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# Place Value with Decimals

The Indian Numerical System and the International Numerical System both have a base 10. The place values ones, tens, hundreds, thousands, signifies this base.
Starting from ones, as we move towards left, at each step, the value increases 10 times.
10 times ones is the place for tens (10)
10 times tens is the place for hundreds (100)
10 times hundreds is the place for thousands (1000)
and so on, as we move towards left, we go on multiplying the place value by 10 at each step.
If we reverse the process of moving towards left, i.e., if we move towards right, the place value will decrease 10 times.

• Moving one step towards right will mean decreasing the value 10 times of each step.
• Moving one step towards right from ones, the value should become $\frac{1}{10}$ of 1.

The place value next to ones, towards right is $\frac{1}{10}$ th. So the place value chart extends as below:

 Ten lakhs Lakhs Ten thousand Thousand Hundreds Tens Ones Decimal One-ten 1000000 100000 10000 1000 100 10 1 . $\frac{1}{10}$

The demarcation between ones and $\frac{1}{10}$ th is marked by a decimal point. Let us what do the numbers written in decimals mean and how they are read. The decimal point in a number separates the whole number from its decimal number part. The part on the left of the decimal point is the whole part, and part on the right of the decimal point is the decimal part. For example, 8.5 means --- 8 is the whole part and .5 is the decimal part. Observe the following table.

 Number To be reads as Whole part Decimal part 2.9 10.3 0.8 7.1 5.0 Two point nine Ten point three Zero point eight Seven point one Five point zero 2 10 0 7 5 .9 .3 .8 .1

A whole number or a natural number can also be represented as a decimal number by putting .0 on the right of the number (just after ones)
For example, 1 can be written as 1.0
15 can be written as 15.0
100 can be written as 100.0
9531can be written as 9531.0

## Expanded Notation of Decimal Fraction

Like whole numbers, the decimal numbers also can be written in an expanded form.
7.4=7 ones + 4tenths or 7 + $\frac{4}{10}$
32.6= 3 tens + 2 ones + 6tenths = 30 + 2+ $\frac{6}{10}$
504.9 = 5 hundreds + 0 tens + 4 ones + 9 tenths = 500 + 0+4 + $\frac{9}{10}$
37508.4 = 3ten- thousands + 7 thousands + 5hundreds +0 tens + 8 ones + 4tenths.
= 30,000 + 7,000 + 500 + 0 + 8 + $\frac{4}{10}$

## Hundredths and Thousandths

By tenth ( $\frac{1}{10}$ ), we mean one part out of ten equal parts, so by hundredths ( $\frac{1}{100}$ ), we mean one part out of 100 equal part . If we divide one unit into 100 equal parts, then taking one of its parts, we mean to represent $\frac{1}{100}$.Taking 2 parts out of 100 equal parts , we mean $\frac{2}{100}$, read as two hundredths.
Similarly, 3 parts out of 100 equal parts means $\frac{3}{100}$ , i.e., three hundredths.
27 parts out of 100 equal parts means $\frac{27}{100}$, i.e., twenty-seven hundredths.
99 parts out of 100 equal parts means $\frac{99}{100}$, i.e., ninety-nine hundredths.
These numbers $\frac{1}{100}$, $\frac{2}{100}$, $\frac{3}{100}$, $\frac{27}{100}$, $\frac{99}{100}$, are written as decimal numbers as 0.0, 0.02, 0.02, 0.27, 0.99 respectively.
55 shaded parts in figure (a) shows $\frac{55}{100}$ or 0.55.
27 shaded parts in figure (b) shows $\frac{27}{100}$ or 0.27.
$\frac{1}{100}$, i.e., one hundredth = 0.01 = 0 tenths and one hundredth.
It is read as zero point zero one.

 Decimal fraction Number name Decimal number To be read as $\frac{1}{100}$ One hundredth 0.01 Zero point zero one $\frac{2}{100}$ Two hundredth 0.02 Zero point zero two $\frac{7}{100}$ Twenty-seven hundredth 0.27 Zero point two seven $\frac{99}{100}$ Ninety-nine hundredth 0.99 Zero point nine nin

Note: 0.27 is read as zero point two seven and not as zero point twenty-seven . In the place value chart, hundredths gets a place on the right of tenths.

Place value, next to tenths towards the right of $\frac{1}{10}$ is for hundredths, i.e., for $\frac{1}{100}$.

So, the place value extends to the right one more column for hundredths, i.e,$\frac{1}{100}$.

## Thousandths

Like .01 for $\frac{1}{100}$, now we introduce a decimal number for $\frac{1}{1000}$, which is one thousandth part of a whole unit. For $\frac{1}{1000}$, we divide 1 unit 1000 equal parts.

We can imagine the size of each thousandth part. One thousandth part is $\frac{1}{10}$th of the hundredth part, so its size will be 10 times smaller to that of.

In the place value chart $\frac{1}{1000}$ will get a place next to $\frac{1}{100}$ towards the right.

Decimal number for decimal fraction $\frac{1}{1000}$ is .001 or 0.001.

Now the extended place value chart is as shown below:

 Place value Lakh Ten thousandths Thousandths Hundreds Tens Ones Decimal Tenths Hundredths Thousandths PV as a fraction 100000 10000 1000 100 10 1 . $\frac{1}{10}$ $\frac{1}{100}$ $\frac{1}{1000}$ PV as decimal 100000 10000 1000 100 10 1 . .1 .01 .0

From the place value table shown above, we observe that as we go on increasing the number of part, each time the size of the parts goes on decreasing and its place value goes on moving farther away from the decimal point towards the right.

For $\frac{1}{10}$ = .1, the place value of tenth is one step right of decimal.

For $\frac{1}{100}$ = .01, the place value of one hundredth is 2 steps right of decimal.

For $\frac{1}{1000}$ = .001, the place value of the thousandth is 3 steps right of decimal and we can extend it further.

Now, 0.001 is written as a fraction as $\frac{1}{1000}$.

0.005 can be written as a fraction as $\frac{5}{1000}$.

0.015 can be written as a fraction as $\frac{15}{1000}$

0.378 can be written as a fraction as $\frac{387}{1000}$.

0.999 can be written as a fraction as $\frac{999}{1000}$.

As discussed earlier, reading a decimal fraction is done by reading its every digit one by one.

For example, 0.832 is to be read as zero point eight three two and not as zero point eight hundred thirty – two.

In an expanded form, we have

0.832 = 0 + 8 tenths + 3 hundredths + 2 thousandths = 0 +$\frac{8}{10}$ +$\frac{3}{100}$ +$\frac{2}{1000}$ .

3056.201 = 3000 + 0 + 50 + 6 + $\frac{2}{10}$ + 0 + $\frac{1}{1000}$.

920.007 = 900 + 20 + 0 + 0 + 0 +$\frac{7}{1000}$.

## Importance of the Place of Decimal

By using decimal point at a wrong place, value of the number may increase or decrease 10 times or 100 times or 1000 times. This can be seen from the example given below:

Rs12 and 75paise can be written as Rs 12.75 If we put the decimal point at a wrong place, this amount may increase to Rs 127.5 which is Rs 127.50 or it may increase to Rs 1,275 by putting decimal point after one more digits.

Think about a cashier who, while entering Rs 8000.00 in an account may make a small mistake of a decimal and enter Rs 800000 instead of Rs 8000.00. The poor cashier will have to bear a loss of Rs 792000 just because of a small mistake.

## Importance of the Place of Zero

‘0’ has no value if put at the extreme left of a whole number.

‘0’ makes no difference to the decimal number if placed at the extreme right.

For example, 0.5 = $\frac{1}{1000}$ =$\frac{1}{1000}$

0.50 = $\frac{1}{1000}$ = $\frac{1}{1000}$

0.500 = $\frac{1}{1000}$ = $\frac{1}{1000}$

Thus, 0.5 = 0.50 = 0.500.

So, one zero at the extreme right or two zeroes at the extreme right of a decimal part of the number do not change the value of the number.

But an extra zero just at the right of the decimal point does make a lot of difference.

For example, 0.1 = $\frac{1}{10}$

An extra zero between the decimal and 1 will make it. .1 Which will mean $\frac{1}{10}$ . reducing the original value times.

Two extra zeroes will reduce the value $\frac{1}{10}$ times.