Converting Common Numeral Systems to BASE 10

A numeral system in a way to represent or express numbers using digits or other symbols in a consistent manner.

The most widely used numeral system by people is decimal, or BASE 10.

Definitions

10^4

In the example above, the number 10 is called the BASE (or radix / root), and the number 4 is called the exponent. The exponent corresponds to the number of times the base is used as a factor (4 x 4 x 4 x 4).

Binary (0, 1) - used by logic gates in most digital circuits

To convert the binary number 1011 to decimal,

 1         0         1         1             # Your binary number
(1x2^3) + (0x2^2) + (1x2^1) + (1x2^0) # Multiply your number by the power of the base that the digit represents
   8    +    0    +    2    +    1    = 11   # Add the numbers together to get the result in BASE 10


Octal (0..7) - used often in computing

To convert the octal number 350 into decimal,

 3         5         0 # Your octal number
(3x8^2) + (5x8^1) + (0x8^0) # Multiply your number by the power of the base that the digit represents
  192   +   40    +    0    = 232 # Add the numbers together to get the result in BASE 10


Hexadecimal, hex (0..9, a..f) - used often in computing

To convert the hex number 1F20 into decimal,

 1   F      2     0 # Your hex number
(1x16^3) + (Fx16^2)  + (2x16^1) + (0x16^0) # Multiply your number by the power of the base that the digit represents
(1x16^3) + (15x16^2) + (2x16^1) + (0x16^0) # Convert the hex numbers into the decimal equivilent
4096     +  3840     +  32      +  0       = 7968 # Add the numbers together to get the result in BASE 10


Further Reading:

http://whatis.techtarget.com/definition/logic-gate-AND-OR-XOR-NOT-NAND-NOR-and-XNOR

Online Toolbox

Converter: http://coderstoolbox.net/number/
- Does string, number, time, network, and bandwidth conversion

Security+ Certification

I recently passed the CompTIA Security+ ce exam. I've never been much on certifications, but I decided to take this test to formalize some of my knowledge. 

About the Test

There were 72 questions and I had 90 minutes to complete the test. This was enough time to complete the test and to fully review all of the questions. A few of the questions were 'simulation' questions and the rest were multiple choice. 

To prepare, I purchased the CertMaster training, which can be bought as an add-on when you purchase your exam voucher.  https://certification.comptia.org/training/certmaster/security

The above training takes ~15 hours to complete the initial 702 practice questions. I took the initial training, reviewed concepts that I did not know, and reviewed missed questions twice before taking the test.

A Hex Upon Numerals

BASE 16

0123456789abcdef

Definition

In mathematics and computing, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or alternatively a, b, c, d, e, f) to represent values ten to fifteen. -- https://en.wikipedia.org/wiki/Hexadecimal

Notes

• Hex numbers uses digits from 0..9 and A..F
• 0xVALU  - Unix/C prefix
• Other than 0xVALU, VALU16 is a common notation
• Each character is the equivalent of a nibble (4 bits)

Converting hex (BASE 16) to decimal (BASE 10)

A byte can be represented as:

n·16^3 n·16^2 n·16^1 n·16^0

OR

n·4096 n·256 n·16 n·1

*Any nonzero number raised to the zero power is equal to one (i.e., n^0=1). This concludes that all nonzero numbers raised to the zero power are equivalent because they all equal 1. Note that it is any nonzero number, since 00 is undefined.

Conversion Examples

0x31 = (3·16^1)+(1·16^0) = (48) + (1) = 49
0xc9 = (12·16^1) + (9·16^0) = (192) + (9) = 201
0x1f = (1·16) + (15·1) = 31

Decimal to Hexadecimal to Binary Conversion Table

Dec Hex Bin
B10 B16  B2 
0 0 00000000
1 1 00000001
2 2 00000010
3 3 00000011
4 4 00000100
5 5 00000101
6 6 00000110
7 7 00000111
8 8 00001000
9 9 00001001
10 a 00001010
11 b 00001011
12 c 00001100
13 d 00001101
14 e 00001110
15 f 00001111
16 10 00010000
17 11 00010001
18 12 00010010
19 13 00010011
20 14 00010100

References

Binary is as easy as 1, 10, 11

Definition

Binary - In mathematics and digital electronics, a binary number is a number expressed in the binary numeral system or base-2 numeral system which represents numeric values using two different symbols: typically 0 (zero) and 1 (one).- https://en.wikipedia.org/wiki/Binary_number

Notes

• BASE 2 Counting method only uses 0 and 1
• One binary digit is called a bit
• Two digits are called a crumb, four digits are called a nibble, and eight digits are called a byte
• Lower Case letters start with 011xxxxx
• Upper Case letters start with 010xxxxx
• When counting, range 0-1, carry the 1 (i.e. 00000000, 00000001, 00000010, 00000011, 00000100, 00000101, 00000110, 00000111, 00001000, 00001001, 00001010 ...)
• 1+1=10

Converting binary (BASE 2) to decimal (BASE 10)

A byte can be represented as:

n·2^7 | n·2^6 | n·2^5 | n·2^4 | n·2^3 | n·2^2 | n·2^1 | n·2^0

OR

n·128 | n·64 | n·32 | n·16 | n·8 | n·4 | n·2 | n·1*

*Any nonzero number raised to the zero power is equal to one (i.e., n^0=1). This concludes that all nonzero numbers raised to the zero power are equivalent because they all equal 1. Note that it is any nonzero number, since 00 is undefined.

Conversion Examples

10 = 1·2^1 | 0·2^0 = 1·2 + 0·1 = 2
111 = 1·2^2 | 1·2^1 | 1·2^0 = 1·4 + 1·2 + 1·1 =7
1011 =  1·2^3 | 0·2^2 | 1·2^1 | 1·2^0 = 1·8 + 0 + 1·2 + 1 = 11

Converting binary (BASE 10) to decimal (BASE 2)

2^4 2^3 2^2 2^1 2^0

16 8 4 2 1

Use the value in the second row to count to the value--you can use each value no more than 1 time (since we count 0..1). For example, we want to convert 22 to binary. To do this we need:

(1 x 16) + (0x8) + (1x4) + (1x2) + (0x1)

10110 = 22

Letter to ASCII to Binary Character Table

Letter	ASCII Code	Binary	Letter	ASCII Code	Binary
a	097	01100001	A	065	01000001
b	098	01100010	B	066	01000010
c	099	01100011	C	067	01000011
d	100	01100100	D	068	01000100
e	101	01100101	E	069	01000101
f	102	01100110	F	070	01000110
g	103	01100111	G	071	01000111
h	104	01101000	H	072	01001000
i	105	01101001	I	073	01001001
j	106	01101010	J	074	01001010
k	107	01101011	K	075	01001011
l	108	01101100	L	076	01001100
m	109	01101101	M	077	01001101
n	110	01101110	N	078	01001110
o	111	01101111	O	079	01001111
p	112	01110000	P	080	01010000
q	113	01110001	Q	081	01010001
r	114	01110010	R	082	01010010
s	115	01110011	S	083	01010011
t	116	01110100	T	084	01010100
u	117	01110101	U	085	01010101
v	118	01110110	V	086	01010110
w	119	01110111	W	087	01010111
x	120	01111000	X	088	01011000
y	121	01111001	Y	089	01011001
z	122	01111010	Z	090	01011010

References 

• http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary
• https://www.mathsisfun.com/numbers/binary-count-fingers.html
• https://en.wikipedia.org/wiki/Binary_number