The equivalent decimals are those decimals which are having the same values even if we add extra zeros at the extreme end of the decimal part.

**For example:**

0.1 is the decimal number which can be also written as 0.10 or 0.100 or 0.1000 and so on

Hence, we can write 0.1 = 0.10 = 0.100 = 0.1000 = 0.10000….

Thus, these are the equivalent decimals. This can be illustrated with the help of following construction.

- First we consider the square as shown in figure 1, then this square is divided into 10 equal parts.
- If we select the one part from the ten parts as shown in figure 2, then it can be written as,

**1/10 i.e. 1 out of ten and hence, 1/10 = 0.1**

- In next step, we divided these ten parts of figure 2, in such way that each part get divided into again 10 equal parts as shown in figure 3. Thus in figure 3, there are 100 parts.

If we select the ten parts from these 100 parts as shown in figure 3, then it can be written as,

**10/ 100 i.e. 10 out of hundred and hence, 10/100 = 0.10**

- If we observed here, it is found the space selected in figure 1 and 3 is same and the decimals are also same.

**1/ 10 = 0.1 = 10/100 = 0.10**

**0.1 = 0.10**

Thus, from this illustration we conclude that, the zeros at the extreme end of the decimal part has no value.

Hence, we can write 0.1 = 0.10 = 0.100 = 0.1000 = 0.10000…

And the zeros on the extreme end are not having any value and does not affects the value of decimal number these zeros are called as **redundant zeros.**

**Similarly,**

- First we consider the square as shown in figure 1, then this square is divided into 10 equal parts.

- If we select the 4 parts from the ten parts as shown in figure 2, then it can be written as,

**4/10 i.e. 4 out of ten and hence, 4/10 = 0.4**

- In next step, we divided these ten parts of figure 2, in such way that each part get divided into again 10 equal parts as shown in figure 3. Thus in figure 3, there are 100 parts.

If we select the 40 parts from these 100 parts as shown in figure 3, then it can be written as,

**40/ 100 i.e. 40 out of hundred and hence, 40/100 = 0.40**

- If we observed here, it is found the space selected in figure 1 and 3 is same and the decimals are also same.

**4/ 10 = 0.4 = 40/100 = 0.40**

**0.4 = 0.40**

Thus, from this illustration we conclude that, the zeros at the extreme end of the decimal part has no value.

Hence, we can write 0.4 = 0.40 = 0.400 = 0.4000 = 0.40000…

- And the zeros on the extreme end are not having any value and does not affects the value of decimal number these zeros are called as
**redundant zeros.** - This happens in case of decimal numbers then that numbers are called as equivalent decimals.

**For example:**

- 13 can be written as, 0.13 = 0.130 = 0.1300 = 0.13000 and so on
- Also, 0.024 can be written as, 0.024 = 0.0240 = 0.02400 and so on.

Hence, when in numerical redundant zeros are given we have to not consider these zeros.

For example:

- 23400 This decimal number is having 2 redundant zero hence its actual value is

0.23400 = 0.234

- Also, 0.008902000 is the decimal number having 3 redundant zeros and its actual value is

0.008902000 = 0.008902