Introduction to number systems and binary | Pre-Algebra | Khan Academy

Summary

This educational video script delves into the evolution of human counting systems, highlighting the transition from basic tallying to complex number systems. It emphasizes the significance of the base 10 (decimal) system, attributed to our 10 fingers, and its efficiency through place value and powers of 10. The script also introduces the binary system, foundational to modern computing, with its simple two-symbol set of 0 and 1, and how it operates on powers of 2. The video promises to explore other systems like hexadecimal in future episodes, aiming to deepen viewers' appreciation for the beauty and utility of numerical representations.

Takeaways

• 🌟 Humans have always counted and sought ways to represent quantities, which led to the development of number systems.
• 🗣️ Early humans might have used simple counting methods, like tallying days since the last rain, before naming numbers.
• 🔢 Every language has unique names for numbers, reflecting the universal need for a system to keep track of quantities.
• 📚 The script introduces the concept of place value, which is crucial for understanding how numbers are represented and calculated.
• 🖐 The base 10 (decimal) system is likely chosen because humans have 10 fingers, making it natural to group and count in tens.
• 🔐 The decimal system uses 10 digits (0-9) to represent numbers, with each position indicating a power of 10, from ones to hundreds, thousands, and so on.
• 💡 The script explains how place value works in the decimal system, using the number 231 as an example to show how each digit contributes to the total value.
• 💻 The base 2 (binary) system is introduced as the foundation of modern computing, with a focus on the on/off states of computer hardware.
• 🔄 The binary system uses only two symbols, 0 and 1, which correspond to the on/off states of computer components like transistors and logic gates.
• 🔢 The script demonstrates how to represent numbers in binary, with a focus on powers of 2, and provides an example of how the decimal number 231 is represented in binary as 11100111.
• 🚀 The video promises to explore other number systems like hexadecimal in future episodes, hinting at the diversity and complexity of numerical representations across different systems.

Q & A

Why did early humans need to count and represent numbers?

Early humans needed to count and represent numbers to keep track of various things, such as the days since it last rained, which was essential for survival and planning.

How did the naming of numbers evolve over time?

Initially, humans used physical objects or gestures to represent numbers. Eventually, they realized the need for standardized names for numbers, leading to the development of numerical words in different languages.

What is the significance of the base 10 number system?

The base 10 number system, also known as the decimal system, is significant because it is based on the number of fingers humans typically have, which is 10. This made it natural to think in terms of bundles of 10.

Why is the base 10 system efficient for humans?

The base 10 system is efficient because it uses place value, allowing us to represent large numbers compactly and perform calculations more easily than if we had to count individual units.

What is the role of the number 231 in illustrating the base 10 system?

The number 231 is used to demonstrate how place value works in the base 10 system, where each digit represents a different power of 10, and the number is the sum of these values.

How does the base 2 number system, or binary, differ from the base 10 system?

The base 2 system, or binary, differs from the base 10 system by using only two symbols, 0 and 1, and having place values that are powers of 2 instead of 10.

Why is the binary system fundamental to modern computing?

The binary system is fundamental to modern computing because it aligns with the on/off states of transistors and logic gates, which are the building blocks of computer hardware.

What is the process of converting a decimal number to binary?

The process of converting a decimal number to binary involves breaking down the number into sums of powers of 2, where each digit in the binary representation indicates the presence or absence of that power of 2.

How is the number 231 represented in binary?

The number 231 is represented in binary as 11100111, which corresponds to one 128, one 64, one 32, zero 16s, zero 8s, one 4, one 2, and one 1.

What other number systems will be explored in future videos according to the script?

In future videos, other number systems such as hexadecimal, which uses 16 digits, will be explored, along with methods for converting between different bases.

Outlines

00:00

📚 The Evolution of Number Systems

This paragraph delves into the historical development of number systems, explaining how early humans began counting and representing quantities. It highlights the cumbersome nature of representing numbers with words and the subsequent invention of numerical symbols. The paragraph emphasizes the significance of the base 10, or decimal, system, which is attributed to the number of human fingers. It illustrates how numbers are constructed in this system through place value, using the example of the number 231. The concept of place value is explored, showing how each position in a number represents a power of 10, leading to the understanding of how numbers are built upon powers of 10.

05:00

💡 Binary System and Modern Computing

The second paragraph focuses on the binary system, which is fundamental to modern computing. It contrasts the decimal system with binary, which uses only two symbols: 0 and 1. The reason for binary's utility in computation is tied to the hardware of computers, which operates in states of on or off, mirroring binary's structure. The explanation extends to how binary places are constructed, with each position representing a power of two rather than ten. The paragraph provides a step-by-step guide to building binary places and culminates in representing the decimal number 231 in binary as 11100111, breaking down its composition to show the equivalence between the two systems. It concludes by hinting at future explorations of other number systems, such as hexadecimal.

Keywords

💡Counting

Counting is the process of determining the number of elements in a set. In the video, counting is depicted as a fundamental human activity, essential for keeping track of time and events. The script uses the example of counting days since the last rain to illustrate the origin of numerical systems. Counting is foundational to the video's theme as it leads to the development of number systems for more efficient representation and manipulation of numbers.

💡Number Systems

A number system is a collection of symbols and rules that represent numbers. The video emphasizes the invention of number systems as a significant advancement, allowing humans to efficiently represent and calculate with numbers. The script explains that number systems evolved to address the limitations of counting, with different cultures developing various systems, each with unique symbols and bases.

💡Base 10 (Decimal System)

The base 10 system, also known as the decimal system, is a positional numeral system that uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The video explains that this system is likely chosen because humans typically have ten fingers, making it natural to group items in tens. The script uses the number 231 as an example to demonstrate how place value works in the decimal system, where each position represents a power of ten.

💡Place Value

Place value is the significance of a digit based on its position within a number. The video script explains how place value works in the decimal system, where each position to the left increases the value by a power of ten. For instance, the number 231 is broken down into 2 hundreds, 3 tens, and 1 one, illustrating the concept of place value.

💡Binary System

The binary system is a base 2 numeral system that uses only two symbols: 0 and 1. The video highlights the binary system as the foundation of modern computing, as it aligns with the on/off states of computer hardware like transistors and logic gates. The script explains how binary numbers are constructed using powers of two, differentiating it from the decimal system's powers of ten.

💡Transistors

Transistors are semiconductor devices used to amplify or switch electronic signals and electrical power. In the context of the video, transistors are mentioned as part of the hardware that enables the binary system to function within computers. They operate in an on/off state, which corresponds to the binary digits of 1 and 0.

💡Logic Gates

Logic gates are fundamental building blocks of a digital circuit, performing basic logical operations on one or more binary inputs. The video script mentions logic gates as components that work with transistors to enable the binary system's computation. They are crucial for the operations that computers perform, which are based on binary.

💡Exponents

Exponents are used to denote the power to which a number is raised. In the video, exponents are explained in relation to place value, where each position in a number represents a different power of the base. For example, the tens place is 10^1, the hundreds place is 10^2, and so on. This concept is used to explain how moving to the left in a number multiplies the value by the base, whether in decimal or binary.

💡Hexadecimal System

The hexadecimal system is a base 16 numeral system that extends beyond the decimal system by including six additional symbols: A, B, C, D, E, and F, representing the values ten to fifteen. The video script teases the exploration of the hexadecimal system in future videos, indicating its relevance in computing and its use for representing larger numbers more compactly than binary.

💡Conversion

Conversion in the context of the video refers to the process of changing a number from one base to another. The script discusses the idea of representing the same number in different bases, such as binary or decimal, and implies that future videos will delve into the methods for converting between these systems. Conversion is essential for understanding and using different number systems effectively.

Highlights

Humans have been counting and representing numbers since ancient times.

Early humans used physical objects to keep track of quantities.

The development of number names was a significant step in representing numbers.

Every language has different names for numbers, reflecting cultural diversity.

The physical representation of numbers was bulky and inefficient.

The invention of number systems revolutionized how humans represent and calculate numbers.

The base 10, or decimal, system is the most familiar to us, likely due to the number of human fingers.

The base 10 system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Place value in the base 10 system allows for efficient representation of numbers.

The number 231 is used to illustrate the concept of place value in the base 10 system.

The base 10 system is based on powers of 10, which is intuitive for humans.

Binary, or base 2, system is fundamental to modern computing.

Binary system uses only two symbols, 0 and 1, which correspond to on and off states in computer hardware.

Binary system's place values are powers of two, unlike the base 10 system.

The number 231 can be represented in binary as 11100111, demonstrating the conversion between number systems.

The video encourages viewers to appreciate the beauty and utility of number systems.

Future videos will explore other number systems like hexadecimal, which uses 16 digits.

Transcripts

- [Voiceover] For as long as human

beings have been around we've

been counting things, and we've been

looking for ways to keep track and

represent those things that we counted.

So, for example if you were

an early human and you were

trying to keep track of the days

since it last rained you might say

okay let's see it didn't rain today so

one day has gone by, and we now use

the word one, but they might have

not used it back then.

Now another day goes by.

Then another day goes by.

Then another day goes by.

Another day goes by.

Another day goes by.

Another day goes by, then it rained.

And so when his friend comes

he says, "Well, how long has it been

since we last rained."

Well you would say, "Well, this is how

many days it's been."

And your friend would say, "Okay,

I think I have a general sense of that."

And at some point they probably

realized that it's useful to have names

for these.

So they would call this one, two, three,

four, five, six, seven.

Obviously every language in the world

has different names for these.

I'm sure there are lost languages

that had other names for them.

But very quickly you start to

realize that this is a pretty bulky

way of representing numbers.

One it takes a long time to write down.

It takes up a lot of space,

and then later if someone wants to

read the number they have to sit here

and count.

It's hard enough with seven,

but you could imagine if there were

what we call 27 of it, or 1000 of it.

Then it would take up, possibly, a whole

page and even when you counted

you might make a mistake.

And to solve this human beings

have invented number systems.

And it's something that we take for granted.

You might say, "Oh, isn't that just the

way you've always counted?

But hopefully over the course

of this video you'll start to appreciate

the beauty of a number system

and to realize our number system isn't

the only number system that is around.

The number system that most

of us are familiar with is the base 10

number system.

Often called the decimal, the decimal

number system.

And why 10?

Well probably because we have 10 fingers.

Or most of us have 10 fingers.

So, it was very natural to think

in terms of bundles of 10 or to have 10

symbols.

So however many bundles you have

you can use your fingers and eventually

your symbols to think about how many there are.

And since we needed 10 symbols

we came up with zero, one, two, three, four,

five, six, seven, eight, nine.

These 10 digits, these are our 10 symbols

that we use in the base 10 system.

To just give us a little bit of a reminder

how we use them imagine the number 231.

So, 231. 231.

What does this represent?

Well, what's neat about number systems

is we have place value.

This place all the way to the right,

this is the ones place.

This is the ones place.

This literally means one, one.

One bundle of one.

So, this is one, one right over here.

This right over here, this is in the 10s place.

This is in the 10s place.

This three here, literally means three 10s.

So this literally means three 10s.

And this two here, this two here is in

the 100s place.

It's in the 100s place.

So, this represent two 100s.

You add them together and once again

I'm still thinking in base 10, you'd

get 231.

This is two 100s plus three 10s plus one.

In our base 10 system notice every

time we move to the left we're thinking

in bundles of 10 of the space

to the right.

So, this is the ones place.

You multiply by 10, you go to the 10s place.

You want to go to the next place

you multiply by 10 again.

You get the 100s place.

If you're familiar with exponents,

one is the same thing as 10 to the zero power.

10 is the same thing as 10 to the first power.

So this is the 10s place. Three tens.

And 100 is the same thing as 10

to the second power.

Obviously we could keep going on and on

and on and on and on.

That is the power of the base 10 system.

So, you might be curious now.

"Well, what if this wasn't 10 here?

What if we did, let's just go as simple

as we can. You can almost view this

as a base one system.

You only have one symbol right over here.

But what if we went to something slightly

more complex, a base two system.

You'd be happy to know that not only

can we do this, but the base two system

often called the binary system.

This is called the decimal system.

The base two system often called

the binary system is the basis of all

modern computing.

It's the underlying mathematics

and operations that computers perform

are based on binary.

And in binary you have two symbols.

You have zero and you have one.

The reason why this is useful for computation

is because all the hardware that we use

to make our modern computers, all

of the transistors and the logic gates

they either result in an on or an off state.

On or an off state.

And so what we do is when you use

your calculator or whatever you might

be operating in base 10, but underlying

everything it is doing the operations

in binary.

But you might say well how do we

actually think in terms of binary?

Well, we can construct similar places here,

but instead of them being powers of 10

they're going to be powers of two.

So, let's set up some places here.

So, all the way on the right

two to the zero power is still one.

So we can still call that the ones place.

Then we can move to the left of that.

We can move to the left of that.

That would be two to the first power.

So we could call that the twos place,

and I can even write it out if I want.

Twos place instead of the 10s place.

Then I could keep going.

Instead of this space being the 10 to

the second or the 100s place, it will be

the two to the second, or the fours place.

And I can keep going.

I encourage you actually to pause

the video and try to build this out

for yourself.

What would this be?

Well this would be two to the third,

or the eights place.

Notice every time we're doing this

we're multiplying by two.

Everytime we go to the left,

just like we multiplied by 10 here.

So notice everywhere you see this 10s

we're now dealing with twos.

Let's keep going.

Let's keep going and then we can actually

represent this number using binary.

So, let's do that.

So, this right over here I've already

used that color.

This right over here, this is two

to the fourth.

We could call that the 16s place.

Then we could have --

I'll reuse some of these colors.

This is two to the fifth.

We could call this the 32s place.

Then we can go two to the sixth.

We can call that, multiply by two again,

or two to the six is 64.

So this is the 64s place.

Tells us how many 64s we have. Zero or one 64s.

We'll see that in a second.

Then we can go over here.

This would be two to the seventh.

That would be the 128s place.

And we can obviously keep going on

and on and on, but this should be enough for

me to represent this number.

In future videos I will show you how

to do that, but let's actually represent

the number.

It turns out that this number

in decimal can be represented

as 11100111 in binary.

What does this mean?

This means you have one 128 plus one 64,

plus one 32, plus no 16s, plus no eights,

plus one four, plus one two, plus one one.

So you can see that these are going

to be the same thing.

Notice, this is one 128.

So it's 128, plus 64, plus 32.

We have zero 16s, zero eights.

So we're not going to add those.

Plus four, one four.

Plus one two.

Plus one one.

And add these together,

and once again when we're doing this,

when I'm writing it this way I'm

kind of using the number system

that we're most familiar with.

We're most used to doing the operations in,

but when you do it you will see that

this is the exact same number as 231.

This is just another representation.

One isn't better than the other.

The only reason why I converted this

is this is what I'm used to thinking in.

It's what I'm used to doing operations in.

So, hopefully you find that pretty interesting.

To me, this kind of opened my mind

to the power of even our decimal system.

In future videos we'll explore other

number systems.

The most used ones, base 10 is

used very heavily, binary and there's

also hexadecimal where you don't have

two digits or not 10 digits, but you have 16 digits.

And we'll explore those in future videos

and how to convert between or rewrite the

the different representations and different bases.