In this explainer, we will learn how to identify the value of digits in numbers with up to 4 decimal places.

### Definition: Decimal

A decimal is a number that contains a decimal point.

The digits to the left of the decimal point represent the whole number part of the number, and the digits to the right of the decimal point represent the decimal part of the number.

We can think of the decimal part of a number as a fraction where the denominator is a power of 10. For example, 35.9 has a whole number part of 35 and a decimal part of 9 tenths.

In the same way that we have place value names for whole numbers, including hundreds, tens, and ones, we have names for the place values to the right of the decimal point in a decimal number. We can see these below. In this explainer, we are looking at decimals that have up to 4 decimal places, that is, numbers that have up to a ten-thousandths value.

Let us remind ourselves of the names for the place value columns.

If we had the decimal 0.36, we could write this in our place value grid as below, showing that 0.36 has 3 tenths and 6 hundredths.

The zeros in a decimal have an important role. Zeros between the decimal point and the digits hold the place value. For example, 0.62, 0.062, and 0.0062 do not have the same value. We can demonstrate this by writing these values as fractions, giving us three distinct fractions:

However, we can add zeros to the right of the final digit in a decimal and this will not affect the value. So, 0.76, 0.760, and 0.7600 all have the same value, since we can write them as fractions and simplify, giving us

We will now look at how to write a decimal number in words.

### How To: Writing a Decimal in Words

- Read the digits to the right of the decimal place as a whole number. For example if we have 0.036, we would say these digits as “thirty-six.” When saying this as a whole number, we can ignore any zeros after the decimal point and before the digits.
- Use the place value name for the last decimal digit on the right. The last decimal digit place value in 0.036 is thousandths.
- Put them together. So, 0.036 in words would be “thirty-six thousandths.”
- If there is a whole number portion of the decimal number, we use “and,” “decimal,” or “point” to signify the decimal point. For example, we say the number 247.036 in words as “two hundred forty-seven and thirty-six thousandths.”

Let us look at an example of how we can write a decimal number with 4 decimal places in words.

### Example 1: Writing a Decimal Number in Words

Write the decimal 43.2875 in words.

### Answer

We can write our decimal 43.2875 into a place value grid as below.

We can say the whole number part of this as forty-three.

Taking the digits to the right of the decimal point, we can say these digits—2, 8, 7, and 5—as we would if it were a whole number, as two thousand, eight hundred seventy-five. We use the place value for the last digit on the right, which is ten-thousandths. Therefore, the decimal part of the number is two thousand, eight hundred seventy-five ten-thousandths.

We can write the whole number part and the decimal part together with the word “and” to signify the decimal point. Therefore, our final answer in words is forty-three and two thousand, eight hundred seventy-five ten-thousandths.

It is important that we remember the place values of each digit in a decimal. It can be useful to write these above a decimal number or write the number into a place value grid. Let us look at an example of finding the value of a specific digit in a decimal number.

### Example 2: Finding the Value of a Digit in a Decimal Number

What value does the digit 5 represent in the number 67.538?

### Answer

Recall that we can write our number in a place value grid as below.

We can see that the digit 5 is in the tenth column. Therefore, the value of the digit 5 is 5-tenths. We could also write this as 0.5.

In the next example, we will move from finding the value of one digit in a decimal to writing out a decimal by writing each number according to its place value. This is known as the expanded form of a decimal, and we can write a decimal in its expanded form by multiplying each digit by its place value and adding them together.

Let us now look at two examples where we change between a standard decimal form and an expanded decimal form.

### Example 3: Writing a Decimal Number in Expanded Form

Write seven thousand, three hundred twenty-six ten-thousandths in expanded form.

### Answer

As we have the word “ten-thousandths” in our number, with no “and” or “decimal” signifying a whole number portion, we know that there is no whole number part of the decimal. The “ten-thousandths” means that the final digit on the right will be in the fourth decimal place value. We can fill in the digits 7, 3, 2, and 6 into a place value grid as below, remembering that we need to write 0 as a place holder to the left of the decimal point.

Since tenths represents 0.1, hundredths represents 0.01, thousandths represents 0.001, and ten-thousandths represents 0.0001, we can add these to our place value grid as below.

So, in expanded form, our answer is

### Example 4: Writing a Decimal in Expanded Form as a Standard Decimal

Write the expanded form decimal as a decimal in standard form.

### Answer

We can begin by filling in the digits 9, 5, 3, 2, and 7 into a place value grid. Since there is no thousandths value, we must write a 0 as a place holder in the thousandths column.

So, our answer as a decimal in standard form is 95.3207.

When we are working with decimals, it is important to remember that a longer decimal number with more digits is not necessarily a larger number than one with fewer digits. For example, 0.0034 is smaller than 0.15, because the highest decimal digit in 0.0034 is 3 thousandths, and the highest decimal digit in 0.15 is 1 tenth, making 0.15 a larger value.

We will now look more closely at how we can compare and order decimal numbers.

### How To: Ordering Decimal Numbers

- Put the numbers to be ordered in a table or grid so that the decimal point and column values are aligned.
- Beginning at the left column, check which numbers have the highest value in a column. If there is a single largest value, that is the largest number.
- If there are two or more values that have the same digit value in a column, check the next right column until a single largest value is found. If there are two or more, continue the process of checking the next right column until a single largest value is found.

### Example 5: Ordering Decimal Numbers

Write the following numbers in order of size, from largest to smallest.

### Answer

To begin ordering decimals, it can be helpful to write the numbers in a column so that they are aligned at the decimal place.

Recall that if we have two numbers, to find the larger one, we firstly check if there is a value in a higher column. For example, 500 is larger than 50 since 500 has five hundreds and 50 has no hundreds. If our values to be compared have the same value in a column, we check the next column on the right to find the first column where there is a larger value.

To begin our answer, we check for the largest value in the columns starting at the left side. Here we see that 50.6 is the only value that has 5 tens, so it is our largest value.

Checking the ones column, we see that 5.60 and 5.006 both have 5 ones. To see which one is largest, we check the next column on the right, the tenths column, to see which has the larger value. Since 5.60 has 6 tenths, it will be larger, and 5.006 will be the next largest. Notice that we can write 5.60 as 5.6.

Our final two values to order are 0.506 and 0.056. Since 0.506 has 5 tenths, it is larger than 0.056, which has 0 tenths.

Therefore, our answer in order from largest to smallest is as follows.

### Example 6: Comparing Decimal Numbers

Match the values to their correct place on the number line.

- 0.105
- 1.005
- 1.5
- 0.51

### Answer

Recall that 0.105 begins with 1 tenth, and so it will be approximately one-tenth of the distance between 0 and 1 on the number line. This means that 0.105 is at the value denoted a.

For the value 1.005, because we have a comparatively small decimal portion of one and five-thousandths, we know that this will appear on the number line very close to the value of 1; hence, it is at the value denoted c.

The number 1.5 appears exactly halfway between 1 and 2 on the number line, and so it is at the value denoted d.

The number 0.51 can be described as fifty-one hundredths, and so it is approximately halfway between 0 and 1 on the number line. This is denoted by the arrow marked b.

Hence, we can put all the values on the number line as follows.

### Key Points

- A decimal consists of a whole number part and a decimal part to the right of the decimal point.
- The column values to the right of a decimal point are tenths, hundredths, thousandths, and ten-thousandths.
- To write a decimal number in words, we read the decimal digits as we would a whole number, along with the place value name for the last decimal digit on the right. If there is a whole number part of the decimal, we link the whole number part and the decimal part with the word “and.”
- To write a decimal in expanded form, we multiply each digit by its place value and add them together.
- It is important to remember that a longer decimal number is not necessarily a larger number than one with fewer digits.