Understanding the Binary system and it's relationship to the digital age

They were trying to design a mathematical way at first to convert electrical circuits to mathematics and then from mathematics they could easily create codes to represent letters as well.

in Binary each place is represented by a zero or a 1

so 0001 would be the decimal system 1
     0010 would be the decimal system 2

The advantage of this is you can use this on electrical circuits, now especially in digital chips. So each circuit is either on which would be a binary 1 or a zero which is off (like a light switch) would be a place holder  as non-activated circuit and depending upon which place it is in would represent different decimal numbers.

Then they figured out a alpha number code beyond this for digital representation of letters as well as numbers. So, when you see an "a" this is a decimal code too but is instead of a numeric code it is an alpha numeric code.

This is binary for 1 through 17  converted into decimal.

0hex	=	0dec	=	0oct		0	0	0	0
1hex	=	1dec	=	1oct		0	0	0	1
2hex	=	2dec	=	2oct		0	0	1	0
3hex	=	3dec	=	3oct		0	0	1	1
4hex	=	4dec	=	4oct	0	1	0	0
5hex	=	5dec	=	5oct	0	1	0	1
6hex	=	6dec	=	6oct	0	1	1	0
7hex	=	7dec	=	7oct	0	1	1	1
8hex	=	8dec	=	10oct	1	0	0	0
9hex	=	9dec	=	11oct	1	0	0	1
Ahex	=	10dec	=	12oct	1	0	1	0
Bhex	=	11dec	=	13oct	1	0	1	1
Chex	=	12dec	=	14oct	1	1	0	0
Dhex	=	13dec	=	15oct	1	1	0	1
Ehex	=	14dec	=	16oct	1	1	1	0
Fhex	=	15dec	=	17oct	1	1	1	1

Binary may be converted to and from hexadecimal somewhat more easily. This is because the radix of the hexadecimal system (16) is a power of the radix of the binary system (2). More specifically, 16 = 2⁴, so it takes four digits of binary to represent one digit of hexadecimal, as shown in the adjacent table.
To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:

3A₁₆ = 0011 1010₂

E7₁₆ = 1110 0111₂

To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra 0 bits at the left (called padding). For example:

1010010₂ = 0101 0010 grouped with padding = 52₁₆

11011101₂ = 1101 1101 grouped = DD₁₆

To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:

C0E7₁₆ = (12 × 16³) + (0 × 16²) + (14 × 16¹) + (7 × 16⁰) = (12 × 4096) + (0 × 256) + (14 × 16) + (7 × 1) = 49,383₁₀

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_{So, if you  go beyond 17 you have to add another place so instead of 1111 for the decimal 17 you would have to go to 00000 a fifth place to represent further. So, each time you run out of a zero and a 1 to represent something you have to add another place.}

_{You might think this is awkward but all digital computers at core start with idea at a machine language point which is below where programmers usually deal with but many programmers can "in an emergency" program in machine language too even though this can be extremely time consuming.}

_{However if things break down in certain ways you either do this or throw away the machinery one or the other.}

begin quote from:

Binary number from wikipedia:

_{Also, from a hexidecimal point of view you can see how the letters A through F are created in a binary way. Each number or letter can be a bit.} 

Decimal pattern	Binary number
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.

Decimal counting

Decimal counting uses the ten symbols 0 through 9. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the first digit. When the available symbols for this position are exhausted, the least significant digit is reset to 0, and the next digit of higher significance (one position to the left) is incremented (overflow), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:

000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)

010, 011, 012, ...

   ...

090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)

100, 101, 102, ...

Binary counting

This counter shows how to count in binary from numbers zero through thirty-one.

Binary counting follows the same procedure, except that only the two symbols 0 and 1 are available. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:

0000,

0001, (rightmost digit starts over, and next digit is incremented)

0010, 0011, (rightmost two digits start over, and next digit is incremented)

0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)

1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...