Mathematics 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
43210-1-2 column position
10110.01₂ = 1*2⁴ + 0*2³ + 1*2² + 1*2¹ + 0*2⁰+ 0*2^-1 + 1*2^-2

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

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:

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

Decimal to Decimal

₁₀

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

₁₀

Decimal to Binary

₁₀

   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

₁₀

₂

Decimal to Hexadecimal

₁₀

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

₁₀

₁₆

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

Arithmetic

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

Examples

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

Hexadecimal Arithmetic - Convert 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