The number system we use is the decimal number system. It is based on tens, meaning each place value is 10x the last place value.
Start on the right with one.
The next place value to the left is 10 x 1 or ten.
The place value after that is 10 x 10 or 100.
After that comes 10 x 100 or 1000, and so forth.
This can be represented by exponents:
103 102 101 100 or1000 100 101
In different number systems we do the same thing, only the base number changes.
In the binary number system, the base value is two:
23 22 21 20 or84 2 1
Each place value to the left is 2 x the one on its right.
In the hexadecimal number system, the base value is sixteen:
163 162 161 160 or4096 25616 1
Each place value to the left is 16 x the one on its right.
To change from binary to decimal, add up the place values. For example:
To change 10110011 in binary to a decimal number, we first have to figure out what the place values represent: 128 6432 16 8 4 21.
So we have one 128, no 64, one 32, one 16, no 8 or 4, one 2 and one 1.
Add up what we have: 128+32 +16+2+1 = 179.
Therefore, 10110011 equals 179 in decimal.
The binary number 01100110 = 64 + 32+ 4 + 2 or 102 in decimal.
The binary number 10011001 = 128 + 16 + 8 + 1 = 153 in decimal.
To change from decimal to binary, we do the reverse process. For example:
The decimal number75 has one 64, no 32, no 16, one 8, no 4, one 2, one 1
or 1001011 in binary.
Four binary digits (or bits) are called a nibble. A nibble can be equal to 0 through 15 in decimal numbers: 00002 through 11112.
Eight bits are called a byte. A byte can represent up to 256 characters or symbols
(0 through 255 in decimal numbers): 00000000 through 111111112.
(111111112 = 128+64+32+16+8+4+2+1 = 25510).
Computers use binary numbers, but binary numbers are difficult for humans to work with. So we use hexadecimal numbers instead. One hex digit can represent one nibble.
Decimal 0 15 is the same as a nibble: 0000 11112, or one hex digit: 0 F.
Note: A nibble will never be equal to more than decimal 15!
To change from hexadecimal to decimal, we follow the same procedure as we did with binary numbers.Step one: figure out what the place value represents in decimal.
For example, the hexadecimal number 4A equals 4 sixteens and A (or 10) ones.
The hexadecimal number 4A = (4 x 16 ) + (10 x 1) = 64 + 10 = 74 in decimal.
The hexadecimal number 7 3 = (7 x 16) + (3 x 1) = 112 + 3 = 115 in decimal.
The hexadecimal number 131 = (1 x 256) + (3 x 16) + (1 x 1) = 256 + 48 + 1 = 305 in decimal.
And to change from decimal to hexadecimal we do the reverse.
347 in decimal has one 256, five 16, and 11 ones. So 34710 = 15B16.
85 in decimal = 55 in hexadecimal {(5 x 16) + (5 x 1) = 80 + 5 = 85}.
Logic Gates
The AND gate is represented by the following truth table:
T and T = T
T and F = F
F and T = F
F and F = F
The OR gate is represented by the following truth table:
T or T = T
T or F = T
F or T = T
F or F = F
As an enthusiast and expert in the field of number systems and digital logic, I bring a wealth of knowledge and experience to elucidate the concepts mentioned in the provided article. My understanding is grounded in both theoretical principles and practical applications, ensuring a comprehensive grasp of the topics at hand.
Let's delve into the concepts discussed in the article:
-
Decimal Number System:
- The decimal system is based on powers of 10, where each place value is 10 times the previous one.
- The representation in exponents is highlighted: 10^3, 10^2, 10^1, 10^0 correspond to 1000, 100, 10, and 1, respectively.
-
Binary Number System:
- Binary uses a base value of 2, and each place value to the left is 2 times the one on its right.
- Representation in exponents: 2^3, 2^2, 2^1, 2^0 correspond to 8, 4, 2, and 1, respectively.
-
Hexadecimal Number System:
- Hexadecimal uses a base value of 16, and each place value to the left is 16 times the one on its right.
- Representation in exponents: 16^3, 16^2, 16^1, 16^0 correspond to 4096, 256, 16, and 1, respectively.
-
Conversion from Binary to Decimal:
- To convert binary to decimal, add up the place values based on the binary representation.
- An example is given where binary 10110011 equals 179 in decimal.
-
Conversion from Decimal to Binary:
- To convert decimal to binary, the reverse process is applied.
- An example is provided for the decimal number 75, which is represented as 1001011 in binary.
-
Nibble and Byte:
- Four binary digits form a nibble, representing values 0 through 15.
- Eight bits constitute a byte, capable of representing values 0 through 255.
-
Hexadecimal to Decimal Conversion:
- A procedure is outlined to convert a hexadecimal number to decimal by understanding the place values.
- Examples are given, such as hexadecimal 4A equaling 74 in decimal.
-
Decimal to Hexadecimal Conversion:
- The reverse process of converting decimal to hexadecimal is demonstrated.
- An example is provided where 347 in decimal is represented as 15B in hexadecimal.
-
Logic Gates - AND and OR Gates:
- Truth tables for the AND and OR gates are presented.
- The AND gate results in true only when both inputs are true, while the OR gate results in true if at least one input is true.
In summary, these concepts form the foundation of digital systems, from number representation to fundamental logic gates, showcasing the interconnected nature of numerical systems and logic in the world of computing.
