Question

# Ben and Carl are remembering how you told them that a 0 at the end of a whole number makes it ten times greater. They remember that 420 is ten times bigger than 42, for example. They wonder if the same thing works with decimals. How do you help them to see what's true and what's not?

Decimals
Ben and Carl are remembering how you told them that a 0 at the end of a whole number makes it ten times greater. They remember that 420 is ten times bigger than 42, for example. They wonder if the same thing works with decimals. Ben argues that putting a 0 at the end of a decimal number makes it ten times greater. Carl says, "No. You've gotta sneak it in right after the decimal point and push everything to the right." How do you help them to see what's true and what's not?

2021-08-03

Step 1
The operations applied to whole numbers may result differently compared to the numbers with decimals.
When '0' is applied at the end of the whole number that means that the number is applied by 10. To understand this, let first learn the concept of places in whole numbers.
If we read the whole number from the Right end,
The first term is at the unit place.
The second term is at the tens place.
The third term is in the hundredth place.
For eg., if we have 100000, then
$$\begin{array}{|c|c|}\hline 1 & 0 & 0 & 0 & 0 & 0 \\ \hline 100\ \text{thousand} & 10\ \text{thousand} & \text{thousand} & \text{hundred} & \text{tens} & \text{unit}\\ \hline \end{array}$$
So now if one more '0' is added at the end, then the number becomes '1000000'.
$$\begin{array}{|c|c|}\hline 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 1\ \text{million} & 100\ \text{thousand} & 10\ \text{thousand} & \text{thousand} & \text{hundred} & \text{tens} & \text{unit}\\ \hline \end{array}$$
And the number is increased by 10 times.
Step 2
Now let us understand the decimal numbers.
Here, there are two sections. One is on the left side of the decimal and one is on the right side of the decimal.
The place values are as follows. Take the example of 456.123
On the left side of the decimal:
6 is at the unit place.
5 is at the tens place.
4 is at the hundredth place. (similar to the whole number)
On the Right Side of the decimal:
1 is at the $$\displaystyle{\left(\frac{{1}}{{10}}\right)}^{{{t}{h}}}$$ place.
2 is at the $$\displaystyle{\left(\frac{{1}}{{100}}\right)}^{{{t}{h}}}$$ place.
3 is at the $$\displaystyle{\left(\frac{{1}}{{1000}}\right)}^{{{t}{h}}}$$ place.
So as we go on the right of the decimal place, the value of the number decreases.
So if '0' is added at the right side of the decimal place, then one more number is added with a further $$\displaystyle{\left(\frac{{1}}{{10}}\right)}^{{{t}{h}}}$$ value to the value of the last number in the end.
For example, if '0' is written at the end of 456.123 then the number becomes '456.1230', and the value of this place is $$\displaystyle{\left(\frac{{1}}{{10000}}\right)}^{{{t}{h}}}.$$
So this is the same number with a more precise value.
Step 3
This can be further understood as follows:
$$\displaystyle{456.1230}={4}\cdot{100}+{5}\cdot{10}+{6}\cdot{1}+{1}\cdot{\left(\frac{{1}}{{10}}\right)}+{2}\cdot{\left(\frac{{1}}{{100}}\right)}+{3}\cdot{\left(\frac{{1}}{{1000}}\right)}+{0}\cdot{\left(\frac{{1}}{{10000}}\right)}$$
What Carl said is also not true.
If the zero is placed right after the decimal shifting rest of the existing number to the right of it, then also it is kept at $$\displaystyle{\left(\frac{{1}}{{10}}\right)}^{{{t}{h}}}$$ place. That will also give the wrong value to the given number.
For example, the present number will become 456.0123.