The Base-Ten Number System and Place Value

Base-Ten Structure and Positional Value

Our number system is called the base-ten system. It is built on groups of 10. This means that every place in a number is worth 10 times as much as the place to its right.

For example, in the number \(352\), the digit 3 is in the hundreds place, the 5 is in the tens place, and the 2 is in the ones place. Each place has a different value because of its position.

This idea is called place value. A digit does not always have the same value. Its value depends on where it is in the number.

In this lesson, you will learn how the base-ten system works, how moving left or right changes a digit’s value, and how this helps you read, write, compare, and round numbers.

1. Understanding the places in base ten

Here are some common place values in whole numbers:

• Ones
• Tens
• Hundreds
• Thousands
• Ten thousands
• Hundred thousands

Each place is 10 times the value of the place to its right.

For example:

• 1 ten = 10 ones
• 1 hundred = 10 tens
• 1 thousand = 10 hundreds

We can also show this pattern with decimals. To the right of the ones place, the places become smaller by a factor of 10 each time.

• Tenths
• Hundredths
• Thousandths

For decimals:

• 1 one = 10 tenths
• 1 tenth = 10 hundredths
• 1 hundredth = 10 thousandths

This means:

Since dividing by 10 is the same as multiplying by one-tenth, we can also say:

2. How a digit’s value changes by position

Look at the digit 7 in different numbers:

• In \(7\), the 7 means 7 ones.
• In \(70\), the 7 means 7 tens, or 70.
• In \(700\), the 7 means 7 hundreds, or 700.

Each time the 7 moves one place to the left, its value becomes 10 times greater.

Now look at decimals:

• In \(0.7\), the 7 means 7 tenths.
• In \(0.07\), the 7 means 7 hundredths.
• In \(0.007\), the 7 means 7 thousandths.

Each time the 7 moves one place to the right, its value becomes one-tenth as great.

Important idea: The digit stays the same, but its value changes because its place changes.

3. Reading numbers by place value

To read a whole number, separate it into periods of three digits from right to left. The periods are usually ones, thousands, and millions. In 6th Grade, you will often work with ones and thousands.

For example, the number \(48,372\) is read as forty-eight thousand, three hundred seventy-two.

The digits mean:

• 4 ten thousands = 40,000
• 8 thousands = 8,000
• 3 hundreds = 300
• 7 tens = 70
• 2 ones = 2

Decimals are read by saying the whole number part, then the decimal point as and, then the decimal part by its place value.

For example, \(12.45\) is read as twelve and forty-five hundredths.

This is because:

• 4 is in the tenths place
• 5 is in the hundredths place
• \(45\) means 45 hundredths

4. Writing numbers in different forms

Place value helps us write numbers in more than one way.

• Standard form: 4,582
• Word form: four thousand, five hundred eighty-two
• Expanded form: \(4,000 + 500 + 80 + 2\)

Expanded form shows the value of each digit clearly.

For decimals, expanded form works the same way.

Example:

Or we can write it with place-value names:

5. Comparing numbers using place value

To compare numbers, start by looking at the greatest place value. If the digits are the same there, move to the next place to the right.

Example: Compare \(5,432\) and \(5,389\).

• Both have 5 thousands.
• Now compare the hundreds: 4 hundreds is greater than 3 hundreds.

So:

The same rule works for decimals.

Example: Compare \(0.56\) and \(0.6\).

Write \(0.6\) as \(0.60\) to line up the place values.

• Tenths: both have 6 tenths? No. \(0.56\) has 5 tenths and \(0.60\) has 6 tenths.

So:

6. Rounding using place value

Rounding means finding the nearest value of a number at a certain place.

To round:

1. Find the place you are rounding to.
2. Look at the digit to the right of that place.
3. If that digit is 5 or more, round up.
4. If that digit is less than 5, keep the digit the same.
5. Change all digits to the right into 0s for whole numbers, or drop them if rounding decimals.

Example: Round \(4,276\) to the nearest hundred.

• The hundreds digit is 2.
• The digit to the right is 7.
• Since 7 is greater than 5, round up.

So:

Example: Round \(6.284\) to the nearest tenth.

• The tenths digit is 2.
• The hundredths digit is 8.
• Since 8 is 5 or more, round up.

So:

7. Worked Examples

Example 1: Find the value of a digit

What is the value of the digit 6 in \(46,218\)?

Step 1: Find the place of the digit 6.

The 6 is in the thousands place.

Step 2: Write its value.

Answer: The value of the 6 is 6,000.

Example 2: Explain how moving changes value

Compare the value of the digit 4 in \(4,500\) and \(450\).

In \(4,500\), the 4 is in the thousands place, so its value is \(4,000\).

In \(450\), the 4 is in the hundreds place, so its value is \(400\).

Since:

the 4 in \(4,500\) is 10 times the value of the 4 in \(450\).

Example 3: Write a decimal in expanded form

Write \(7.305\) in expanded form.

Look at each digit:

• 7 is in the ones place, so it means 7
• 3 is in the tenths place, so it means 0.3
• 0 is in the hundredths place, so it means 0
• 5 is in the thousandths place, so it means 0.005

So the expanded form is:

Example 4: Compare and round

Which number is greater, \(23.47\) or \(23.5\)? Then round the greater number to the nearest one.

Step 1: Compare the numbers.

Write \(23.5\) as \(23.50\).

• Ones: both have 23
• Tenths: 4 tenths in \(23.47\), 5 tenths in \(23.50\)

So:

The greater number is \(23.5\).

Step 2: Round \(23.5\) to the nearest one.

• The ones digit is 3.
• The tenths digit is 5.
• Since it is 5, round up.

So:

8. Key patterns to remember

• Each place to the left is worth 10 times more.
• Each place to the right is worth one-tenth as much.
• A digit’s value depends on its position in the number.
• Expanded form helps show the value of each digit.
• To compare numbers, start at the greatest place value.
• To round, look at the digit to the right of the place you are rounding to.

Brief Summary

The base-ten system is built on groups of 10. This means every time a digit moves one place to the left, its value becomes 10 times greater, and every time it moves one place to the right, its value becomes one-tenth as great.

Understanding place value helps you read and write numbers, show numbers in expanded form, compare numbers, and round numbers correctly. When you know what each place means, numbers make much more sense.

Powers of Ten and Exponents

Our number system is called the base-ten system. That means it is built on groups of 10. Every place value is 10 times the value of the place to its right.

For example, in the number 4,382, the 8 is in the tens place, so it means 8 tens, or 80. The 3 is in the hundreds place, so it means 3 hundreds, or 300. Each time you move one place to the left, the value becomes 10 times greater.

This is where powers of ten come in. Powers of ten help us write repeated multiplication by 10 in a short way.

An exponent tells how many times a number is multiplied by itself. When the base is 10, powers of ten are especially important in place value.

Here are some powers of ten:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1{,}000\)
• \(10^4 = 10{,}000\)

Let’s look closely at what these mean:

• \(10^1\) means 10 is used as a factor 1 time: \(10\)
• \(10^2\) means \(10 \times 10 = 100\)
• \(10^3\) means \(10 \times 10 \times 10 = 1{,}000\)
• \(10^4\) means \(10 \times 10 \times 10 \times 10 = 10{,}000\)

Notice a pattern: each time the exponent goes up by 1, you multiply by 10, and the number gets one more zero.

We can show this pattern like this:

This connects directly to place value. Multiplying by 10 moves a digit one place to the left in value. Dividing by 10 moves a digit one place to the right in value.

When we multiply or divide by powers of ten, we can think about the decimal point shifting.

• Multiplying by 10 moves the decimal point 1 place to the right.
• Multiplying by \(10^2 = 100\) moves the decimal point 2 places to the right.
• Multiplying by \(10^3 = 1{,}000\) moves the decimal point 3 places to the right.

For division:

• Dividing by 10 moves the decimal point 1 place to the left.
• Dividing by \(10^2 = 100\) moves the decimal point 2 places to the left.
• Dividing by \(10^3 = 1{,}000\) moves the decimal point 3 places to the left.

Remember: every whole number has a decimal point, even if you do not see it. For example, 45 means \(45.0\), and 700 means \(700.0\).

Worked Example 1: Writing powers of ten

Write \(10^3\) as a standard number.

Step 1: The exponent 3 means multiply 10 by itself 3 times.

Step 2: Multiply.

So, \(10^3 = 1{,}000\).

Worked Example 2: Multiplying by a power of ten

Find \(36 \times 10^2\).

Step 1: Rewrite \(10^2\) as 100.

Step 2: Multiplying by 100 moves the decimal point 2 places to the right.

Think of 36 as \(36.0\).

So,

Worked Example 3: Dividing by a power of ten

Find \(572 \div 10^2\).

Step 1: Rewrite \(10^2\) as 100.

Step 2: Dividing by 100 moves the decimal point 2 places to the left.

Think of 572 as \(572.0\).

So,

Worked Example 4: Using decimals

Find \(4.8 \times 10^3\).

Step 1: Rewrite \(10^3\) as 1,000.

Step 2: Multiplying by 1,000 moves the decimal point 3 places to the right.

You can imagine adding zeros if needed: \(4.8 = 4.800\).

Then move the decimal point 3 places:

So,

Important Patterns to Remember

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1{,}000\)
• \(10^4 = 10{,}000\)

• Multiply by \(10^1\): move the decimal 1 place right
• Multiply by \(10^2\): move the decimal 2 places right
• Multiply by \(10^3\): move the decimal 3 places right

• Divide by \(10^1\): move the decimal 1 place left
• Divide by \(10^2\): move the decimal 2 places left
• Divide by \(10^3\): move the decimal 3 places left

Common Mistakes to Watch Out For

• Do not just count zeros in the original number. Use the exponent to know how many places to move the decimal point.
• Do not forget that whole numbers have a decimal point at the end. For example, \(8 = 8.0\).
• When moving the decimal point, add zeros if needed. For example, \(6.3 \times 10^2 = 630\).

Try Thinking About These

• \(10^2\) means \(10 \times 10\), so it equals 100.
• \(7 \times 10^3\) means \(7 \times 1{,}000 = 7{,}000\).
• \(9.5 \div 10\) means move the decimal 1 place left, so the answer is 0.95.

Summary

Powers of ten are numbers like \(10^1\), \(10^2\), and \(10^3\). The exponent tells how many times 10 is multiplied by itself.

Multiplying by a power of ten moves the decimal point to the right. Dividing by a power of ten moves the decimal point to the left.

These ideas help us understand place value, because each place in the base-ten system is 10 times the value of the place to its right.

Reading, Writing, and Expanding Whole Numbers

Whole numbers are built using the base-ten number system. This means each place in a number is worth 10 times as much as the place to its right.

When you understand place value, you can do three important things:

• Read a whole number correctly
• Write a whole number in standard form and word form
• Expand a whole number to show the value of each digit

This lesson will help you understand how numbers are organized and how each digit has a value based on its position.

1. Understanding Place Value

In a whole number, each digit has a place. Starting from the right, the places are:

Here is what each place is worth:

• ones = \(1\)
• tens = \(10\)
• hundreds = \(100\)
• thousands = \(1{,}000\)
• ten thousands = \(10{,}000\)
• hundred thousands = \(100{,}000\)
• millions = \(1{,}000{,}000\)

For example, in the number \(45{,}382\):

• The \(4\) is in the ten-thousands place, so its value is \(40{,}000\).
• The \(5\) is in the thousands place, so its value is \(5{,}000\).
• The \(3\) is in the hundreds place, so its value is \(300\).
• The \(8\) is in the tens place, so its value is \(80\).
• The \(2\) is in the ones place, so its value is \(2\).

2. Standard Form

Standard form means writing the number using digits.

Example:

\(45{,}382\) is in standard form.

This is the way numbers are usually written in math problems, charts, and real life.

3. Word Form

Word form means writing the number using words.

To read large numbers, it helps to separate them into periods. A period has three digits. Starting from the right, the periods are:

• ones period
• thousands period
• millions period

For example, look at \(45{,}382\). We can think of it as:

\(45\) thousand and \(382\)

So the word form is:

forty-five thousand, three hundred eighty-two

When writing whole numbers in word form:

• Write the value of each period from left to right.
• Use the period name, such as thousand or million.
• Do not use digits.
• Be careful with hyphenated numbers like twenty-one or forty-five.

4. Expanded Form

Expanded form shows the value of each digit in a number.

For \(45{,}382\), we break the number into parts based on place value:

This helps us see exactly how the number is built.

Sometimes expanded form can also be written using multiplication:

Both forms show place value clearly.

5. How to Read and Write Whole Numbers

Here is a simple step-by-step method:

1. Start at the left side of the number.
2. Read the digits in each period.
3. Say the period name: thousand, million, and so on.
4. Move to the next period on the right.

Example: \(3{,}206{,}014\)

• \(3\) is in the millions period, so we say three million.
• \(206\) is in the thousands period, so we say two hundred six thousand.
• \(014\) is in the ones period, so we say fourteen.

So the number is read as:

three million, two hundred six thousand, fourteen

6. Zeros Matter

Zeros are very important in whole numbers. Even when a zero does not add value by itself, it holds a place so the other digits stay in the correct positions.

For example, compare these numbers:

• \(4{,}507\)
• \(457\)

In \(4{,}507\), the zero is in the tens place. It shows that there are no tens, but the \(5\) is still in the hundreds place and the \(7\) is still in the ones place.

Expanded form for \(4{,}507\) is:

Usually, we leave out the zero part and write:

Worked Example 1

Write \(6{,}341\) in word form and expanded form.

Step 1: Identify each digit's place value.

• \(6\) thousands = \(6{,}000\)
• \(3\) hundreds = \(300\)
• \(4\) tens = \(40\)
• \(1\) ones = \(1\)

Step 2: Write the word form.

six thousand, three hundred forty-one

Step 3: Write the expanded form.

Worked Example 2

Write eighty-two thousand, nine hundred five in standard form and expanded form.

Step 1: Break the words into place values.

• eighty-two thousand = \(82{,}000\)
• nine hundred five = \(905\)

Step 2: Combine the values.

So the standard form is:

\(82{,}905\)

Step 3: Write the expanded form.

Notice that there are no tens, so we do not need to include \(0\) tens.

Worked Example 3

Read the number \(507{,}018\), then write it in word form and expanded form.

Step 1: Look at the periods.

• \(507\) thousand
• \(018\)

Step 2: Write the word form.

five hundred seven thousand, eighteen

Step 3: Write the expanded form.

Notice that there are no ten-thousands, no thousands digit other than \(7\), no hundreds, and no tens digit except \(1\) ten.

Worked Example 4

Write two million, forty thousand, three hundred six in standard form and expanded form.

Step 1: Think about each period.

• two million = \(2{,}000{,}000\)
• forty thousand = \(40{,}000\)
• three hundred six = \(306\)

Step 2: Add the parts together.

So the standard form is:

\(2{,}040{,}306\)

Step 3: Write the expanded form.

7. Common Mistakes to Avoid

• Mixing up place values: In \(34{,}562\), the \(3\) means \(30{,}000\), not \(3{,}000\).
• Forgetting zeros: one thousand six is \(1{,}006\), not \(1{,}6\) or \(106\).
• Skipping a place in expanded form: Make sure each digit is matched to the correct place.
• Reading by digits instead of place value: Read \(12{,}345\) as twelve thousand, three hundred forty-five, not one two three four five.

8. Helpful Tips

• Use commas to separate periods in large numbers.
• Say the period names to help read numbers correctly.
• Check each digit's place before writing expanded form.
• If a place has zero, remember it still holds the spot.

Summary

Whole numbers are made of digits, and each digit has a value based on its place. You can write a number in standard form using digits, in word form using words, and in expanded form by showing the value of each digit.

When you read or write large numbers, pay close attention to place value and periods like thousands and millions. Expanded form is a great way to see how a number is built from its parts.

Lesson: Reading, Writing, and Expanding Decimals

Decimals are a way to write numbers that are less than one whole, or numbers that have a whole-number part and a fractional part together. They are part of the base-ten system, just like whole numbers.

When you read, write, and expand decimals, you are using place value. Each place is worth 10 times as much as the place to its right, and 1 tenth of the place to its left.

In this lesson, you will learn how to:

• read decimals to the thousandths place,
• write decimals in standard form and word form,
• write decimals in expanded form.

1. Understanding decimal place value

The decimal point separates the whole-number part from the decimal part.

To the left of the decimal point are whole-number places:

• ones
• tens
• hundreds

To the right of the decimal point are decimal places:

• tenths
• hundredths
• thousandths

Here is the place value pattern:

For example, in the number \(4.583\):

• \(4\) is in the ones place,
• \(5\) is in the tenths place,
• \(8\) is in the hundredths place,
• \(3\) is in the thousandths place.

This means:

2. Reading decimals

To read a decimal:

1. Read the whole-number part.
2. Say and for the decimal point.
3. Read the digits to the right of the decimal as a whole number.
4. Say the name of the place value of the last digit.

Example: Read \(2.47\)

• The whole-number part is \(2\): two
• The decimal point is read as and
• The digits to the right are \(47\): forty-seven
• The last digit is in the hundredths place

So \(2.47\) is read as two and forty-seven hundredths.

Example: Read \(0.306\)

• The whole-number part is \(0\): zero
• The digits to the right are \(306\): three hundred six
• The last digit is in the thousandths place

So \(0.306\) is read as zero and three hundred six thousandths.

Important: When reading decimals, the place of the last digit tells you the name of the decimal part.

3. Writing decimals in word form

To write a decimal in word form, write the whole-number part, then use and, then write the decimal part as a fraction name based on the last place.

For example:

• \(5.2\) is five and two tenths
• \(8.19\) is eight and nineteen hundredths
• \(3.405\) is three and four hundred five thousandths

Zeros in decimals matter because they show place value. Compare:

• \(0.5\) = five tenths
• \(0.05\) = five hundredths
• \(0.005\) = five thousandths

These are different numbers because the digit \(5\) is in different places.

4. Writing decimals in standard form from words

To write a decimal from word form:

1. Write the whole-number part.
2. Write the decimal point for the word and.
3. Use the place value named at the end to decide how many decimal places are needed.

Examples:

• Six and three tenths means \(6.3\)
• Nine and twenty-five hundredths means \(9.25\)
• Zero and eight thousandths means \(0.008\)

Notice that in \(0.008\), there are zeros in the tenths and hundredths places. Those zeros are needed so the \(8\) lands in the thousandths place.

5. Writing decimals in expanded form

Expanded form shows the value of each digit. You can write it as a sum.

For whole numbers, you may have written:

Decimals work the same way.

Example:

This is because:

• \(7\) is 7 ones,
• \(2\) in the tenths place is \(0.2\),
• \(4\) in the hundredths place is \(0.04\).

You can also think of decimal parts as fractions:

So another way to expand \(5.603\) is:

There is no hundredths part in this number, so the hundredths digit is \(0\).

6. Worked examples

Example 1: Read and write in word form

Number: \(4.8\)

• Whole-number part: \(4\) → four
• Decimal part: \(8\) is in the tenths place

Answer: four and eight tenths

Example 2: Write the decimal in standard form

Words: twelve and thirty-four hundredths

• Whole-number part: \(12\)
• Thirty-four hundredths means \(34\) in the hundredths places

Answer:

Example 3: Write in expanded form

Number: \(3.572\)

• \(3\) is in the ones place → \(3\)
• \(5\) is in the tenths place → \(0.5\)
• \(7\) is in the hundredths place → \(0.07\)
• \(2\) is in the thousandths place → \(0.002\)

Expanded form:

Example 4: Read, write, and expand a decimal with zeros

Number: \(6.045\)

• Whole-number part: \(6\) → six
• Digits to the right of the decimal: \(045\) → forty-five
• The last digit is in the thousandths place

Word form: six and forty-five thousandths

Expanded form:

Notice that the \(0\) in the tenths place does not add value, but it is important because it keeps the \(4\) in the hundredths place and the \(5\) in the thousandths place.

7. Common mistakes to avoid

• Do not read the decimal point as “point” in word form. In math class, \(3.4\) should be read as three and four tenths, not “three point four,” when using place-value words.
• Look at the last decimal place. That tells whether to say tenths, hundredths, or thousandths.
• Do not ignore zeros in the middle of a decimal. In \(2.08\), the \(8\) is in the hundredths place, not the tenths place.
• Expanded form shows value, not just digits. For \(1.25\), write \(1 + 0.2 + 0.05\), not \(1 + 2 + 5\).

8. Quick practice ideas

• Read \(0.9\), \(2.16\), and \(7.008\) aloud in word form.
• Write these in standard form: five and six tenths, one and two hundredths, nine and nineteen thousandths.
• Write these in expanded form: \(8.3\), \(4.62\), \(0.407\).

Summary

Decimals use place value, just like whole numbers. To read a decimal, read the whole number, say and, then read the decimal part and name the place value of the last digit. To write a decimal in expanded form, show the value of each digit as a sum.

If you remember the places tenths, hundredths, and thousandths, you can read, write, and expand decimals with confidence.

The Role of Zero in Base-Ten

In our base-ten number system, the value of a digit depends on where it is placed. This is called place value.

For example, in the number \(352\), the \(3\) means \(3\) hundreds, the \(5\) means \(5\) tens, and the \(2\) means \(2\) ones.

Zero is a very important digit in base-ten. It has two main jobs:

• Zero as a value: It can mean “none” or “nothing.”
• Zero as a placeholder: It can hold a place so the other digits keep their correct values.

Understanding these two jobs helps us read, write, compare, and work with numbers correctly.

1. Zero as a value

Sometimes zero means there is no amount of something.

For example, if you have \(0\) apples, that means you have no apples. In a number sentence, \(7 - 7 = 0\), the zero means nothing is left.

On its own, zero is a number. It is not the same as “empty space.” It has a real value: \(0\).

2. Zero as a placeholder

In larger numbers, zero often works as a placeholder. This means it shows that a certain place has no value, while keeping the other digits in the correct positions.

Look at the number \(504\).

• The \(5\) is in the hundreds place, so it means \(500\).
• The \(0\) is in the tens place, so it means there are no tens.
• The \(4\) is in the ones place, so it means \(4\).

So,

Without the zero, the number would be \(54\), which is very different. The zero keeps the \(5\) in the hundreds place.

Why placeholders matter

Placeholders are important because moving a digit changes its value.

Compare these numbers:

• \(54\) means \(5\) tens and \(4\) ones.
• \(504\) means \(5\) hundreds, \(0\) tens, and \(4\) ones.
• \(5004\) means \(5\) thousands, \(0\) hundreds, \(0\) tens, and \(4\) ones.

The zeros show which places are empty, so the \(5\) and \(4\) stay in the correct places.

Zero in whole numbers

In whole numbers, zero can appear in the middle or at the end.

When zero is in the middle, it usually acts as a placeholder. For example, in \(306\), the zero shows there are no tens.

When zero is at the end of a whole number, it also shows place value. In \(80\), the zero shows there are \(8\) tens and \(0\) ones.

So,

and

The zeros help show whether the \(8\) means \(8\), \(80\), or \(800\).

Zero in decimals

Zero is also important in decimals. It can be a value or a placeholder there too.

In \(4.05\):

• The \(4\) is in the ones place.
• The \(0\) is in the tenths place, meaning there are no tenths.
• The \(5\) is in the hundredths place.

So,

The zero is important because \(4.05\) is not the same as \(4.5\).

• \(4.05\) means 4 ones, 0 tenths, and 5 hundredths.
• \(4.5\) means 4 ones and 5 tenths.

Since \(5\) tenths is greater than \(5\) hundredths, \(4.5\) is greater than \(4.05\).

Important note about zeros at the end of decimals

Sometimes a zero at the end of a decimal does not change the value.

For example,

This is because \(0.5\) means 5 tenths, and \(0.50\) means 50 hundredths. These are equal amounts.

But zeros inside a decimal number can be very important. For example, \(0.05\) and \(0.5\) are not equal.

Reading numbers with zero

When reading numbers, we still use place value to understand what zero is doing.

• \(407\) is read as four hundred seven.
• \(5,020\) is read as five thousand twenty.
• \(3.08\) is read as three and eight hundredths.

We do not usually say the zero when reading whole numbers, but we must understand that it is holding a place.

Comparing numbers with zero

Zero helps us compare numbers correctly because it affects place value.

Compare \(402\) and \(42\).

• \(402\) has \(4\) hundreds.
• \(42\) has \(4\) tens.

Since hundreds are larger than tens, \(402 > 42\).

Compare \(0.7\) and \(0.07\).

• \(0.7\) means 7 tenths.
• \(0.07\) means 7 hundredths.

Since tenths are larger than hundredths,

Worked Example 1: Zero as a value

Question: What does the zero mean in \(9 - 9 = 0\)?

Step 1: Think about the subtraction.

If you take \(9\) away from \(9\), nothing is left.

Answer: The zero means no amount. Here, zero is being used as a value.

Worked Example 2: Zero as a placeholder in a whole number

Question: What does the zero do in \(608\)?

Step 1: Name the places.

• \(6\) is in the hundreds place.
• \(0\) is in the tens place.
• \(8\) is in the ones place.

Step 2: Write the value of each digit.

Answer: The zero shows there are no tens. It is a placeholder that keeps the \(6\) in the hundreds place and the \(8\) in the ones place.

Worked Example 3: Why zero changes a number

Question: Explain the difference between \(45\) and \(405\).

Step 1: Break apart \(45\).

This means 4 tens and 5 ones.

Step 2: Break apart \(405\).

This means 4 hundreds, 0 tens, and 5 ones.

Answer: The zero in \(405\) holds the tens place. Because of that zero, the \(4\) is in the hundreds place, not the tens place. So \(405\) is much greater than \(45\).

Worked Example 4: Zero in a decimal

Question: Which is greater: \(2.5\) or \(2.05\)?

Step 1: Look at the decimal places.

• \(2.5\) means 2 ones and 5 tenths.
• \(2.05\) means 2 ones, 0 tenths, and 5 hundredths.

Step 2: Compare tenths and hundredths.

5 tenths is greater than 5 hundredths.

So,

Answer: \(2.5\) is greater. The zero in \(2.05\) shows there are no tenths, so the \(5\) is in the hundredths place.

Key ideas to remember

• Zero can mean none or nothing.
• Zero can also be a placeholder in a number.
• A placeholder zero keeps other digits in the correct place values.
• Without zero, a number can mean something completely different.
• In decimals, zero can help show whether a digit is in the tenths or hundredths place.
• A zero at the end of a decimal, like in \(0.50\), does not change the value.

Summary

Zero is one of the most important digits in the base-ten system. It is both a number with value \(0\) and a placeholder that helps show place value.

In numbers like \(504\), \(80\), and \(4.05\), zero tells us which places are empty so the other digits keep their correct values. When you understand the role of zero, you can read, write, compare, and understand numbers much more accurately.

Comparing and Ordering Rational Numbers means deciding which numbers are greater, smaller, or equal, and putting numbers in order from least to greatest or greatest to least.

In 6th grade, you will often compare whole numbers and decimals. The key idea is to use place value. Each digit has a value based on its position.

For example, in the number \(4.27\):

• The \(4\) is in the ones place.
• The \(2\) is in the tenths place.
• The \(7\) is in the hundredths place.

When comparing numbers, always start by looking at the digit with the greatest place value. If those digits are the same, move one place to the right until you find a difference.

Comparison symbols help show the relationship between numbers:

• \(>\) means greater than
• \(<\) means less than
• \(=\) means equal to

For example, \(8 > 5\), \(2.4 < 2.9\), and \(6.00 = 6\).

Important idea: Adding zeros to the end of a decimal does not change its value. So \(3.5 = 3.50 = 3.500\).

This is helpful because sometimes writing extra zeros makes decimals easier to compare.

Steps for comparing whole numbers and decimals:

1. Look at the greatest place value.
2. If the digits are different, the number with the greater digit is greater.
3. If the digits are the same, move to the next place on the right.
4. Keep going until you find a difference.
5. If all compared digits are the same, the numbers are equal.

When ordering numbers, compare them two at a time and then place them in the correct sequence.

Worked Example 1: Comparing whole numbers

Compare \(4{,}582\) and \(4{,}625\).

Start with the thousands place. Both numbers have \(4\) thousands.

Move to the hundreds place. The first number has \(5\) hundreds, and the second number has \(6\) hundreds.

Since \(5 < 6\), we know:

$$4{,}582 < 4{,}625$$

So \(4{,}625\) is greater.

Worked Example 2: Comparing decimals

Compare \(2.48\) and \(2.53\).

Start with the ones place. Both numbers have \(2\) ones.

Move to the tenths place. \(2.48\) has \(4\) tenths, and \(2.53\) has \(5\) tenths.

Since \(4 < 5\), we know:

$$2.48 < 2.53$$

So \(2.53\) is greater.

Worked Example 3: Comparing decimals with different numbers of digits

Compare \(6.7\) and \(6.68\).

It helps to write \(6.7\) as \(6.70\).

Now compare:

$$6.70 \quad \text{and} \quad 6.68$$

Start with the ones place. Both have \(6\) ones.

Move to the tenths place. \(6.70\) has \(7\) tenths, and \(6.68\) has \(6\) tenths.

Since \(7 > 6\), we know:

$$6.70 > 6.68$$

So:

$$6.7 > 6.68$$

Worked Example 4: Ordering a set of decimals

Order these numbers from least to greatest:

$$3.09,\; 3.9,\; 3.12,\; 3.105$$

First, write them so they have the same number of decimal places:

$$3.090,\; 3.900,\; 3.120,\; 3.105$$

Now compare from left to right.

All have \(3\) ones, so move to the tenths place.

• \(3.090\) has \(0\) tenths
• \(3.900\) has \(9\) tenths
• \(3.120\) has \(1\) tenth
• \(3.105\) has \(1\) tenth

The smallest is \(3.090\), or \(3.09\).

Now compare \(3.120\) and \(3.105\). They both have \(1\) tenth, so check the hundredths place.

\(3.120\) has \(2\) hundredths, and \(3.105\) has \(0\) hundredths.

Since \(0 < 2\), we get:

$$3.105 < 3.120$$

And \(3.900\), or \(3.9\), is the greatest.

So the order from least to greatest is:

$$3.09,\; 3.105,\; 3.12,\; 3.9$$

Tips to remember:

• Compare digits from the greatest place value to the least.
• Do not just count how many digits a decimal has.
• Use zeros at the end of decimals to help line up place values.
• A number like \(4.50\) is equal to \(4.5\).

A common mistake is thinking that \(0.9 < 0.27\) because \(27\) is bigger than \(9\). But this is not correct.

Write \(0.9\) as \(0.90\). Then compare:

$$0.90 \quad \text{and} \quad 0.27$$

At the tenths place, \(9 > 2\), so:

$$0.9 > 0.27$$

This shows why place value is so important.

Let’s review the main idea: whole numbers and decimals are compared by checking digits from left to right, starting with the greatest place value. If needed, write extra zeros at the end of decimals so the place values are easier to line up.

Once you can compare two numbers correctly, you can also order a whole list of numbers with confidence.

Rounding Strategies and Midpoints

Rounding helps us replace a number with a nearby number that is easier to read, compare, or use in everyday life. For example, a store might round a price, or we might round a distance to the nearest mile.

To round well, you need to understand place value and midpoints. A midpoint is the number exactly halfway between two benchmark numbers.

In this lesson, you will learn how to round whole numbers and decimals by using place value and by thinking about where a number sits on a number line.

1. What does rounding mean?

When we round, we choose the nearest benchmark number. A benchmark number is a number at the place you are rounding to.

For example, if you round to the nearest ten, the benchmark numbers are multiples of 10, such as 20, 30, 40, and 50. If you round to the nearest tenth, the benchmark numbers are numbers like 3.1, 3.2, 3.3, and 3.4.

The rounded number should be the benchmark number that is closest to the original number.

2. Why midpoints matter

A midpoint is the value exactly halfway between two benchmark numbers. Midpoints help us decide whether to round down or round up.

Here is the key rounding rule:

• If the number is less than the midpoint, round down.
• If the number is greater than the midpoint, round up.
• If the number is exactly at the midpoint, round up.

For example, when rounding to the nearest ten, the number 45 is the midpoint between 40 and 50.

Since 45 is exactly halfway, it rounds up to 50.

3. Rounding on a number line

A number line is a great tool for rounding because it shows which benchmark numbers a value is between.

To round using a number line:

1. Find the two benchmark numbers around the number.
2. Find the midpoint between those benchmarks.
3. Decide whether the number is closer to the lower benchmark or the higher benchmark.

Suppose you want to round 67 to the nearest ten.

The two benchmark numbers are 60 and 70.

The midpoint is:

$$\frac{60+70}{2}=65$$

Because 67 is greater than 65, it is closer to 70. So 67 rounds to 70.

4. A quick place-value strategy

You can also round by looking at the digit to the right of the place you are rounding to.

• If that digit is 0, 1, 2, 3, or 4, round down.
• If that digit is 5, 6, 7, 8, or 9, round up.

This works because the digit 5 represents the midpoint or more.

For whole numbers:

• Round 482 to the nearest ten: look at the ones digit, which is 2. Round down to 480.
• Round 486 to the nearest ten: look at the ones digit, which is 6. Round up to 490.

For decimals:

• Round 4.23 to the nearest tenth: look at the hundredths digit, which is 3. Round down to 4.2.
• Round 4.28 to the nearest tenth: look at the hundredths digit, which is 8. Round up to 4.3.

5. How to round whole numbers

Let us review the steps for whole numbers.

1. Decide which place to round to.
2. Find the digit in that place.
3. Look at the digit to its right.
4. If that digit is 5 or more, increase the rounding digit by 1.
5. If that digit is 4 or less, keep the rounding digit the same.
6. Change all digits to the right into 0.

Example: Round 3,472 to the nearest hundred.

• The hundreds digit is 4.
• The digit to the right is 7.
• Since 7 is greater than or equal to 5, round up.
• The 4 becomes 5, and the digits to the right become 0.

So,

$$3,472 \approx 3,500$$

6. How to round decimals

Decimals can be rounded in the same way as whole numbers. The only difference is that the places are tenths, hundredths, and thousandths.

Remember these decimal places:

• ones: the digit just left of the decimal point
• tenths: first digit right of the decimal point
• hundredths: second digit right of the decimal point
• thousandths: third digit right of the decimal point

Example: Round 6.347 to the nearest hundredth.

• The hundredths digit is 4.
• The digit to the right is 7.
• Since 7 is 5 or more, round up.
• The 4 becomes 5.

So,

$$6.347 \approx 6.35$$

7. Worked Examples

Example 1: Round a whole number to the nearest ten

Round 83 to the nearest ten.

The benchmark numbers are 80 and 90.

The midpoint is 85.

Since 83 is less than 85, it is closer to 80.

$$83 \approx 80$$

Example 2: Round a whole number to the nearest hundred

Round 650 to the nearest hundred.

The benchmark numbers are 600 and 700.

The midpoint is:

$$\frac{600+700}{2}=650$$

Since 650 is exactly at the midpoint, we round up.

$$650 \approx 700$$

Example 3: Round a decimal to the nearest tenth

Round 2.46 to the nearest tenth.

The tenths benchmark numbers around 2.46 are 2.4 and 2.5.

The midpoint between them is 2.45.

Since 2.46 is greater than 2.45, it is closer to 2.5.

$$2.46 \approx 2.5$$

Example 4: Round a decimal to the nearest hundredth

Round 9.125 to the nearest hundredth.

The hundredths benchmark numbers are 9.12 and 9.13.

The midpoint is 9.125.

Since the number is exactly at the midpoint, we round up.

$$9.125 \approx 9.13$$

8. Common mistakes to avoid

• Do not look at all the digits. Only look at the digit to the right of the place you are rounding to.
• Do not forget the midpoint rule. If the number is exactly halfway, round up.
• Do not change digits to the left. Only the rounding digit may change, and digits to the right become 0 in whole numbers or are removed in decimals.
• Be careful with decimals. The place names matter. Nearest tenth and nearest hundredth are not the same.

9. Helpful thinking questions

When you round, ask yourself:

• What place am I rounding to?
• What are the two benchmark numbers?
• What is the midpoint?
• Is my number below the midpoint, above it, or exactly on it?

10. Summary

Rounding means finding the nearest benchmark number. Midpoints are important because they tell you when to round down or up.

You can round by using a number line or by checking the digit to the right of the place you are rounding to. If that digit is 5 or more, round up. If it is 4 or less, round down.

With practice, rounding whole numbers and decimals becomes quick and accurate.

Number Line Scaling and Intervals

A number line is a straight line with numbers placed in order from least to greatest. We use number lines to locate, compare, and estimate numbers.

Sometimes a number line shows every counting number, like 0, 1, 2, 3, and so on. But many number lines do not label every point. Instead, you must figure out the scale and the interval size.

Understanding scale and intervals helps you place exact whole numbers and decimals correctly. This is very important when numbers are large, small, or close together.

What do scale and interval mean?

• Scale tells how the number line is counting.
• Interval is the amount between two marks that are next to each other.

For example, if the marks on a number line go 0, 1, 2, 3, then the interval is 1. If the marks go 0, 5, 10, 15, then the interval is 5.

If the number line shows decimals like 0.0, 0.1, 0.2, 0.3, then the interval is 0.1.

How to find the interval size

When some numbers are labeled, you can find the interval by using two labeled marks.

1. Find the difference between the two labeled numbers.
2. Count how many equal spaces are between them.
3. Divide the difference by the number of spaces.

This gives the value of one interval.

In math form:

Main idea: Equal spaces on a number line must represent equal amounts.

Worked Example 1: Finding the interval on a whole-number line

A number line has a mark labeled 10 and, 4 equal spaces later, a mark labeled 30. What is the interval size?

Step 1: Find the difference.

$$30 - 10 = 20$$

Step 2: Count the equal spaces.

There are 4 spaces.

Step 3: Divide.

$$\frac{20}{4} = 5$$

So the interval size is 5.

That means the number line counts like this:

$$10,\ 15,\ 20,\ 25,\ 30$$

How to identify a number on a number line

Once you know the interval, move from one labeled point by adding or subtracting the interval each time.

If the point is to the right, add. If the point is to the left, subtract.

Worked Example 2: Identifying a missing whole number

Suppose a number line has equal marks. One mark is labeled 40 and the next labeled mark, 5 spaces later, is 65. What number is at the third mark after 40?

Step 1: Find the difference.

$$65 - 40 = 25$$

Step 2: Divide by the number of spaces.

$$\frac{25}{5} = 5$$

So each interval is 5.

Step 3: Count forward from 40.

• 1st mark after 40: \(45\)
• 2nd mark after 40: \(50\)
• 3rd mark after 40: \(55\)

The third mark after 40 is 55.

Number lines with decimals

Number lines do not only show whole numbers. They can also show tenths, hundredths, and other decimal amounts.

Remember:

• Tenths change by \(0.1\)
• Hundredths change by \(0.01\)

For example:

• \(1.0, 1.1, 1.2, 1.3\) has interval \(0.1\)
• \(2.45, 2.46, 2.47\) has interval \(0.01\)

Worked Example 3: Finding a decimal on a number line

A number line starts at \(2.0\) and ends at \(2.5\). There are 5 equal spaces between these labels. What is the interval size, and what is the second mark after \(2.0\)?

Step 1: Find the difference.

$$2.5 - 2.0 = 0.5$$

Step 2: Divide by the number of spaces.

$$\frac{0.5}{5} = 0.1$$

So each interval is 0.1.

Step 3: Count forward.

• 1st mark after \(2.0\): \(2.1\)
• 2nd mark after \(2.0\): \(2.2\)

The second mark after \(2.0\) is 2.2.

Worked Example 4: A smaller decimal interval

On a number line, one mark is labeled \(3.40\) and the mark 6 equal spaces later is labeled \(3.46\). What is the interval size? What number is 4 marks after \(3.40\)?

Step 1: Find the difference.

$$3.46 - 3.40 = 0.06$$

Step 2: Divide by the number of spaces.

$$\frac{0.06}{6} = 0.01$$

So each interval is 0.01.

Step 3: Count forward by hundredths.

• 1st mark: \(3.41\)
• 2nd mark: \(3.42\)
• 3rd mark: \(3.43\)
• 4th mark: \(3.44\)

The number 4 marks after \(3.40\) is 3.44.

How to plot a number on a number line

To plot a number means to place it in the correct spot.

Use these steps:

1. Find the interval size.
2. Decide which labeled numbers the point is between.
3. Count the intervals carefully.
4. Place the number exactly at that mark.

For example, if a number line counts by \(0.2\), then the marks might be:

$$1.0,\ 1.2,\ 1.4,\ 1.6,\ 1.8,\ 2.0$$

To plot \(1.6\), start at \(1.0\) and count by \(0.2\):

$$1.0 \to 1.2 \to 1.4 \to 1.6$$

So \(1.6\) goes on the third mark after \(1.0\).

Common mistakes to avoid

• Counting marks instead of spaces: The interval is based on the number of spaces between labeled points.
• Forgetting the decimal place value: \(0.1\) and \(0.01\) are not the same.
• Assuming the interval is 1: Always check the scale first.
• Skipping careful counting: Count one interval at a time.

Helpful strategy

If you are not sure, write the numbers that would go on each mark. This helps you see the pattern clearly.

For example, if the interval is \(0.05\), you can list:

$$0.20,\ 0.25,\ 0.30,\ 0.35,\ 0.40$$

Seeing the pattern makes it easier to identify missing numbers.

Summary

• A scale shows how a number line is counting.
• An interval is the amount between two neighboring marks.
• To find the interval, subtract the labeled numbers and divide by the number of equal spaces.
• Once you know the interval, add or subtract to find missing whole numbers or decimals.
• Always count spaces carefully and pay attention to place value.

When you understand number line scaling and intervals, you can read, identify, and plot numbers much more accurately.