Represent and compare whole numbers up to one billion using powers of ten

Ontario Grade 7 Mathematics

Number

Represent and compare numbers up to one billion

A whole number can describe a count, such as the number of people at an event. The position of each digit tells how much that digit is worth. Powers of ten provide a compact way to show these place values. In this lesson, you will connect the digits in a number to their values, then use that understanding to compare numbers up to one billion.

What you will learn

Explain how a digit’s place tells its value.
Represent a whole number as a sum of digits multiplied by powers of ten.
Compare whole numbers up to one billion using place value and powers of ten.

1. Build place value from groups of ten

Start with a familiar idea: ten ones make one ten, and ten tens make one hundred. Each move one place to the left makes a digit’s value ten times as large. For example, the digit 4 means 4 ones in 4, but 40 in 40 and 400 in 400.

A place-value chart lines up digits so you can see their places. From right to left, the places are ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, ten millions, hundred millions, and billions. A comma can help separate groups of three digits. In 37,205,418, the groups are 37 million, 205 thousand, and 418.

A power of ten is a number made by multiplying ten by itself a certain number of times. The small raised number is called an exponent. It tells how many factors of ten are used. For this lesson, the key powers are

10^0=1

,

10^1=10

,

10^2=100

, and so on. The ones place matches

10^0

, the tens place matches

10^1

, and each place to the left uses the next power.

This gives a useful rule: a digit’s value equals the digit multiplied by the power of ten for its place. A zero digit contributes zero, but it still holds its place. For example, the zero in 4,052 shows that there are no hundreds. Without it, 452 would have a different value.

\text{digit value}=\text{digit}\times 10^{\text{place exponent}}

Moving one place left makes a place value ten times as large.
A digit is worth its face value multiplied by the power of ten for its place.
Zeros hold places even though they contribute no value.

2. Represent numbers with powers of ten

To represent a number using powers of ten, write each digit multiplied by the power for its place. Then add the parts. This is called an expanded form using powers of ten. It shows how the number is built from its place values.

For instance, in 6,304, the 6 is in the thousands place, the 3 is in the hundreds place, the 0 is in the tens place, and the 4 is in the ones place. The powers for those places are

10^3

,

10^2

,

10^1

, and

10^0

. The zero term can be shown or left out because adding zero does not change the total.

You can also work in the other direction. If a number is written as a sum of digit-times-power parts, calculate each part and add them. Check that the powers move down by one as you move from left to right through consecutive places. A missing place must still be accounted for by a zero digit.

The largest whole number named in this lesson is one billion. It is a 1 followed by nine zeros, and it is written as 1,000,000,000. This is also

10^9

. Numbers in this lesson do not go beyond one billion.

1,000,000,000=10^9

Match each digit with the power for its position.
Add the place-value parts to recover the original number.
One billion is $10^9$ .

3. Compare whole numbers by place

To compare two whole numbers, first compare how many digits they have, not counting commas. If one number has more digits, it is greater. For example, any seven-digit whole number is greater than any six-digit whole number, because its leftmost digit is in a higher place.

If the numbers have the same number of digits, compare from the left. Find the first place where the digits differ. The number with the larger digit in that place is greater. You do not need to compare the places to its right once you have found that first difference.

Powers of ten help explain why this method works. A digit in a higher place contributes a larger amount than all the places to its right can change when the leading digits first differ. You can also compare expanded forms by checking their highest place with different digits.

Use the symbols

>

for greater than,

<

for less than, and

=

for equal to. For example,

540,000>504,000

because both numbers have six digits and the first different digit is in the ten-thousands place: 4 is greater than 0.

a>b \Longleftrightarrow a is greater than b

A number with more digits is greater than a number with fewer digits.
For equal-length numbers, compare from the left and stop at the first difference.
Use $>$ , $<$ , or $=$ to record the comparison.

4. Guided practice and self-check

When you practise, first identify what the question asks: a number representation or a comparison. For a representation, mark each digit’s place before writing powers. For a comparison, check digit counts, then compare from the left if needed.

Try these without looking at the examples first. Write 82,006 in expanded form using powers of ten. Then decide which is greater: 709,300 or 710,003. Finally, explain what the zero digits tell you in each number.

Check your work by evaluating the expanded form and returning to the original number. For a comparison, point to the first place where the digits differ. A clear explanation names that place and tells which digit is larger.

Mark the places before writing powers.
Check an expanded form by adding its terms.
Explain a comparison using the first different place.

Place names and matching powers of ten

Place	Power of ten	Value of one digit
Ones	$10^0$	1
Tens	$10^1$	10
Hundreds	$10^2$	100
Thousands	$10^3$	1,000
Ten thousands	$10^4$	10,000
Hundred thousands	$10^5$	100,000
Millions	$10^6$	1,000,000
Ten millions	$10^7$	10,000,000

Worked example

Write a number using powers of ten

Write 48,207 as an expanded sum using powers of ten.

Identify each place
Read the digits from left to right. The 4 is in the ten-thousands place, the 8 in the thousands place, the 2 in the hundreds place, the 0 in the tens place, and the 7 in the ones place.
Match places to powers
The places use powers from $10^4$ down to $10^0$ . Multiply each digit by its matching power. The zero term may be included to show the tens place clearly.
$4\times10^4+8\times10^3+2\times10^2+0\times10^1+7\times10^0$
Check the total
The parts are 40,000, 8,000, 200, 0, and 7. Their sum returns the starting number, so the representation is correct.
$40,000+8,000+200+0+7=48,207$

Answer:

48,207=4\times10^4+8\times10^3+2\times10^2+0\times10^1+7\times10^0

Check: The place-value parts add to 48,207.

Worked example

Compare numbers with the same number of digits

Compare 6,405,120 and 6,450,012. Use place value to explain your choice.

Compare digit counts
Both numbers have seven digits, so digit count does not decide which is greater. Compare from the left.
Find the first difference
The millions digits are both 6. The hundred-thousands digits are both 4. At the ten-thousands place, the first number has 0 and the second has 5. Since 5 is greater than 0, the second number is greater.
$0\times10^4<5\times10^4$
Record the comparison
The first difference decides the comparison. The digits to the right cannot make the first number catch up.
$6,405,120<6,450,012$

Answer: 6,450,012 is greater.

Check: Both have seven digits, and the second has the larger digit at the first place where they differ.

Worked example

Compare a number with one billion

Which is greater: 980,000,000 or 1,000,000,000? Explain using powers of ten.

Compare digit counts
The first number has nine digits. The second has ten digits. A number with ten digits is greater than a number with nine digits.
Connect one billion to a power
One billion is a 1 in the billions place, which matches $10^9$ . The other number has 9 in the hundred-millions place and no billions digit.
$1,000,000,000=1\times10^9$
State the comparison
The billions place makes one billion greater than 980 million.
$980,000,000<1,000,000,000$

Answer: 1,000,000,000 is greater.

Check: One billion has ten digits, while 980,000,000 has nine.

Common mistakes and how to avoid them

Using the same power of ten for every digit.

Correction: Each position has its own power. Moving one place left increases the exponent by one.

Removing a zero when writing a number.

Correction: A zero marks an empty place. Keep it in the number so the other digits stay in the correct positions.

Comparing only the last digits.

Correction: Compare equal-length numbers from the left. The first place where digits differ decides which number is greater.

Lesson summary

Each place in a whole number matches a power of ten.
Represent a number by multiplying each digit by its place’s power and adding the results.
Compare digit counts first. For equal counts, compare digits from left to right.
One billion is 1,000,000,000, or $10^9$ .

Check your understanding

Question 1

Which expression represents 3,506 using powers of ten?

$3\times10^3+5\times10^2+0\times10^1+6\times10^0$
$3\times10^2+5\times10^3+6\times10^1$
$3\times10^3+5\times10^1+6\times10^0$
correctIndex: 0

Show answer and explanation

3\times10^3+5\times10^2+0\times10^1+6\times10^0

The 3 is in the thousands place, the 5 in the hundreds place, the 0 in the tens place, and the 6 in the ones place.

Question 2

Which number is greater: 8,099,999 or 8,100,000?

8,099,999
8,100,000
They are equal
correctIndex: 1

Show answer and explanation

8,100,000

The millions digits match. At the hundred-thousands place, the first number has 0 and the second has 1, so 8,100,000 is greater.

Question 3

What is

10^6

?

10,000
100,000
1,000,000
correctIndex: 2

Show answer and explanation

1,000,000

10^6

means six factors of ten, which is 1,000,000.

Key terms

Place value: The value a digit has because of its position in a number.
Power of ten: A number formed by multiplying ten by itself a stated number of times.
Exponent: The small raised number that tells how many factors are used in a power.
Expanded form: A way to show a number as the sum of the values of its digits.

Continue through Grade 7 Math

View the complete Grade 7 Math Ontario Grade 7 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B1.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

B1.1 · Represent and compare whole numbers up to one billion using powers of ten

Ontario Grade 7 Mathematics

Represent and compare numbers up to one billion

What you will learn

1. Build place value from groups of ten

2. Represent numbers with powers of ten

3. Compare whole numbers by place

4. Guided practice and self-check

Place names and matching powers of ten

Write a number using powers of ten

Compare numbers with the same number of digits

Compare a number with one billion

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 7 Math

About this lesson and its review