Base-Ten Number System and Place Value

Digits, Numerals, and Quantities

In math, we use symbols to show numbers. It is important to understand the difference between a digit, a numeral, and a quantity.

These words are connected, but they do not mean the same thing. Once you know the difference, it becomes much easier to read, write, and compare numbers.

What is a digit?

A digit is one of the ten symbols we use in our base-ten number system:

$$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$$

These are like the building blocks of all whole numbers and decimals. Every larger number is made by putting digits together in different places.

For example:

• In the number 7, there is 1 digit.
• In the number 42, there are 2 digits: 4 and 2.
• In the number 305, there are 3 digits: 3, 0, and 5.

What is a numeral?

A numeral is a written number. It can have one digit or many digits.

For example:

• 7 is a numeral.
• 42 is a numeral.
• 305 is a numeral.
• 1,248 is a numeral.

So, a digit is a single symbol, but a numeral can be made of one or more digits.

What is a quantity?

A quantity is the amount a number represents. It tells how many.

For example, if you see 5 apples, the quantity is 5. The numeral is 5, and the digit used is also 5.

If you see 12 books, the quantity is 12 books. The numeral is 12, and the digits are 1 and 2.

This means:

• Digit = a symbol like 3 or 8
• Numeral = a written number like 38 or 208
• Quantity = the amount, like 38 stickers or 208 marbles

Why does place matter?

In our base-ten system, the value of a digit depends on its place. The same digit can represent different amounts in different numerals.

Look at the digit 4 in these numerals:

• 4
• 40
• 400

Even though the digit is the same, the quantity changes because the place changes.

$$4 = 4 \text{ ones}$$

$$40 = 4 \text{ tens} = 40$$

$$400 = 4 \text{ hundreds} = 400$$

This is why digits are symbols, but numerals show a full number, and quantities show the amount that number means.

Worked Example 1: Finding digits and the numeral

Look at the number 58.

• The numeral is 58.
• The digits are 5 and 8.
• The quantity is 58 things, such as 58 coins.

This numeral has 2 digits, but it represents 1 quantity: 58.

Worked Example 2: Same digits, different numerals

Compare 24 and 42.

Both numerals use the same digits: 2 and 4.

But they do not represent the same quantity.

In 24:

• 2 is in the tens place, so it means 20.
• 4 is in the ones place, so it means 4.

$$24 = 20 + 4$$

In 42:

• 4 is in the tens place, so it means 40.
• 2 is in the ones place, so it means 2.

$$42 = 40 + 2$$

So, the digits are the same, but the numerals and quantities are different.

Worked Example 3: Understanding zero as a digit

Look at the numeral 507.

The digits are 5, 0, and 7.

The 0 is important. It shows that there are no tens.

$$507 = 5 \text{ hundreds} + 0 \text{ tens} + 7 \text{ ones}$$

$$507 = 500 + 0 + 7$$

The quantity is 507.

Without the 0, the numeral would be 57, which is a very different quantity.

Worked Example 4: Matching a quantity to a numeral

A jar has 136 beads.

What is the quantity, what is the numeral, and what are the digits?

• The quantity is 136 beads.
• The numeral is 136.
• The digits are 1, 3, and 6.

We can also break the numeral into place values:

$$136 = 100 + 30 + 6$$

This shows that the digits work together to represent one quantity.

Important ideas to remember

• There are only 10 digits: 0 through 9.
• A numeral is a written number made from digits.
• A quantity is the amount the numeral represents.
• The place of a digit changes its value.
• The digit 0 is important because it can show that a place has no value.

Quick check

1. In the numeral 83, what are the digits?
   Answer: 8 and 3
2. Is 409 a digit or a numeral?
   Answer: It is a numeral.
3. What quantity does the numeral 12 represent?
   Answer: A quantity of 12 things.
4. Do 35 and 53 represent the same quantity?
   Answer: No, because the digits are in different places.

Summary

A digit is a single symbol from 0 to 9. A numeral is a written number made from one or more digits. A quantity is the amount that the numeral shows.

When you read or write numbers, always ask yourself: What are the digits? What numeral do they make? What quantity does it represent? This helps you understand numbers clearly and use place value correctly.

Base-Ten Positional Structure

Our number system is called the base-ten system. It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

In this system, a digit’s value depends on its place in the number. This is called place value.

For example, in the number \(352\), the digit 3 does not mean just 3 ones. It means 3 hundreds, or \(300\), because it is in the hundreds place.

This is the big idea of positional structure: moving one place to the left makes a value 10 times greater, and moving one place to the right makes a value one-tenth as much.

Here is a place value chart for whole numbers and decimals:

• Thousands
• Hundreds
• Tens
• Ones
• Tenths
• Hundredths
• Thousandths

You can think of the places like steps. Each step to the left is 10 times bigger. Each step to the right is one-tenth the size.

For example:

• 1 ten = 10 ones
• 1 hundred = 10 tens = 100 ones
• 1 thousand = 10 hundreds
• 1 one = 10 tenths
• 1 tenth = 10 hundredths

This pattern continues forever in both directions.

Important rule: The same digit can have different values depending on where it is placed.

Look at the digit 5 in these numbers:

• \(5\) means 5 ones
• \(50\) means 5 tens, or \(50\)
• \(500\) means 5 hundreds, or \(500\)
• \(0.5\) means 5 tenths
• \(0.05\) means 5 hundredths

The digit stays the same, but its value changes because its position changes.

We can also write numbers in expanded form. Expanded form shows the value of each digit.

For example, the number \(4,382\) can be written as

$$4,382 = 4,000 + 300 + 80 + 2$$

This shows that:

• 4 is in the thousands place
• 3 is in the hundreds place
• 8 is in the tens place
• 2 is in the ones place

Decimals follow the same pattern. The decimal point separates the ones place from places smaller than one.

To the right of the decimal point:

• the first place is tenths
• the second place is hundredths
• the third place is thousandths

For example, in \(2.47\):

• 2 is in the ones place, so it means 2
• 4 is in the tenths place, so it means \(0.4\)
• 7 is in the hundredths place, so it means \(0.07\)

Its expanded form is

$$2.47 = 2 + 0.4 + 0.07$$

Another way to think about place value is by comparing the same digit in different places.

In \(7,000\), the 7 means 7 thousands. In \(700\), the 7 means 7 hundreds. Since a thousand is 10 times a hundred, the 7 in \(7,000\) is 10 times the value of the 7 in \(700\).

We can write that idea like this:

$$7,000 = 10 \times 700$$

Now let’s work through some examples.

Worked Example 1: Find the value of a digit

What is the value of the digit 6 in \(6,241\)?

Step 1: Find the place of the digit 6.

The 6 is in the thousands place.

Step 2: Write its value.

$$6 \text{ thousands} = 6,000$$

Answer: The value of the 6 is \(6,000\).

Worked Example 2: Write a number in expanded form

Write \(9,305\) in expanded form.

Look at each digit:

• 9 is in the thousands place: \(9,000\)
• 3 is in the hundreds place: \(300\)
• 0 is in the tens place: \(0\)
• 5 is in the ones place: \(5\)

So,

$$9,305 = 9,000 + 300 + 5$$

We usually do not need to write the \(0\) tens.

Worked Example 3: Understand the 10-times relationship

Compare the digit 4 in \(4,800\) and \(480\).

In \(4,800\), the 4 is in the thousands place, so its value is \(4,000\).

In \(480\), the 4 is in the hundreds place, so its value is \(400\).

Now compare:

$$4,000 = 10 \times 400$$

Answer: The 4 in \(4,800\) is 10 times the value of the 4 in \(480\).

Worked Example 4: Place value with decimals

What is the value of the digit 3 in \(5.34\)?

The 3 is in the tenths place.

So its value is

$$3 \text{ tenths} = 0.3$$

We can also describe the whole number:

$$5.34 = 5 + 0.3 + 0.04$$

Answer: The value of the 3 is \(0.3\).

Helpful patterns to remember

• Each place to the left is 10 times greater.
• Each place to the right is one-tenth as much.
• A digit’s value depends on its position.
• The decimal point separates whole numbers from parts smaller than one.

Common mistakes to avoid

• Do not say a digit’s value is just the digit itself. In \(823\), the 8 means \(800\), not 8.
• Do not forget zero placeholders. In \(507\), the 0 shows there are no tens.
• Do not mix up tenths and hundredths. In \(0.6\), the 6 is tenths. In \(0.06\), the 6 is hundredths.

Quick check questions

1. What is the value of the 7 in \(7,421\)?
2. What is the value of the 2 in \(3.29\)?
3. Write \(6,040\) in expanded form.
4. Is the 5 in \(500\) ten times the value of the 5 in \(50\)?

Answers

1. \(7,000\)
2. \(0.2\)
3. \(6,000 + 40\)
4. Yes, because \(500 = 10 \times 50\).

Summary

In the base-ten number system, the value of a digit depends on where it is in the number. Each place to the left is 10 times greater than the place to its right, and each place to the right is one-tenth of the place to its left.

Understanding this pattern helps you read numbers, write numbers in expanded form, and compare the value of digits in whole numbers and decimals.

Place Value through Billions

Our number system is called the base-ten system. That means numbers are built using groups of 10. Every time you move one place to the left, the value becomes 10 times greater. Every time you move one place to the right, the value becomes 10 times smaller.

Place value helps us understand what each digit in a number is worth. The same digit can have different values depending on where it is in the number.

For example, in the number \(5,555\), each digit is a 5, but they do not all mean the same thing:

• The 5 in the thousands place means \(5{,}000\)
• The 5 in the hundreds place means \(500\)
• The 5 in the tens place means \(50\)
• The 5 in the ones place means \(5\)

To work with very large numbers, we group digits into periods. Each period has 3 places.

• Ones period: ones, tens, hundreds
• Thousands period: thousands, ten thousands, hundred thousands
• Millions period: millions, ten millions, hundred millions
• Billions period: billions

Here is the place value chart through billions:

$$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{Billions} & \text{Hundred Millions} & \text{Ten Millions} & \text{Millions} & \text{Hundred Thousands} & \text{Ten Thousands} & \text{Thousands} & \text{Hundreds} & \text{Tens} & \text{Ones} \\ \hline \end{array} $$

You can also think of the places like this:

$$ 1{,}000{,}000{,}000 \quad 100{,}000{,}000 \quad 10{,}000{,}000 \quad 1{,}000{,}000 \quad 100{,}000 \quad 10{,}000 \quad 1{,}000 \quad 100 \quad 10 \quad 1 $$

Important pattern: each place is 10 times the place to its right.

• 10 ones = 1 ten
• 10 tens = 1 hundred
• 10 hundreds = 1 thousand
• 10 thousands = 1 ten thousand
• 10 hundred millions = 1 billion

When reading large numbers, read the number in periods from left to right. Say the period name after each group of 3 digits.

For example, in \(4,238,517,906\):

• 4 is in the billions period
• 238 is in the millions period
• 517 is in the thousands period
• 906 is in the ones period

So we read it as four billion, two hundred thirty-eight million, five hundred seventeen thousand, nine hundred six.

Notice: We do not say commas when we read the number. The commas just help us see the periods.

We can write numbers in different forms:

• Standard form: the usual way we write a number with digits
• Word form: writing the number with words
• Expanded form: showing the value of each digit

For example, the number \(3,406,020,105\) can be written as:

• Standard form: \(3,406,020,105\)
• Word form: three billion, four hundred six million, twenty thousand, one hundred five
• Expanded form:

$$ 3{,}000{,}000{,}000 + 400{,}000{,}000 + 6{,}000{,}000 + 20{,}000 + 100 + 5 $$

Zeroes are very important in place value. A zero holds a place when there is no value in that spot.

In \(3,406,020,105\):

• There are 0 ten millions
• There are 0 hundred thousands
• There are 0 thousands
• There are 0 tens

Those zeroes help the other digits stay in the correct places.

How to find the value of a digit

1. Find where the digit is located.
2. Name the place.
3. Multiply the digit by the value of that place.

For example, if a 7 is in the hundred millions place, its value is:

$$ 7 \times 100{,}000{,}000 = 700{,}000{,}000 $$

Worked Example 1

What is the value of the digit 6 in \(6,214,389\)?

Step 1: Find the place of the digit 6.

The 6 is in the millions place.

Step 2: Find its value.

$$ 6 \times 1{,}000{,}000 = 6{,}000{,}000 $$

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

Worked Example 2

Read and write this number in word form: \(45,072,300\)

First, separate the periods:

• 45 million
• 072 thousand
• 300

Now read each part:

• 45 million = forty-five million
• 072 thousand = seventy-two thousand
• 300 = three hundred

Answer: forty-five million, seventy-two thousand, three hundred

Worked Example 3

Write \(508,030,019\) in expanded form.

Look at each digit and its place:

• 5 hundred millions = \(500{,}000{,}000\)
• 0 ten millions = skip
• 8 millions = \(8{,}000{,}000\)
• 0 hundred thousands = skip
• 3 ten thousands = \(30{,}000\)
• 0 thousands = skip
• 0 hundreds = skip
• 1 ten = \(10\)
• 9 ones = \(9\)

So the expanded form is:

$$ 500{,}000{,}000 + 8{,}000{,}000 + 30{,}000 + 10 + 9 $$

Worked Example 4

In the number \(2,741,605,830\), what is the value of the digit 4, and what place is it in?

Read the places from left to right:

• 2 = billions
• 7 = hundred millions
• 4 = ten millions
• 1 = millions

So the digit 4 is in the ten millions place.

Its value is:

$$ 4 \times 10{,}000{,}000 = 40{,}000{,}000 $$

Answer: The 4 is in the ten millions place, and its value is 40,000,000.

Helpful Tips

• Use commas to separate periods every 3 digits.
• Start reading from the left.
• Say the period names: billion, million, thousand.
• Remember that zeroes hold places.
• To find a digit’s value, combine the digit with its place.

Common Mistakes to Avoid

• Do not confuse place with value. The place might be millions, but the value could be \(7{,}000{,}000\).
• Do not skip period names when reading large numbers.
• Do not forget zeroes. They keep digits in the correct positions.
• Be careful with numbers that have empty places, like \(204,005,010\).

Summary

Place value tells what each digit is worth in a number. In the base-ten system, each place to the left is 10 times greater than the one to its right.

Through billions, the places are ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, and billions. We can read, write, and understand large numbers by using periods, place names, and the value of each digit.

Decimal Place Value through Thousandths

We already know that in our base-ten number system, each place to the left is 10 times as great as the place to its right. For example, in the number 352, the 3 means 3 hundreds, the 5 means 5 tens, and the 2 means 2 ones.

Decimals help us show numbers that are less than one whole. They use the same base-ten pattern, but now we move to the right of the decimal point.

The decimal point is very important. It separates the whole-number part from the part smaller than one.

To the right of the decimal point, the places are:

• tenths
• hundredths
• thousandths

Here is the place-value chart:

$$ \begin{array}{c|c|c|c|c|c|c} \text{Hundreds} & \text{Tens} & \text{Ones} & \text{.} & \text{Tenths} & \text{Hundredths} & \text{Thousandths} \\ \hline & & & & & & \end{array} $$

Notice the pattern:

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

Each place to the right is 10 times smaller than the place to its left.

We can also write these places as fractions:

• tenths: \(\frac{1}{10}\)
• hundredths: \(\frac{1}{100}\)
• thousandths: \(\frac{1}{1000}\)

That means:

• \(0.1\) means 1 tenth
• \(0.01\) means 1 hundredth
• \(0.001\) means 1 thousandth

Reading decimals is easier when you read the whole-number part first, say “and” for the decimal point, and then read the decimal part by its last place value.

For example:

• \(2.4\) is read as two and four tenths
• \(5.36\) is read as five and thirty-six hundredths
• \(7.208\) is read as seven and two hundred eight thousandths

Zeros can be important in decimals because they help show place value.

For example, \(0.5\), \(0.50\), and \(0.500\) all have the same value, but they show the number in different place-value forms.

However, \(0.5\) and \(0.05\) are not the same.

• \(0.5\) means 5 tenths
• \(0.05\) means 5 hundredths

The position of the digit changes its value.

Example 1: Name the value of each digit

Look at the number \(4.583\).

The digits mean:

• 4 is in the ones place, so it means 4 ones
• 5 is in the tenths place, so it means 5 tenths
• 8 is in the hundredths place, so it means 8 hundredths
• 3 is in the thousandths place, so it means 3 thousandths

We can write it as:

$$ 4.583 = 4 + \frac{5}{10} + \frac{8}{100} + \frac{3}{1000} $$

Example 2: Write a decimal from words

Write six and twenty-four hundredths as a decimal.

Step 1: The whole-number part is 6.

Step 2: “Twenty-four hundredths” means \(\frac{24}{100}\), which is \(0.24\).

Step 3: Put them together.

$$ 6.24 $$

So, six and twenty-four hundredths = \(6.24\).

Example 3: Write a decimal in expanded form

Write \(3.407\) in expanded form.

Look at each digit:

• 3 ones
• 4 tenths
• 0 hundredths
• 7 thousandths

Expanded form is:

$$ 3.407 = 3 + \frac{4}{10} + \frac{0}{100} + \frac{7}{1000} $$

We usually do not need to write the zero part, so we can also write:

$$ 3.407 = 3 + \frac{4}{10} + \frac{7}{1000} $$

This example shows that a zero can hold a place so the 7 stays in the thousandths place.

Example 4: Compare two decimals

Which is greater: \(0.56\) or \(0.506\)?

To compare decimals, line up the place values.

We can rewrite \(0.56\) as \(0.560\).

Now compare:

$$ 0.560 $$

$$ 0.506 $$

• Tenths: both have 5 tenths
• Hundredths: 6 hundredths is greater than 0 hundredths

So:

$$ 0.56 > 0.506 $$

Helpful tips for decimals

• Say the place names in order: ones, tenths, hundredths, thousandths.
• The farther right a digit is, the smaller its value becomes.
• Use zeros as placeholders when needed.
• When comparing decimals, line up the decimal points.
• Read the last decimal place to know the name of the number part.

Common mistakes to avoid

• Do not forget that \(0.1\) is 1 tenth, not 1 one.
• Do not think a longer decimal is always greater. For example, \(0.7\) is greater than \(0.65\).
• Be careful with zeros. In \(2.03\), the 3 is in the hundredths place, not the tenths place.

Summary

Decimals show parts of a whole. The places to the right of the decimal point are tenths, hundredths, and thousandths. Each place is 10 times smaller than the place to its left. By knowing place value, you can read decimals, write them, expand them, and compare them correctly.

Standard, Word, and Expanded Notation are three different ways to show the same number.

When you understand all three forms, you can read numbers, write numbers, and see the value of each digit more clearly. This helps you with place value and with understanding how our base-ten number system works.

In this lesson, you will learn what each notation means, how to change a number from one form to another, and how place value helps you do it correctly.

1. What is standard notation?

Standard notation is the way we usually write a number using digits.

For example, the number 4,582 is written in standard notation.

2. What is word notation?

Word notation means writing the number using words.

For example, 4,582 in word notation is four thousand five hundred eighty-two.

3. What is expanded notation?

Expanded notation shows the value of each digit in the number.

For example, in 4,582:

• The 4 is in the thousands place, so it means 4,000.
• The 5 is in the hundreds place, so it means 500.
• The 8 is in the tens place, so it means 80.
• The 2 is in the ones place, so it means 2.

So the expanded notation is:

$$4,582 = 4,000 + 500 + 80 + 2$$

Why place value matters

Each digit has a value based on its place. A digit does not always mean the same amount.

For example, in the number 3,303:

• The first 3 means 3,000.
• The second 3 means 300.
• The last 3 means 3.

Even though the digit is the same, its value changes because its place changes.

Place value chart

A place value chart can help you see each digit's value.

For the number 47,216:

• 4 ten-thousands
• 7 thousands
• 2 hundreds
• 1 ten
• 6 ones

This means:

$$47,216 = 40,000 + 7,000 + 200 + 10 + 6$$

And in words, it is forty-seven thousand two hundred sixteen.

How to write a number in word notation

1. Look at the number from left to right.
2. Break it into periods, such as thousands and ones.
3. Write the value of each period in words.
4. Add the period name, like thousand, when needed.

Example: 12,405

• 12 thousand
• 405 is four hundred five

So the word notation is twelve thousand four hundred five.

Important note: A zero holds a place, but it does not add value in expanded notation.

In 12,405, the 0 is in the tens place. That means there are 0 tens, so we usually do not write + 0 in expanded notation.

$$12,405 = 10,000 + 2,000 + 400 + 5$$

How to write a number in expanded notation

1. Find the value of each digit by its place.
2. Write each value as an addend.
3. Add the values together.

Example: 6,731

• 6 thousands = 6,000
• 7 hundreds = 700
• 3 tens = 30
• 1 one = 1

So:

$$6,731 = 6,000 + 700 + 30 + 1$$

How to write a number from expanded notation to standard notation

Sometimes you are given the parts and must put them together.

Example:

$$30,000 + 5,000 + 600 + 20 + 9$$

Think about each place:

• 3 in the ten-thousands place
• 5 in the thousands place
• 6 in the hundreds place
• 2 in the tens place
• 9 in the ones place

So the standard notation is 35,629.

Worked Examples

Example 1: Write 2,468 in word notation and expanded notation.

Step 1: Read the number.

• 2 is in the thousands place
• 4 is in the hundreds place
• 6 is in the tens place
• 8 is in the ones place

Word notation: two thousand four hundred sixty-eight

Expanded notation:

$$2,468 = 2,000 + 400 + 60 + 8$$

Example 2: Write 50,307 in word notation and expanded notation.

This example has zeros, so be careful.

• 5 is in the ten-thousands place = 50,000
• 0 is in the thousands place = 0
• 3 is in the hundreds place = 300
• 0 is in the tens place = 0
• 7 is in the ones place = 7

Word notation: fifty thousand three hundred seven

Expanded notation:

$$50,307 = 50,000 + 300 + 7$$

We do not need to write the zero-value places in expanded notation.

Example 3: Write the standard notation for the number "nine thousand eighty-four."

Break the words into place values:

• nine thousand = 9,000
• eighty = 80
• four = 4

There are no hundreds, so the hundreds digit is 0.

$$9,000 + 80 + 4 = 9,084$$

Standard notation: 9,084

Example 4: Write the standard and word notation for

$$70,000 + 4,000 + 90 + 6$$

Step 1: Put the values into place value order.

• 7 in the ten-thousands place
• 4 in the thousands place
• 0 in the hundreds place
• 9 in the tens place
• 6 in the ones place

Standard notation: 74,096

Word notation: seventy-four thousand ninety-six

Common mistakes to avoid

• Forgetting place value: In 3,482, the 3 means 3,000, not 3.
• Skipping zero places: In 6,045, the 0 in the hundreds place and the 4 in the tens place matter. The number is not 645.
• Mixing up word order: 14,032 is fourteen thousand thirty-two, not fourteen hundred thirty-two.
• Writing too many parts in expanded notation: You usually leave out the parts that are 0.

Tips for success

• Read the number slowly by place value.
• Use commas to help separate thousands from ones.
• Ask yourself, “What is each digit worth?”
• When writing from words to digits, make sure every place is filled, even if a digit is 0.

Quick check

Try these on your own:

1. Write 8,125 in word notation and expanded notation.
2. Write sixty-two thousand five hundred one in standard notation.
3. Write 90,040 in word notation and expanded notation.

Answers

1. eight thousand one hundred twenty-five; $$8,000 + 100 + 20 + 5$$
2. 62,501
3. ninety thousand forty; $$90,000 + 40$$

Summary

A number can be written in standard notation using digits, in word notation using words, and in expanded notation by showing the value of each digit.

Place value is the key to changing a number from one form to another. When you know what each digit is worth, you can read, write, and understand numbers more easily.

Powers of Ten and Exponent Foundations

Our number system is called the base-ten system. That means it is built on groups of 10. Every time we move one place to the left in a number, the value becomes 10 times greater. Every time we move one place to the right, the value becomes 10 times smaller.

This lesson will help you understand powers of ten and how they connect to place value. You will also learn what an exponent means in simple cases like 10, 100, and 1,000.

1. Place value and powers of ten

Look at the places in a whole number:

ones, tens, hundreds, thousands, ten thousands

Each place is made by multiplying by 10:

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

We can write these values using powers of ten:

• 1 = \(10^0\)
• 10 = \(10^1\)
• 100 = \(10^2\)
• 1{,}000 = \(10^3\)
• 10{,}000 = \(10^4\)

The small raised number is called an exponent. The exponent tells how many 10s are being multiplied together.

For example:

$$10^3 = 10 \times 10 \times 10 = 1{,}000$$

Here are more examples:

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

2. What happens when you multiply by 10, 100, or 1,000?

When you multiply a number by 10, each digit moves one place to the left. That makes the number 10 times as great.

When you multiply by 100, each digit moves two places to the left. When you multiply by 1,000, each digit moves three places to the left.

You can think of it like this:

• Multiply by \(10 = 10^1\)  move 1 place left
• Multiply by \(100 = 10^2\)  move 2 places left
• Multiply by \(1{,}000 = 10^3\)  move 3 places left

Example place value chart idea:

If the digit 4 is in the ones place, it is worth 4.

If the digit 4 moves to the tens place, it is worth 40.

If the digit 4 moves to the hundreds place, it is worth 400.

Each move left multiplies its value by 10.

3. What happens when you divide by 10, 100, or 1,000?

When you divide by 10, each digit moves one place to the right. That makes the number 10 times smaller.

When you divide by 100, each digit moves two places to the right. When you divide by 1,000, each digit moves three places to the right.

• Divide by \(10 = 10^1\)  move 1 place right
• Divide by \(100 = 10^2\)  move 2 places right
• Divide by \(1{,}000 = 10^3\)  move 3 places right

This is true for whole numbers and decimals.

4. Powers of ten and decimals

Decimals are also part of the base-ten system. To the right of the decimal point, each place is divided by 10.

The places are:

ones, tenths, hundredths, thousandths

Each step to the right makes the value 10 times smaller:

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

For example:

• \(0.1\) is one tenth
• \(0.01\) is one hundredth
• \(0.001\) is one thousandth

So dividing by powers of ten can move a whole number into the decimal places.

5. A helpful pattern

When multiplying or dividing by powers of ten, many students notice the digits seem to move. Another way to think about it is that the decimal point moves in the opposite direction.

• Multiply by 10  decimal point moves 1 place right
• Multiply by 100  decimal point moves 2 places right
• Divide by 10  decimal point moves 1 place left
• Divide by 100  decimal point moves 2 places left

For whole numbers, the decimal point is at the end, even if you do not write it. For example, 45 means \(45.0\).

Worked Example 1: Writing a power of ten

Write 1,000 as a power of ten.

Step 1: Count how many 10s are multiplied together.

$$1{,}000 = 10 \times 10 \times 10$$

Step 2: There are 3 tens, so the exponent is 3.

$$1{,}000 = 10^3$$

Answer: \(1{,}000 = 10^3\)

Worked Example 2: Multiplying by a power of ten

Find \(36 \times 100\).

Since \(100 = 10^2\), multiplying by 100 moves each digit 2 places to the left.

36 becomes 3,600.

$$36 \times 100 = 3{,}600$$

Answer: \(3{,}600\)

Worked Example 3: Dividing by a power of ten

Find \(4{,}800 \div 10\).

Dividing by 10 moves each digit 1 place to the right.

$$4{,}800 \div 10 = 480$$

You can also think: the decimal point in \(4{,}800.0\) moves one place left to make \(480.0\).

Answer: \(480\)

Worked Example 4: Dividing into decimals

Find \(7 \div 100\).

Dividing by \(100 = 10^2\) moves the decimal point 2 places to the left.

Start with \(7.0\). Move the decimal point two places left:

$$7.0 \div 100 = 0.07$$

Answer: \(0.07\)

6. Common mistakes to avoid

• Do not just add zeros every time. This works for some whole-number multiplication problems, but not for division or all decimal problems.
• Watch the direction. Multiplying by powers of ten makes numbers greater, so digits move left. Dividing makes numbers smaller, so digits move right.
• Remember the decimal point. Every whole number has one, even if you do not see it.

7. Quick practice ideas

1. Write each as a power of ten: 10, 100, 1,000.
2. Find: \(52 \times 10\)
3. Find: \(52 \times 1{,}000\)
4. Find: \(900 \div 100\)
5. Find: \(6 \div 10\)

Answers:

• \(10 = 10^1\), \(100 = 10^2\), \(1{,}000 = 10^3\)
• \(52 \times 10 = 520\)
• \(52 \times 1{,}000 = 52{,}000\)
• \(900 \div 100 = 9\)
• \(6 \div 10 = 0.6\)

Summary

Powers of ten help us describe place value patterns in our base-ten number system. The exponent in a power of ten tells how many times 10 is used as a factor.

Multiplying by \(10\), \(100\), or \(1{,}000\) makes a number greater and shifts digits left. Dividing by these powers of ten makes a number smaller and shifts digits right. This pattern works with whole numbers and decimals.

Comparing and Ordering Numbers means deciding which numbers are greater, which are less, and putting numbers in the correct order. In 5th Grade, you compare whole numbers and decimals by using place value.

Place value tells us the value of each digit based on where it is in a number. For example, in the number \(4{,}582\), the digit 4 is in the thousands place, the 5 is in the hundreds place, the 8 is in the tens place, and the 2 is in the ones place.

When we compare numbers, we often use these symbols:

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

For example:

$$45 > 12$$

This means 45 is greater than 12.

To order numbers, we put them in a sequence:

• Least to greatest: smallest to largest
• Greatest to least: largest to smallest

How to compare whole numbers

When comparing whole numbers, start by looking at the digit with the greatest place value, which is usually the digit farthest to the left.

1. Look at how many digits each number has.
2. If one number has more digits, it is greater.
3. If the numbers have the same number of digits, compare digits from left to right.
4. As soon as one digit is greater than the other in the same place, that number is greater.

For example, compare \(3{,}482\) and \(3{,}529\).

• Both numbers have 4 digits.
• Compare the thousands digits: both are 3.
• Compare the hundreds digits: 4 and 5.
• Since \(4 < 5\), we know \(3{,}482 < 3{,}529\).

How to compare decimals

Decimals are compared in a very similar way. The important idea is to line up the decimal points and compare digits from left to right.

Each place to the right of the decimal has a value:

• tenths
• hundredths
• thousandths

For example, in \(6.37\):

• 6 is in the ones place
• 3 is in the tenths place
• 7 is in the hundredths place

Sometimes it helps to add zeros to the end of a decimal. Adding zeros to the right of a decimal does not change its value.

For example:

$$0.5 = 0.50 = 0.500$$

This is useful when comparing decimals with different numbers of digits.

Steps for comparing decimals

1. Line up the decimal points.
2. If needed, add zeros to make the same number of decimal places.
3. Compare digits from left to right.
4. The first place where the digits are different tells which number is greater.

Worked Example 1: Comparing whole numbers

Compare \(6{,}205\) and \(6{,}250\).

Step 1: Both numbers have 4 digits.

Step 2: Compare from left to right.

• Thousands: 6 and 6 — same
• Hundreds: 2 and 2 — same
• Tens: 0 and 5 — different

Since \(0 < 5\), we know:

$$6{,}205 < 6{,}250$$

Worked Example 2: Ordering whole numbers

Order these numbers from least to greatest:

\(4{,}901, 4{,}109, 4{,}190, 4{,}019\)

All of the numbers have 4 digits, and all have 4 in the thousands place. So compare the hundreds digits:

• \(4{,}019\) has 0 hundreds
• \(4{,}109\) has 1 hundred
• \(4{,}190\) has 1 hundred
• \(4{,}901\) has 9 hundreds

Now compare the two numbers with 1 hundred:

• \(4{,}109\): tens digit is 0
• \(4{,}190\): tens digit is 9

So the order is:

$$4{,}019 < 4{,}109 < 4{,}190 < 4{,}901$$

Worked Example 3: Comparing decimals

Compare \(3.7\) and \(3.65\).

First, line up the decimal points. It helps to write \(3.7\) as \(3.70\).

Now compare:

$$3.70 \text{ and } 3.65$$

• Ones: 3 and 3 — same
• Tenths: 7 and 6 — different

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

$$3.70 > 3.65$$

So:

$$3.7 > 3.65$$

Worked Example 4: Ordering decimals

Order these decimals from greatest to least:

\(2.08, 2.8, 2.18, 2.80\)

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

$$2.08, 2.80, 2.18, 2.80$$

Now compare from left to right:

• All have 2 in the ones place.
• Compare the tenths digits: 0, 8, 1, 8

The numbers with 8 tenths are greatest: \(2.8\) and \(2.80\).

These are equal because:

$$2.8 = 2.80$$

Next is \(2.18\), and last is \(2.08\).

So from greatest to least:

$$2.80 = 2.8 > 2.18 > 2.08$$

Helpful tips

• Always compare digits in the same place value.
• For whole numbers, a number with more digits is greater.
• For decimals, line up the decimal points first.
• Adding zeros to the end of a decimal does not change its value.
• When ordering, compare carefully one pair at a time.

Common mistakes to avoid

• Do not compare decimals by the number of digits only. For example, \(0.9\) is greater than \(0.35\), even though 35 looks bigger than 9.
• Do not forget to line up decimal points.
• Do not stop too early. If the first digits are the same, keep comparing the next place value.

Let’s look at one quick check:

Compare \(0.9\) and \(0.35\).

Write \(0.9\) as \(0.90\).

Now compare:

• Tenths: 9 and 3

Since \(9 > 3\), we know:

$$0.90 > 0.35$$

So \(0.9\) is greater than \(0.35\).

Summary

To compare and order numbers, use place value. For whole numbers, compare the number of digits first, then compare digits from left to right. For decimals, line up decimal points, add zeros if needed, and compare each place value carefully. These steps will help you decide which number is greater, less, or equal, and put numbers in the correct order.

Rounding and Benchmark Estimation help us work with numbers quickly and sensibly. Sometimes we do not need the exact number. We may only need a number that is close. That is when rounding and estimating are useful.

For example, if 198 people came to an event, you might say about 200 people came. If a toy costs \(\$19.75\), you might think of it as about \(\$20\). These are both examples of using numbers that are close to the exact amount.

In this lesson, you will learn how to round numbers to a place value and how to use benchmarks to estimate.

What does rounding mean?

Rounding means changing a number to the nearest ten, hundred, thousand, or another place value. The rounded number is easier to use, but it stays close to the original number.

To round correctly, we use place value. We look at the digit in the place we are rounding to, and then we check the digit just to the right of it.

Rounding rule:

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

When we round down, the digit in the rounding place stays the same. When we round up, that digit increases by 1. All digits to the right of the rounding place become 0 for whole numbers.

How to round a whole number

1. Find the place you are rounding to.
2. Look at the digit just to the right.
3. Use the rounding rule.
4. Change all digits to the right into 0.

Worked Example 1: Round 347 to the nearest ten.

The tens digit is 4. The digit to the right is the ones digit, which is 7.

Since 7 is 5 or more, we round up. The tens digit 4 becomes 5, and the ones digit becomes 0.

$$347 \approx 350$$

Worked Example 2: Round 347 to the nearest hundred.

The hundreds digit is 3. The digit to the right is the tens digit, which is 4.

Since 4 is less than 5, we round down. The hundreds digit stays 3, and the tens and ones digits become 0.

$$347 \approx 300$$

Notice: The same number can round to different answers depending on the place value.

• Nearest ten: \(347 \approx 350\)
• Nearest hundred: \(347 \approx 300\)

Rounding larger numbers

The same steps work for bigger numbers. You still look at the digit in the place you want, then check the digit just to the right.

Worked Example 3: Round 6,782 to the nearest thousand.

The thousands digit is 6. The digit to the right is the hundreds digit, which is 7.

Since 7 is 5 or more, we round up. The 6 becomes 7, and the rest of the digits become 0.

$$6{,}782 \approx 7{,}000$$

If the number had been 6,212, we would look at the hundreds digit, which is 2. Since 2 is less than 5, it would round down to \(6{,}000\).

Rounding decimals

We can also round decimals. The steps are the same, but instead of changing digits to the right into 0, we stop at the place we are rounding to.

Common decimal places are:

• tenths: first place after the decimal point
• hundredths: second place after the decimal point

Worked Example 4: Round \(4.67\) to the nearest tenth.

The tenths digit is 6. The digit to the right is the hundredths digit, which is 7.

Since 7 is 5 or more, we round up. The 6 tenths becomes 7 tenths.

$$4.67 \approx 4.7$$

If we rounded \(4.67\) to the nearest whole number, we would look at the tenths digit, which is 6. Since 6 is 5 or more, \(4.67\) rounds to \(5\).

What is benchmark estimation?

A benchmark is a familiar number that helps us estimate. Benchmarks are usually easy-to-use numbers such as:

• 0, 5, 10
• 25, 50, 75, 100
• multiples of 10 or 100
• friendly decimals like \(0.5\), \(1.0\), or \(2.5\)

We use benchmarks to decide what a number is close to. This helps us estimate quickly without finding an exact answer.

For example:

• \(48\) is close to \(50\)
• \(198\) is close to \(200\)
• \(74\) is close to \(75\)
• \(0.48\) is close to \(0.5\)

Using benchmarks to estimate sums

When adding, we can round each addend to a nearby benchmark first.

Example: Estimate \(198 + 301\).

Round \(198\) to \(200\). Round \(301\) to \(300\).

Then add the benchmark numbers:

$$200 + 300 = 500$$

So, \(198 + 301\) is about \(500\).

Using benchmarks to estimate differences

When subtracting, we can round both numbers to nearby benchmarks.

Example: Estimate \(612 - 289\).

Round \(612\) to \(600\). Round \(289\) to \(300\).

Then subtract:

$$600 - 300 = 300$$

So, \(612 - 289\) is about \(300\).

Using benchmarks with decimals

Benchmarks also help with decimal numbers.

Example: Estimate \(2.9 + 4.1\).

\(2.9\) is close to \(3\), and \(4.1\) is close to \(4\).

$$3 + 4 = 7$$

So, \(2.9 + 4.1\) is about \(7\).

Why estimation matters

• It helps you check if an exact answer makes sense.
• It helps you solve problems faster.
• It helps in real life, like shopping, measuring, and counting.

If you calculate \(198 + 301 = 499\), your estimate of \(500\) shows that 499 makes sense because it is very close.

Helpful tips for rounding

• Always find the place value first.
• Only look at the digit directly to the right.
• Remember: \(5\) means round up.
• For whole numbers, digits to the right become 0.
• For decimals, stop at the place you rounded to.

Helpful tips for benchmark estimation

• Choose numbers that are easy to work with.
• Pick benchmarks that are close to the original numbers.
• An estimate does not need to be exact. It should be reasonable.

Quick practice ideas

• Round \(563\) to the nearest ten and nearest hundred.
• Round \(8,249\) to the nearest thousand.
• Round \(3.26\) to the nearest tenth.
• Estimate \(49 + 52\) using benchmarks.
• Estimate \(201 - 98\) using benchmarks.

Summary

Rounding means finding the nearest number at a given place value. To round, look at the digit to the right: if it is 5 or more, round up; if it is 4 or less, round down.

Benchmark estimation means using friendly numbers, like 10, 50, 100, or 0.5, to find an answer that is close. Rounding and benchmark estimation help you work quickly and check whether an answer is reasonable.

Zeros as Placeholders

In our base-ten number system, the place of a digit tells its value. A digit can mean different amounts depending on where it is in the number.

For example, in the number \(345\), the \(3\) means \(300\), the \(4\) means \(40\), and the \(5\) means \(5\).

But what happens when there is no digit in a place? That is where zero becomes very important. A zero can act as a placeholder. This means it holds a place so the other digits stay in the correct positions.

If zero did not hold that place, the value of the whole number would change.

Why placeholders matter

Look at these two numbers:

$$15 \qquad 105$$

In \(15\), the \(1\) is in the tens place, so it means \(10\).

In \(105\), the \(1\) is in the hundreds place, so it means \(100\). The zero shows that there are 0 tens. It keeps the \(1\) and the \(5\) in the right places.

So:

• \(15 = 1\) ten and \(5\) ones
• \(105 = 1\) hundred, \(0\) tens, and \(5\) ones

Even though zero means “none” in that place, it is still important because it protects the value of the other digits.

Zero as a placeholder in whole numbers

Each place in a whole number is 10 times the value of the place to its right.

• ones
• tens
• hundreds
• thousands
• ten thousands

Zero can appear in any of these places to show that there are none of that unit.

For example:

• \(302\) means \(3\) hundreds, \(0\) tens, and \(2\) ones
• \(4{,}008\) means \(4\) thousands, \(0\) hundreds, \(0\) tens, and \(8\) ones
• \(50{,}070\) means \(5\) ten-thousands, \(0\) thousands, \(0\) hundreds, \(7\) tens, and \(0\) ones

Without the zeros, these numbers would be completely different.

Compare:

• \(302\) is not the same as \(32\)
• \(4{,}008\) is not the same as \(408\)
• \(50{,}070\) is not the same as \(5{,}070\)

Zero as a placeholder in decimals

Zero is also important in decimal numbers. In decimals, the places to the right of the decimal point are:

• tenths
• hundredths
• thousandths

A zero can hold a decimal place just like it holds a whole-number place.

For example, in \(3.05\):

• the \(3\) is in the ones place
• the \(0\) is in the tenths place
• the \(5\) is in the hundredths place

This means \(3.05\) is 3 ones, 0 tenths, and 5 hundredths.

If the zero were missing, the number would be \(3.5\), which means 3 ones and 5 tenths. That is a much larger number than \(3.05\).

Compare them:

$$3.05 < 3.5$$

The zero keeps the \(5\) in the hundredths place instead of the tenths place.

Zeros at the end of decimals

Sometimes you may see a zero at the end of a decimal, like \(2.50\).

This means:

• \(2\) ones
• \(5\) tenths
• \(0\) hundredths

The value of \(2.50\) is the same as \(2.5\). The ending zero does not change the value, but it can help show place value clearly.

So:

$$2.5 = 2.50$$

But be careful: a zero inside a decimal number can change the value by holding a place.

For example:

$$4.06 \neq 4.6$$

In \(4.06\), the \(6\) is in the hundredths place. In \(4.6\), the \(6\) is in the tenths place.

How to read numbers with zeros

When you see a zero in a number, ask yourself:

1. What place is the zero in?
2. What digit is it helping keep in the correct place?
3. How would the number change if the zero were removed?

This helps you see why the zero matters.

Worked Example 1: Whole number with one zero

What does \(507\) mean?

Step 1: Name each place.

• \(5\) is in the hundreds place
• \(0\) is in the tens place
• \(7\) is in the ones place

Step 2: Write its value.

$$507 = 500 + 0 + 7$$

So \(507\) means 5 hundreds, 0 tens, and 7 ones.

The zero is a placeholder. It shows that there are no tens.

Worked Example 2: Larger whole number with more than one zero

What does \(6{,}040\) mean?

Step 1: Name each place.

• \(6\) is in the thousands place
• \(0\) is in the hundreds place
• \(4\) is in the tens place
• \(0\) is in the ones place

Step 2: Write its value.

$$6{,}040 = 6{,}000 + 0 + 40 + 0$$

So \(6{,}040\) means 6 thousands, 0 hundreds, 4 tens, and 0 ones.

The zeros keep the \(6\) in the thousands place and the \(4\) in the tens place.

Worked Example 3: Decimal with a zero inside the number

What does \(8.03\) mean?

Step 1: Name each place.

• \(8\) is in the ones place
• \(0\) is in the tenths place
• \(3\) is in the hundredths place

Step 2: Write its value.

$$8.03 = 8 + 0.0 + 0.03$$

So \(8.03\) means 8 ones, 0 tenths, and 3 hundredths.

If the zero were removed, the number would be \(8.3\), which is very different. In \(8.3\), the \(3\) is in the tenths place.

Worked Example 4: Comparing decimals

Which is greater: \(5.07\) or \(5.7\)?

Step 1: Line up the place values.

$$5.07 \qquad 5.70$$

We can write \(5.7\) as \(5.70\) to show the place values clearly.

Step 2: Compare by place.

• Ones: both have \(5\)
• Tenths: \(0\) in \(5.07\), but \(7\) in \(5.70\)

Since \(7\) tenths is greater than \(0\) tenths, \(5.70\) is greater.

So:

$$5.07 < 5.7$$

The zero in \(5.07\) holds the tenths place, so the \(7\) stays in the hundredths place.

Common mistakes to avoid

• Do not ignore zeros in the middle of a number. They often change the value.
• Do not think \(2.08\) and \(2.8\) are the same. In \(2.08\), the \(8\) is in the hundredths place.
• Remember that zeros at the end of a decimal usually do not change the value, like \(4.5 = 4.50\).
• Remember that zeros in whole numbers help keep digits in the correct places, like \(700\) and \(70\).

Quick check ideas

Try asking yourself these questions when you read a number:

• How many hundreds, tens, or ones are there?
• How many tenths or hundredths are there?
• Is the zero showing “none” in a place?
• What digit would move if the zero were taken away?

Summary

Zero is more than “nothing.” In place value, zero can be a placeholder that keeps other digits in the correct positions.

In whole numbers, zero shows that a place has no value, such as \(0\) tens or \(0\) hundreds. In decimals, zero can hold the tenths or hundredths place so digits do not shift.

When you understand zeros as placeholders, you can read, write, compare, and understand numbers much more accurately.

Magnitude and the Number Line

Numbers can tell us how much, how far, or where something is. The size of a number is called its magnitude. A number with greater magnitude is farther from zero on a number line.

A number line helps us see numbers in order. It can be horizontal, going left to right, or vertical, going up and down. Number lines help us compare numbers, place numbers correctly, and understand distance between numbers.

In this lesson, you will learn how to:

• read horizontal and vertical number lines,
• plot whole numbers and decimals,
• compare numbers by their position, and
• understand magnitude as distance from zero.

1. Reading a Number Line

A number line has points placed in a straight line. The numbers increase in one direction and decrease in the other direction.

• On a horizontal number line, numbers get larger as you move right.
• On a vertical number line, numbers get larger as you move up.

Here is a simple horizontal number line:

$$0 \quad 1 \quad 2 \quad 3 \quad 4 \quad 5$$

Since 5 is to the right of 3, we know that \(5 > 3\).

Here is a simple vertical number line:

$$5$$

$$4$$

$$3$$

$$2$$

$$1$$

$$0$$

Since 4 is above 2, we know that \(4 > 2\).

2. Equal Spaces Matter

On a number line, the spaces between marks must be equal. If each step is 1, then the line might show 0, 1, 2, 3, 4. If each step is 0.1, then the line might show 0.0, 0.1, 0.2, 0.3, 0.4.

Before plotting a number, always ask:

• What number is at the start?
• What number is at the end?
• How much does each step increase by?

3. Plotting Whole Numbers

To plot a number means to mark its place on the number line.

If the number line shows whole numbers from 0 to 10, then each whole number goes on its matching mark. For example, 7 is placed at the seventh step after 0.

Whole numbers farther to the right or higher up have greater value.

Worked Example 1: Plot a whole number

Plot \(6\) on a horizontal number line from 0 to 8.

1. Find 0 and 8 on the line.
2. Count the equal steps: 1, 2, 3, 4, 5, 6.
3. Place a point at 6.

So, the point for \(6\) is six equal spaces to the right of 0.

4. Plotting Decimals

Decimals can also be shown on a number line. A decimal is part of a whole.

For example:

• \(0.5\) means one-half, so it is halfway between 0 and 1.
• \(1.2\) means 1 whole and 2 tenths, so it is a little to the right of 1.
• \(3.7\) means 3 wholes and 7 tenths, so it is between 3 and 4.

If each interval is divided into 10 equal parts, each small step is one tenth, or \(0.1\).

Worked Example 2: Plot a decimal

Plot \(2.4\) on a number line marked by tenths.

1. Find 2 and 3 on the number line.
2. Since \(2.4\) is between 2 and 3, look in that section.
3. Count 4 tenths after 2: \(2.1, 2.2, 2.3, 2.4\).
4. Place the point at the fourth small mark after 2.

So, \(2.4\) is between 2 and 3, closer to 2 than to 3.

5. Comparing Numbers on a Number Line

A number line makes comparing easy.

• On a horizontal line, the number farther right is greater.
• On a vertical line, the number farther up is greater.

Examples:

• \(8 > 5\) because 8 is to the right of 5.
• \(1.9 < 2.3\) because 1.9 is to the left of 2.3.
• \(4.5 > 4.2\) because 4.5 is farther right than 4.2.

6. Understanding Magnitude

Magnitude means the size of a number. On a number line, we can think about magnitude by looking at how far the number is from 0.

For whole numbers and decimals greater than or equal to 0:

• The farther a number is from 0, the greater its magnitude.
• A number close to 0 has smaller magnitude.

Examples:

• \(9\) has greater magnitude than \(4\) because 9 is farther from 0.
• \(3.6\) has greater magnitude than \(3.1\).
• \(0.8\) has smaller magnitude than \(2.2\).

You can also think of magnitude as distance from zero. If a point is farther from zero on the number line, it has greater magnitude.

Worked Example 3: Compare magnitudes

Which number has greater magnitude: \(1.7\) or \(1.2\)?

1. Both numbers are between 1 and 2.
2. On a number line, \(1.7\) is to the right of \(1.2\).
3. That means \(1.7\) is farther from 0.

So, \(1.7\) has greater magnitude.

7. Finding the Distance Between Numbers

The distance between two numbers on a number line is the difference between them.

For example, the distance from 2 to 5 is:

$$5 - 2 = 3$$

The distance from 1.1 to 1.6 is:

$$1.6 - 1.1 = 0.5$$

This tells us how far apart the numbers are on the number line.

Worked Example 4: Find a decimal distance

What is the distance between \(3.2\) and \(4.0\)?

1. Write a subtraction sentence.
2. $$4.0 - 3.2 = 0.8$$
3. The numbers are 0.8 units apart.

So, the distance between \(3.2\) and \(4.0\) is \(0.8\).

8. Tips for Success

• Always check what each mark on the number line stands for.
• Count equal spaces carefully.
• For decimals, decide which whole numbers the decimal is between first.
• Remember: right means greater on a horizontal line, and up means greater on a vertical line.
• A number farther from 0 has greater magnitude.

Quick Check

• Which is greater, \(4.8\) or \(4.3\)?
• Would \(2.5\) be placed closer to 2 or to 3?
• Which has greater magnitude, \(0.9\) or \(2.1\)?
• What is the distance between \(5\) and \(9\)?

Answers:

• \(4.8\)
• Exactly halfway between 2 and 3
• \(2.1\)
• \(9 - 5 = 4\)

Summary

A number line shows numbers in order using equal spaces. On a horizontal line, numbers increase to the right. On a vertical line, numbers increase upward.

You can plot whole numbers and decimals by finding the correct interval and counting equal parts. Magnitude means the size of a number, and on a number line it can be understood as distance from 0. The farther a number is from 0, the greater its magnitude.