                                                    000
                                                    000

Question

       8.           9.           10.             11.

       10101        10101        1011            _____
       +1011        -1011        *101        101/10010
                           
 

Hexadecimal Arithmetic - Most of us do basic decimal math in our head, so convert column digits to decimal, do arithmetic, convert back to hexadecimal.

Examples

 ACE                    167B
+BAD                    -ACE
167B                     BAD

 CDE                    1C89
+FAB                    -FAB
1C89                     CDE

 FF                     100
 +1                      -1
100                      FF

Question

      11.            12.       

      34             B2
     +E9            -45