Understanding Binary Numbers in Computers

Study Notes

Binary Numbers in Computers

Binary is the basis of all computer operations and calculations. It's used by virtually every type of modern electronic device with a processor. All data found within these devices comes down to ones and zeros. This system is fundamental because it allows us to represent information in a very compact form – just one bit can store two pieces of information!

Binary Representation

Representing something using only zeroes and ones might seem limiting, but there are far more possibilities in binary than we commonly think. Each number has its own sequence of zeros and ones. A single digit in binary represents a power of two, starting from the rightmost digit which represents 1, and each successive digit represents twice the value of the previous digit; for example, '1' equals 1 times 2^0, '10' equals 2 time s2^-1. Thus, any positive whole number can be represented using this base-two numeral system. For instance, take the number five:

Five in binary is:  101
The sequence starts like so:    2+4=6;   8+4=12;   16+4=20;   32+4=36...

So you see how important those first few bits are when interpreting the rest of them? That's why humans need ways of writing large integers in binary format quickly.

Binary Arithmetic

Every operation performed inside your computer — addition, multiplication, division, etc. — is done in binary. There are special rules for performing arithmetical functions on binary numbers. When adding two binary digits together, the result is determined by their sum modulo 2. If both digits are even, so will be their total. Likewise if neither is odd, they equal zero. But if one is odd and the other is even, they produce an odd result.

Addition becomes simple once you understand how carry works - you have just two choices: either carry over one unit; don't carry anything.

Multiplication requires shifting & logical AND operations between multiple binary digits, while dividing involves borrowing and shifting until reaching a remainder of 0.

Binary Logic Gates

Logic gates feed off input values and generate output based on predefined Boolean logic expressions. They determine whether certain conditions are met before allowing other actions to happen. These gate types operate using specific combinations of inputs and outputs.

AND gate: Only allows an action once all necessary conditions are present simultaneously. In order to trigger an event associated with the output wire connected to the OR gate, every condition must pass through it successfully.

OR gate: Once even just one requirement amongst several possible triggers occurs, it enables further steps forward in execution.

NOT Gate (also known as Inverter): Negates or reverses the state of its input signal into its output signal while retaining the polarity of its original voltage level.

NAND gate: Produces true whenever none of its input signals are true. Otherwise, it produces false.

Binary to Decimal Conversion

A second form of arithmetic most people know nothing about is converting between bases such as hexadecimals or decimals. These conversions also involve powers and place values similar to those in traditional decimal systems.

To convert from binary to decimal, simply multiply each position's value according to its exponent, then add up results after reversing direction leftward: Start at the end of string where powers decremented start being counted backwards till you reach zero; Multiply each character by its matching exponentative factor; Add up products sequentially moving back towards right side due incrementation process. Here's our earlier binary code for five broken out step by step:

[ \begin{align*} & {5 = 1 \times 2^{2} + 1 \times 2^{1} + 0 \times 2^{0}} \ & \text{or} \ & {5 = 4 + 1 + 0 }\ & {= 5}\ \end{align*}]

And voilà, the number 5 in decimal form is obtained! Now you may wonder what happens when you cross larger boundaries… Well let me tell ya: You get overflow errors!.

Binary Coding Schemes

Computer programs often transmit packets containing instructions and data via networks like Ethernet cables or radio waves using standardized messaging protocols called frames, segments, or packets depending upon different layers of OSI model. As mentioned above, these messages are encoded in digital form as sequences of electrical pulses representing zeroes and ones that flow along wires, optical fibres, airwaves—whatever medium connects pairs of electronics.

Description

Learn the basics of binary representation, arithmetic, logic gates, conversions to decimal, and coding schemes used in computer operations. Explore how data is processed and transmitted in the form of ones and zeros in electronic devices.