Hello. I'm Professor Von Schmohawk and welcome to Why U.
In our lecture on the decimal number system,
we explained that this is an example of "positional notation".
In positional notation, the column occupied by a digit
determines the multiplier for that digit.
Since the decimal system is base-10,
each column multiplier differs by a factor of ten from the adjacent columns.
The multiplier for the right-most digit is one.
The digits to the left are multiplied by ten, one-hundred, and so on.
The value of a decimal number is the sum of all its digits times their multipliers.
For example, the value of the decimal number one-thousand eight-hundred-seventy-nine is
one times one-thousand,
plus eight times one-hundred,
plus seven times ten,
plus nine times one.
Using decimal notation, we can represent any integer.
But decimal notation is also capable of representing fractional quantities,
by adding a decimal point immediately to the right of the one’s digit.
We can then add as many digits as we like to the right of the decimal point.
Each of these column multipliers is one-tenth the size of the column to the left,
continuing the pattern of multipliers.
The first digit to the right of the decimal point is multiplied by one-tenth.
The next digit is multiplied by one-hundredth, and so on.
Just as with integers, the value of a decimal number
is the sum of all its digits times their multipliers.
For example, the number one eight seven nine point two five,
represents one thousand,
plus eight hundreds,
plus seven tens,
plus nine ones,
plus two tenths,
plus five one-hundredths.
Using a number line, we can see how each digit adds different size divisions to the number.
For example, the decimal number three point four two five
represents three units,
plus four tenths of a unit,
plus two hundredths of a unit,
plus five thousandths of a unit.
Any fraction can be converted to a decimal number
by simply dividing the numerator by the denominator.
We can do this using long division, or with a calculator.
The result of this division will be a decimal number which is equivalent to the fraction.
This decimal number will either repeat or terminate.
A terminating decimal number has a finite number of digits after the decimal point.
A repeating decimal number has a sequence of digits after the decimal point
which repeats forever.
Repeating decimals can be written with three dots, called an "ellipsis",
to indicate that the pattern repeats forever,
or a bar can be placed over the digits which repeat.
For instance, we can indicate a repeating three by placing a bar over the three.
Likewise, a bar over a sequence of digits indicates that the sequence repeats forever.
Some unit fractions when written in decimal form,
can be represented with a finite number of digits, while others cannot.
For instance, the fractions one-half, one-fourth, one-fifth, one-eighth, and one-tenth
can be written with a finite number of digits.
On the other hand, the decimal equivalent of one-third
is a decimal point, followed by an infinite number of threes.
One-sixth is a decimal point followed by a one, and then an infinite number of sixes.
One-seventh is a decimal point followed by the infinitely repeating sequence
one, four, two, eight, five, seven.
And one-ninth is a decimal point followed by an infinite number of ones.
You may be wondering why some fractions can be represented with a finite number of digits,
while others cannot.
To see why, let’s look at the fraction one-half on the number line.
One-half can be represented by exactly five tenths.
So one-half is equivalent to the decimal number zero point five.
On the other hand, the fraction one-third is larger than three tenths,
but smaller than four tenths.
If we subdivide these divisions into hundredths,
we see that one-third falls somewhere between thirty-three hundredths
and thirty-four hundredths.
If we subdivide these divisions into thousandths,
we see that one-third falls somewhere between three-hundred-thirty-three thousandths
and three-hundred-thirty-four thousandths.
If we subdivide these divisions into ten-thousandths,
we see that one-third still falls between two divisions.
It is larger than point three three three three
but smaller than point three three three four.
We can repeat this process indefinitely.
No matter how close we zoom in, the fraction one-third will never fall exactly on a division.
We will just keep adding threes after the decimal point forever.
So although some rational numbers like one-half can be represented by a finite number of digits,
others like one-third cannot.
We have seen that by dividing the numerator by the denominator,
any fraction can be converted to an equivalent decimal number
which will either terminate or repeat.
In the next lecture,
we will see how to convert any terminating or repeating decimal number to a fraction.