Name: 代数学3 - 10進数、2進数、8進数、16進数 (Pre-Algebra 3 - Decimal, Binary, Octal & Hexadecimal)
Uploaded: 2021-01-14T07:50:03.000Z
Duration: 14 min 1 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

Hello. I'm Professor Von Schmohawk and welcome to Why U.

In the previous lectures, we explored some examples of the earliest number systems

which were used primarily for counting objects.

These counting numbers are called “natural numbers”.

The natural numbers start at one and can count to arbitrarily large quantities.

As we have seen, Roman numerals are one of many possible ways to represent natural numbers.

The Roman system was eventually replaced with the modern decimal number system

which uses “positional notation” and only ten numeric symbols.

The decimal number system was found to be superior to the ancient Roman system

because of the simple rules it uses to create numbers.

In the decimal system there are ten numeric symbols, 0 through 9, called “digits”.

these digits represent the quantity of ones

In positional notation, the column occupied by a digit

determines the “multiplier” for that digit.

the value of the right-most digit is multiplied by 1.

The digit in the next column to the left is multiplied by 10.

The next digit is multiplied by 100 and so on.

The value of a number is the sum of all its digits times their multipliers.

For example, the value of the decimal number 1879

In any positional notation, each column's multiplier differs from the adjacent column

by a constant multiple called the “base” of the number system.

In the decimal system, each column multiplier is ten times the previous column.

Therefore the decimal system is called a “base-10” number system.

There are an infinite number of columns in the decimal number system

with each column multiplier being ten times bigger than the column to the right.

However, when writing a number, the zeros in front are normally not written.

the tens column counts the number of times

that the ones column has passed from 9 to 0.

In other words, the tens column registers the number of tens which we have counted.

At that point the ones and tens columns start over at 0

The positional notation system is simple.

How is it that we ended up with a number system based on multiples of ten?

There is not any good reason for choosing ten over some other number

other than the fact that people have ten fingers

and probably originally communicated quantities using their fingers.

But what if we were cartoon characters with four digits on each hand?

everyone uses a number system based on multiples of eight instead of ten?

How would a base-8 or “octal” number system work?

In octal there are only eight numeric symbols instead of ten as in decimal.

Instead of 0 through 9 the symbols 0 through 7 are used.

Counting in octal is very similar to counting in decimal.

the highest quantity which can be represented in the ones column is 7.

Counting an eighth item causes the ones column to start over at 0

So the second column counts the number of eights.

Therefore in octal the number following 7 is 10

which looks just like the decimal number ten.

After octal "10" comes octal  "11",  "12",  and so on

The second column has now counted two “eights” or sixteen.

We continue like this until we get to the highest number we can represent with two digits

At that point, the ones and eights columns start over at 0 and the third column increments.

The 1 in the third column represents eight “eights” or sixty-four.

Each column multiplier is eight times the previous one.

Every number which can be written in decimal can also be written in octal

the way the quantities are represented is completely different.

It is easy to convert an octal number to decimal

when you consider how positional notation works.

Let's take for example, the octal number "1750".

As in decimal, the value of the octal number is the sum of all its digits times their multipliers.

which adds up to the quantity which in decimal is called one-thousand.

You may sometimes see a small subscript 8 or 10 after an octal or decimal number

in case there may be some confusion about which base is being used.

Digital computers use electronic circuits called “flip-flops” to represent numbers.

Each flip-flop can store a single bit which can represent either a 0 or a 1.

Multiple bits can be combined to represent a base-2 or “binary” number.

In the binary number system 0 and 1 are the only two numeric symbols.

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代数学3 - 10進数、2進数、8進数、16進数 (Pre-Algebra 3 - Decimal, Binary, Octal & Hexadecimal)

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