Binary Numbers and Base Systems as Fast as Possible

Summary

The video explains how computers use binary (1s and 0s) through electricity to operate. It introduces the concept of positional notation in number systems, explaining how base 10 (decimal) works, then compares it to base 2 (binary). The video explores other numbering systems, like base 12 and base 36, and discusses how alphanumeric characters are used in systems like URL shorteners. It also touches on the historical preference for base 10 due to human anatomy, specifically our 10 fingers. Finally, the video promotes learning platforms like Lynda.com for further education.

Takeaways

• 🔌 Modern computers use binary, a system of 1s and 0s, to process information.
• 🔢 Our familiar decimal system, or base 10, uses ten symbols (0-9) and positional notation to represent numbers.
• 📚 Positional notation allows us to reuse the same digits in different positions to represent larger numbers, each position having a value ten times greater than the one to its right.
• 🛑 Binary, or base 2, operates on the same principles but uses only two symbols, with each new digit having a value twice as much as the one to its right.
• 💡 Binary counting is as simple as performing multiplication and addition, but it forms the basis for more complex computer operations.
• 🌐 There are many base systems beyond decimal and binary, such as base 8 or base 12, which can be more efficient for certain calculations.
• 🤔 The use of base 10 is likely due to humans having 10 fingers, a historical artifact rather than a practical choice.
• 🔤 When dealing with bases higher than 10, letters are used to represent numerals, leading to alphanumeric systems.
• 🔗 URL shorteners use a combination of numerals and letters to represent large numbers, allowing for the creation of short links from long URLs.
• 🎓 The video also promotes lynda.com as a valuable resource for learning new software and skills through structured, high-quality video tutorials.

Q & A

How do modern-day computers use electricity to function?

Modern-day computers use electricity by turning it on or off inside a microchip, represented by the binary symbols 1 and 0.

What is the significance of the binary system in computing?

The binary system is significant in computing because it provides a simple and efficient way to represent data using only two states, on and off, which correspond to the digits 1 and 0.

How does the decimal system (base 10) work?

The decimal system works by using 10 symbols (0-9) and a positional notation where each new digit to the left has a value ten times greater than the digit to its right.

Why do we use base 10 for our numerical system?

We use base 10 likely because humans have 10 fingers, which made counting and representing numbers in base 10 a natural choice.

What is positional notation and how does it apply to different base systems?

Positional notation is a method of representing numbers where each position has a value that is a multiple of the base. This principle applies to all base systems, whether it's base 10, binary, or any other base.

How is a binary number calculated?

A binary number is calculated by multiplying each digit by 2 raised to the power of its position, starting from 0 on the right, and then summing these values.

What are the advantages of using base systems other than base 10 for everyday math?

Base systems like base 8 and base 12 are advantageous for everyday math because they are more easily divisible than base 10, which can simplify calculations.

Why do we use letters to represent numerals in base systems with more than 10 digits?

Letters are used to represent numerals higher than 9 in base systems with more than 10 digits because there are not enough single symbols to represent all the values needed.

How do URL shorteners work?

URL shorteners work by representing a very large number using a combination of numerals and letters from the alphabet, allowing a long URL to be shortened into a more manageable form.

What is the maximum value that can be represented with 10 alphanumeric digits in a base 62 system?

With 10 alphanumeric digits in a base 62 system, the maximum value that can be represented is 14 million.

What is the connection between the concept of positional notation and the way URL shorteners encode large numbers?

The connection between positional notation and URL shorteners is that both use a base system to encode values, with URL shorteners using a base that includes letters to represent large numbers in a compact form.

Outlines

00:00

🔢 Understanding Binary and Base 10 Systems

This paragraph introduces the concept of how modern computers use electricity, which is represented as binary (1 and 0) inside a microchip. The author explains how binary works by comparing it to our familiar base 10 system. They describe positional notation in the base 10 system, where each new digit represents a value 10 times greater than the previous one. This logic applies to binary as well, where the two symbols (0 and 1) follow the same principle. The paragraph concludes by explaining how binary calculations work through multiplication and addition of the digit positions.

05:01

✍️ A Brief History of Number Systems and Alphanumerics

The second part delves into the history and reasoning behind the use of the base 10 system, which likely originated from humans having 10 fingers. While other systems like base 8 or base 12 are more efficient for division, base 10 became the standard. The author briefly touches on the superiority of the metric system before transitioning into how higher base systems, such as base 12 and beyond, utilize letters to represent numerals greater than 9. They explain that alphanumeric systems are used for large numbers, like in URL shorteners, allowing for an immense range of possible values with fewer characters.

Keywords

💡Binary

Binary is a base-2 numeral system that uses only two symbols, 0 and 1, to represent all numerical values. In the context of the video, binary is the fundamental language of modern-day computers, where electricity is turned on (1) or off (0) to perform operations. The video explains that binary is a simple system of multiplication and addition, which is essential for understanding how computers process information.

💡Decimal

Decimal, also known as base-10, is the numerical base system most commonly used by humans. It involves ten symbols: 0 through 9. The video explains that we use decimal because it is convenient for us, given that we have ten fingers. The script uses decimal as a point of comparison to introduce the concept of binary and other base systems.

💡Positional Notation

Positional notation is a system where each digit in a number has a value that depends on its position. In the video, it is explained that in the decimal system, each new digit to the left has a value ten times greater than the digit to its right. This concept is crucial for understanding how numbers are represented and calculated in any base system, including binary.

💡Bits and Bytes

Bits and bytes are units of data in computing. A bit is the basic unit, representing a single binary digit (0 or 1). A byte consists of eight bits. The video mentions these terms in the context of expanding the discussion from simple binary numbers to the more complex data structures used in computers.

💡Boolean Logic

Boolean logic, named after George Boole, is a system of logical operations used in computer science. It deals with true or false values and is fundamental to how computers process information. The video script alludes to Boolean logic as part of the complexity that arises when moving beyond basic binary operations.

💡ASCII

ASCII (American Standard Code for Information Interchange) is a character encoding standard for electronic communication. The video briefly mentions ASCII as part of the progression from binary to more complex systems, where characters and symbols are represented numerically in computers.

💡Base Systems

Base systems refer to the different numeral systems used to represent numbers, such as base-2 (binary), base-8 (octal), base-10 (decimal), and base-16 (hexadecimal). The video discusses various base systems to illustrate the flexibility and logic behind number representation, with binary being the most relevant to computer science.

💡Alphanumeric

Alphanumeric characters are any letter or number used in communication. In the video, alphanumeric is mentioned in the context of representing numbers beyond 9 in base systems that use more than ten symbols, such as base-12. It also relates to the use of letters in URL shorteners to represent large numbers.

💡URL Shorteners

URL shorteners are services that convert a long URL into a shorter version. The video explains that these shortened URLs use alphanumeric characters to represent large numbers, which is possible due to the vast number of combinations available in higher base systems.

💡Metric System

The metric system is an internationally adopted decimal system of measurement. The video humorously compares the transition to a new base system to the United States' resistance to adopting the metric system, highlighting the difficulty of changing established systems.

Highlights

Modern-day computers use binary, represented by 1s and 0s, to process information.

Decimal system uses 10 symbols, while binary uses only two.

Positional notation is a system where each digit's value is ten times greater than the digit to its right.

Binary counting involves simple multiplication and addition.

The complexity of computing increases with concepts like bits, bytes, boolean logic, and ASCII.

Other base systems like base 2, 3, 4, and 5 operate on the same principles as the decimal system.

Base 10 is used likely because humans have 10 fingers.

Base 8 and base 12 are more divisible and could be superior for everyday math.

Switching from base 10 would be as difficult as changing the imperial system to metric.

Base 12 and higher systems use letters to represent numerals higher than 9, forming alphanumeric characters.

URL shorteners use a combination of numerals and letters to represent large numbers.

Base 36 can represent numbers up to 14 million with only four digits.

With 10 digits, base 62 can represent up to 839 quadrillion possible values.

Lynda.com is recommended for learning new software and skills with high-quality video tutorials.

Lynda.com offers a wide variety of subjects and software tutorials.

Access to Lynda.com starts at $25 a month, with a free 7-day trial available.

Transcripts

modern-day computers use electricity to

work and inside of a microchip

electricity is turned either on or off

which is represented by the symbols 1

and 0. this is called binary you've

probably heard of binary already and

that that's how computers work but do

you know how binary works well you're

about to find out but first we need to

understand exactly how our numerical

base system known as decimal or base 10

works the way it does so there are 10

count em 10 symbols that we use for all

of our numbers starting from 0 we can

count all the way up to 9 before we run

out of symbols to use now we could just

keep adding symbols at this point but

that would get out of hand very quickly

i mean can you imagine having to

memorize a specific symbol for every

single number that's ridiculous and

that's why we reuse the same symbols

over and over again in a very clever

system called positional notation so in

the base 10 system as soon as we get to

10 or an exponent of 10 we need to add

another digit to the left of our current

digit because there are 10 symbols each

new digit has to have a value 10 times

greater than the digit to its right so

that's using 10 symbols but what if you

had only two symbols to work with well

then everything that i said still

applies with just two symbols each new

digit needs to have a value two times

greater than the digit to its right

so a sequence like this would equal 1

times 128 plus 1 times 16 plus 1 times 8

plus 1 times 2 plus 1 which is

155

and that's how you count in binary it's

actually really simple it's just

multiplication and addition now it gets

a lot more complicated from here with

bits and bytes and boolean logic and

ascii and the list just goes on and on

so let's return to base systems there

are a lot of ways to write numbers other

than decimal and binary you've got base

two base three base four base five i

could go on they all work with the same

principles of positional notation so you

might be wondering with all these

numbering systems to choose from why do

we use base 10 that's a good question

this goes all the way back to roman

numerals and egyptian hieroglyphs it's

likely that we use base 10 simply

because we have

10 fingers also known as digits other

base systems like base 8 and base 12 are

actually superior for simple everyday

math since 8 and 12 are much more easily

divisible than 10 but it's definitely

too late to change our minds about using

base 10 we'll probably be stuck with it

forever switching away from it now would

be even harder than trying to convince

america to drop the imperial system and

finally switch to metric you know like

the rest of the civilized world like

yeah the metric system is superior but

who's going to tell america what to do

now if you're going to be using base 12

or any other base system with more than

10 digits it's standard to use letters

to represent numerals higher than 9. so

10 is a 11 is b 12 is c and so on this

is called alphanumeric you know those

url shorteners that you see on twitter

and elsewhere have you ever wondered how

they work all those jumbled characters

really just represent a very large

number by using numerals and every

letter of the alphabet you can get all

the way up to base 36 using lowercase

and uppercase letters gives you base 62

and with that you can get all the way up

to 14 million with only four digits with

just 10 digits you can get up to

839 quadrillion possible values that's a

lot of shortened urls so you just

learned about positional notation binary

numbers numeral based systems

alphanumeric characters and url

shorteners i hope you enjoyed it and if

you're in the mood for more learning

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