1.3 Place value and decimal representation

Achievable Praxis Core: Math (5733)

1. Number and quantity

Our Praxis Core: Math course is currently in development and is a work-in-progress.

Place value and decimal representation

8 min read

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The number system you use every day is built on two ideas:

Place value (a digit’s value depends on its position)
Base $10$ (also called the decimal system)

In a base $10$ system, each place represents a power of $10$ . Starting from the rightmost digit (the ones place), each place value is $10$ times the place to its right. Thinking in powers of $10$ helps you organize, name, and compare both whole numbers and decimals.

Place value and powers of $10$

Place value tells you what each digit is worth based on where it sits in the number. In base $10$ :

Each step to the left multiplies the place value by $10$ .
Each step to the right divides the place value by $10$ .

That’s why shifting digits left or right changes a number’s size so much.

Whole numbers use positive powers of $10$ (tens, hundreds, thousands). Decimals use negative powers of $10$ (tenths, hundredths, thousandths). Together, these place values let you represent both large and small quantities on the same number line.

Helpful ideas:

Each place value is a power of $10$
Moving left multiplies by $10$ ; moving right divides by $10$
Whole-number places use positive exponents; decimal places use negative exponents
The decimal point separates the whole-number part from the fractional part

Each place in a number corresponds to a power of $10$ . The exponent on $10$ tells how many times $10$ is used as a factor for that place.

$1 0^{0} = 1$
$1 0^{1} = 10$
$1 0^{2} = 100$
$1 0^{3} = 1000$

For decimals, negative exponents represent fractional values:

$1 0^{- 1} = \frac{1}{10} = 0.1$
$1 0^{- 2} = \frac{1}{100} = 0.01$
$1 0^{- 3} = \frac{1}{1000} = 0.001$

Example: Place value with positive powers of $10$ Write $5, 738$ in terms of powers of $10$ .

$5$ is in the thousands place: $5 \times 1 0^{3} = 5000$

$7$ is in the hundreds place: $7 \times 1 0^{2} = 700$

$3$ is in the tens place: $3 \times 1 0^{1} = 30$

$8$ is in the ones place: $8 \times 1 0^{0} = 8$

$> 5, 738 = 5 \times 1 0^{3} + 7 \times 1 0^{2} + 3 \times 1 0^{1} + 8 \times 1 0^{0} = 5000 + 700 + 30 + 8$

Answer: $5, 738 = 5000 + 700 + 30 + 8$

Example: Place value with negative powers of $10$ Write $0.053$ in terms of powers of $10$ .

(spoiler)

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$3$ is in the thousandths place: $3 \times 1 0^{- 3} = 0.003$

$> 0.053 = 5 \times 1 0^{- 2} + 3 \times 1 0^{- 3} = 0.05 + 0.003$

Answer: $0.053 = 0.05 + 0.003$

Understanding the decimal point

The decimal point separates the whole-number part of a number from the fractional part.

Places to the left of the decimal point use positive powers of $10$ (ones, tens, hundreds, thousands).
Places to the right use negative powers of $10$ (tenths, hundredths, thousandths).

Reading a number by place value makes it easier to break it apart and understand its size. This is especially useful when comparing numbers, working with decimals, or writing numbers in expanded form.

Key place values:

Ones: $1 0^{0}$
Tenths: $1 0^{- 1}$
Hundredths: $1 0^{- 2}$
Thousandths: $1 0^{- 3}$

Example: Combining positive and negative powers of $10$ Consider the number $324.056$ .

$3$ is in the hundreds place: $3 \times 1 0^{2} = 300$

$2$ is in the tens place: $2 \times 1 0^{1} = 20$

$4$ is in the ones place: $4 \times 1 0^{0} = 4$

$0$ is in the tenths place: $0 \times 1 0^{- 1} = 0.0$

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$6$ is in the thousandths place: $6 \times 1 0^{- 3} = 0.006$

$324.056 = 300 + 20 + 4 + 0.0 + 0.05 + 0.006$

Answer: $324.056 = 300 + 20 + 4 + 0.05 + 0.006$

Multiplying and dividing by powers of $10$

When you multiply or divide by powers of $10$ , the digits stay the same, but the decimal point shifts.

Multiplying by $10$ , $100$ , or $1000$ moves the decimal point to the right.
Dividing by $10$ , $100$ , or $1000$ moves the decimal point to the left.

This happens because each factor of $10$ changes every place value by one step.

Example: Multiply by $100$ Find the value of $23.4 \times 100$ .

$100 = 1 0^{2}$ , so we move the decimal point two places to the right

$23.4 \to 2340$

Answer: $23.4 \times 100 = 2340$

Example: Divide by $1000$ Find the value of $2456 \div 1000$ .

(spoiler)

$1000 = 1 0^{3}$ , so we move the decimal point three places to the left

$2456 \to 2.456$

Answer: $2456 \div 1000 = 2.456$

Example: More practice with powers of $10$ Multiply or divide each expression by the given power of $10$ .

$0.75 \times 100$ $0.004 \times 100$ $34, 500 \div 1000$ $5.6 \div 1000$

(spoiler)

$0.75 \times 100 = 75$
$0.004 \times 100 = 0.4$
$34, 500 \div 1000 = 34.5$
$5.6 \div 1000 = 0.0056$

Answer: $75, 0.4, 34.5, 0.0056$

Naming decimal numbers

You name decimals using the same idea as naming whole numbers, but you name the decimal part by its place value. When a number has both a whole-number part and a decimal part, the word “and” is typically used between them.

To choose the correct place-value name, look at the last digit on the right. For example:

If the last digit is in the thousandths place, you use “thousandths.”
If the last digit is in the hundredths place, you use “hundredths.”

Example: Reading decimal numbers Read each decimal in words.

$324.056$ is “three hundred twenty-four and fifty-six thousandths.”

$0.056$ is “fifty-six thousandths.”

$0.5$ is “five tenths.”

$100.25$ is “one hundred and twenty-five hundredths.”

Example: Reading a specific decimal Write $0.375$ in words.

(spoiler)

The last digit is in the thousandths place, so the number is read as “three hundred seventy-five thousandths.”

Answer: “three hundred seventy-five thousandths”

Ordering numbers

To order numbers, compare place values starting from the leftmost digit.

For whole numbers, compare thousands, hundreds, tens, and ones.
For decimals, keep comparing to the right: tenths, hundredths, thousandths, and beyond.

A reliable method for decimals is to line numbers up by the decimal point. If needed, add zeros to the right so each number has the same number of decimal places. Then compare digits place by place until you find a difference.

Example: Ordering decimals Order $0.25$ , $0.56$ , and $0.3$ from least to greatest.

Compare tenths: $0.25$ has $2$ , $0.3$ has $3$ , and $0.56$ has $5$

Therefore, $0.25 < 0.3 < 0.56$

Answer: $0.25 < 0.3 < 0.56$

Sidenote

Comparing decimal numbers

When comparing decimal numbers, start with the largest place value and move to the right. If two numbers match in one place, compare the next digit to the right until the tie is broken.

Example: compare the following numbers

$0.562$ $0.578$ $0.489$ $0.487$

$0.562$ and $0.578$ both have $5$ in the tenths place
Compare hundredths: $6$ vs $7$ → $0.578 > 0.562$
Compare $0.489$ and $0.487$ : ties in tenths and hundredths, but $9 > 7$ in the thousandths

Final order: $0.487 < 0.489 < 0.562 < 0.578$

Expanded form

Expanded form breaks a number into a sum of its place values, often written using powers of $10$ . You can do this for both whole numbers and decimals.

Example: Expanded form of a whole number Express $4, 309$ in terms of powers of $10$ .

(spoiler)

$4, 309 = 4 \times 1 0^{3} + 3 \times 1 0^{2} + 0 \times 1 0^{1} + 9 \times 1 0^{0} = 4000 + 300 + 0 + 9$

Answer: $4, 309 = 4000 + 300 + 9$

Example: Expanded form of a decimal Express $0.402$ in terms of powers of $10$ .

(spoiler)

$0.402 = 4 \times 1 0^{- 1} + 0 \times 1 0^{- 2} + 2 \times 1 0^{- 3} = 0.4 + 0 + 0.002$

Answer: $0.402 = 0.4 + 0.002$

Ordering mixed numbers

Example: Ordering mixed numbers Order $2 \frac{3}{4}$ , $2 \frac{2}{3}$ , and $2 \frac{5}{8}$ from least to greatest.

(spoiler)

Convert fractional parts to denominator $24$ :

$2 \frac{3}{4} = 2 \frac{18}{24}$

$2 \frac{2}{3} = 2 \frac{16}{24}$

$2 \frac{5}{8} = 2 \frac{15}{24}$ So

$> 2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Answer: $2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Each digit’s value depends on its position and corresponds to a power of $10$ .
Moving left multiplies a number by $10$ , while moving right divides by $10$ .
Positive exponents represent whole-number place values, and negative exponents represent decimal place values.
Multiplying and dividing by powers of $10$ shifts the decimal point without changing the digits.
Decimals are named based on the place value of the final digit to the right.
Numbers are ordered by comparing place values from left to right, aligning decimal points when needed.
Mixed numbers are ordered by comparing whole-number parts first, then fractional parts using a common denominator.

Place value and decimal representation

The number system you use every day is built on two ideas:

Place value (a digit’s value depends on its position)
Base $10$ (also called the decimal system)

Place value and powers of $10$

Place value tells you what each digit is worth based on where it sits in the number. In base $10$ :

Each step to the left multiplies the place value by $10$ .
Each step to the right divides the place value by $10$ .

That’s why shifting digits left or right changes a number’s size so much.

Helpful ideas:

Each place value is a power of $10$
Moving left multiplies by $10$ ; moving right divides by $10$
Whole-number places use positive exponents; decimal places use negative exponents
The decimal point separates the whole-number part from the fractional part

Each place in a number corresponds to a power of $10$ . The exponent on $10$ tells how many times $10$ is used as a factor for that place.

$1 0^{0} = 1$
$1 0^{1} = 10$
$1 0^{2} = 100$
$1 0^{3} = 1000$

For decimals, negative exponents represent fractional values:

$1 0^{- 1} = \frac{1}{10} = 0.1$
$1 0^{- 2} = \frac{1}{100} = 0.01$
$1 0^{- 3} = \frac{1}{1000} = 0.001$

Example: Place value with positive powers of $10$ Write $5, 738$ in terms of powers of $10$ .

$5$ is in the thousands place: $5 \times 1 0^{3} = 5000$

$7$ is in the hundreds place: $7 \times 1 0^{2} = 700$

$3$ is in the tens place: $3 \times 1 0^{1} = 30$

$8$ is in the ones place: $8 \times 1 0^{0} = 8$

$> 5, 738 = 5 \times 1 0^{3} + 7 \times 1 0^{2} + 3 \times 1 0^{1} + 8 \times 1 0^{0} = 5000 + 700 + 30 + 8$

Answer: $5, 738 = 5000 + 700 + 30 + 8$

Example: Place value with negative powers of $10$ Write $0.053$ in terms of powers of $10$ .

(spoiler)

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$3$ is in the thousandths place: $3 \times 1 0^{- 3} = 0.003$

$> 0.053 = 5 \times 1 0^{- 2} + 3 \times 1 0^{- 3} = 0.05 + 0.003$

Answer: $0.053 = 0.05 + 0.003$

Understanding the decimal point

The decimal point separates the whole-number part of a number from the fractional part.

Places to the left of the decimal point use positive powers of $10$ (ones, tens, hundreds, thousands).
Places to the right use negative powers of $10$ (tenths, hundredths, thousandths).

Key place values:

Ones: $1 0^{0}$
Tenths: $1 0^{- 1}$
Hundredths: $1 0^{- 2}$
Thousandths: $1 0^{- 3}$

Example: Combining positive and negative powers of $10$ Consider the number $324.056$ .

$3$ is in the hundreds place: $3 \times 1 0^{2} = 300$

$2$ is in the tens place: $2 \times 1 0^{1} = 20$

$4$ is in the ones place: $4 \times 1 0^{0} = 4$

$0$ is in the tenths place: $0 \times 1 0^{- 1} = 0.0$

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$6$ is in the thousandths place: $6 \times 1 0^{- 3} = 0.006$

$324.056 = 300 + 20 + 4 + 0.0 + 0.05 + 0.006$

Answer: $324.056 = 300 + 20 + 4 + 0.05 + 0.006$

Multiplying and dividing by powers of $10$

When you multiply or divide by powers of $10$ , the digits stay the same, but the decimal point shifts.

Multiplying by $10$ , $100$ , or $1000$ moves the decimal point to the right.
Dividing by $10$ , $100$ , or $1000$ moves the decimal point to the left.

This happens because each factor of $10$ changes every place value by one step.

Example: Multiply by $100$ Find the value of $23.4 \times 100$ .

$100 = 1 0^{2}$ , so we move the decimal point two places to the right

$23.4 \to 2340$

Answer: $23.4 \times 100 = 2340$

Example: Divide by $1000$ Find the value of $2456 \div 1000$ .

(spoiler)

$1000 = 1 0^{3}$ , so we move the decimal point three places to the left

$2456 \to 2.456$

Answer: $2456 \div 1000 = 2.456$

Example: More practice with powers of $10$ Multiply or divide each expression by the given power of $10$ .

$0.75 \times 100$ $0.004 \times 100$ $34, 500 \div 1000$ $5.6 \div 1000$

(spoiler)

$0.75 \times 100 = 75$
$0.004 \times 100 = 0.4$
$34, 500 \div 1000 = 34.5$
$5.6 \div 1000 = 0.0056$

Answer: $75, 0.4, 34.5, 0.0056$

Naming decimal numbers

To choose the correct place-value name, look at the last digit on the right. For example:

If the last digit is in the thousandths place, you use “thousandths.”
If the last digit is in the hundredths place, you use “hundredths.”

Example: Reading decimal numbers Read each decimal in words.

$324.056$ is “three hundred twenty-four and fifty-six thousandths.”

$0.056$ is “fifty-six thousandths.”

$0.5$ is “five tenths.”

$100.25$ is “one hundred and twenty-five hundredths.”

Example: Reading a specific decimal Write $0.375$ in words.

(spoiler)

The last digit is in the thousandths place, so the number is read as “three hundred seventy-five thousandths.”

Answer: “three hundred seventy-five thousandths”

Ordering numbers

To order numbers, compare place values starting from the leftmost digit.

For whole numbers, compare thousands, hundreds, tens, and ones.
For decimals, keep comparing to the right: tenths, hundredths, thousandths, and beyond.

Example: Ordering decimals Order $0.25$ , $0.56$ , and $0.3$ from least to greatest.

Compare tenths: $0.25$ has $2$ , $0.3$ has $3$ , and $0.56$ has $5$

Therefore, $0.25 < 0.3 < 0.56$

Answer: $0.25 < 0.3 < 0.56$

Sidenote

Comparing decimal numbers

When comparing decimal numbers, start with the largest place value and move to the right. If two numbers match in one place, compare the next digit to the right until the tie is broken.

Example: compare the following numbers

$0.562$ $0.578$ $0.489$ $0.487$

$0.562$ and $0.578$ both have $5$ in the tenths place
Compare hundredths: $6$ vs $7$ → $0.578 > 0.562$
Compare $0.489$ and $0.487$ : ties in tenths and hundredths, but $9 > 7$ in the thousandths

Final order: $0.487 < 0.489 < 0.562 < 0.578$

Expanded form

Expanded form breaks a number into a sum of its place values, often written using powers of $10$ . You can do this for both whole numbers and decimals.

Example: Expanded form of a whole number Express $4, 309$ in terms of powers of $10$ .

(spoiler)

$4, 309 = 4 \times 1 0^{3} + 3 \times 1 0^{2} + 0 \times 1 0^{1} + 9 \times 1 0^{0} = 4000 + 300 + 0 + 9$

Answer: $4, 309 = 4000 + 300 + 9$

Example: Expanded form of a decimal Express $0.402$ in terms of powers of $10$ .

(spoiler)

$0.402 = 4 \times 1 0^{- 1} + 0 \times 1 0^{- 2} + 2 \times 1 0^{- 3} = 0.4 + 0 + 0.002$

Answer: $0.402 = 0.4 + 0.002$

Ordering mixed numbers

Example: Ordering mixed numbers Order $2 \frac{3}{4}$ , $2 \frac{2}{3}$ , and $2 \frac{5}{8}$ from least to greatest.

(spoiler)

Convert fractional parts to denominator $24$ :

$2 \frac{3}{4} = 2 \frac{18}{24}$

$2 \frac{2}{3} = 2 \frac{16}{24}$

$2 \frac{5}{8} = 2 \frac{15}{24}$ So

$> 2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Answer: $2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Achievable Praxis Core: Math (5733)

1. Number and quantity

Our Praxis Core: Math course is currently in development and is a work-in-progress.

Place value and decimal representation

8 min read

Font

Discuss

Feedback

The number system you use every day is built on two ideas:

Place value (a digit’s value depends on its position)
Base $10$ (also called the decimal system)

Place value and powers of $10$

Place value tells you what each digit is worth based on where it sits in the number. In base $10$ :

Each step to the left multiplies the place value by $10$ .
Each step to the right divides the place value by $10$ .

That’s why shifting digits left or right changes a number’s size so much.

Helpful ideas:

Each place value is a power of $10$
Moving left multiplies by $10$ ; moving right divides by $10$
Whole-number places use positive exponents; decimal places use negative exponents
The decimal point separates the whole-number part from the fractional part

Each place in a number corresponds to a power of $10$ . The exponent on $10$ tells how many times $10$ is used as a factor for that place.

$1 0^{0} = 1$
$1 0^{1} = 10$
$1 0^{2} = 100$
$1 0^{3} = 1000$

For decimals, negative exponents represent fractional values:

$1 0^{- 1} = \frac{1}{10} = 0.1$
$1 0^{- 2} = \frac{1}{100} = 0.01$
$1 0^{- 3} = \frac{1}{1000} = 0.001$

Example: Place value with positive powers of $10$ Write $5, 738$ in terms of powers of $10$ .

$5$ is in the thousands place: $5 \times 1 0^{3} = 5000$

$7$ is in the hundreds place: $7 \times 1 0^{2} = 700$

$3$ is in the tens place: $3 \times 1 0^{1} = 30$

$8$ is in the ones place: $8 \times 1 0^{0} = 8$

$> 5, 738 = 5 \times 1 0^{3} + 7 \times 1 0^{2} + 3 \times 1 0^{1} + 8 \times 1 0^{0} = 5000 + 700 + 30 + 8$

Answer: $5, 738 = 5000 + 700 + 30 + 8$

Example: Place value with negative powers of $10$ Write $0.053$ in terms of powers of $10$ .

(spoiler)

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$3$ is in the thousandths place: $3 \times 1 0^{- 3} = 0.003$

$> 0.053 = 5 \times 1 0^{- 2} + 3 \times 1 0^{- 3} = 0.05 + 0.003$

Answer: $0.053 = 0.05 + 0.003$

Understanding the decimal point

The decimal point separates the whole-number part of a number from the fractional part.

Places to the left of the decimal point use positive powers of $10$ (ones, tens, hundreds, thousands).
Places to the right use negative powers of $10$ (tenths, hundredths, thousandths).

Key place values:

Ones: $1 0^{0}$
Tenths: $1 0^{- 1}$
Hundredths: $1 0^{- 2}$
Thousandths: $1 0^{- 3}$

Example: Combining positive and negative powers of $10$ Consider the number $324.056$ .

$3$ is in the hundreds place: $3 \times 1 0^{2} = 300$

$2$ is in the tens place: $2 \times 1 0^{1} = 20$

$4$ is in the ones place: $4 \times 1 0^{0} = 4$

$0$ is in the tenths place: $0 \times 1 0^{- 1} = 0.0$

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$6$ is in the thousandths place: $6 \times 1 0^{- 3} = 0.006$

$324.056 = 300 + 20 + 4 + 0.0 + 0.05 + 0.006$

Answer: $324.056 = 300 + 20 + 4 + 0.05 + 0.006$

Multiplying and dividing by powers of $10$

When you multiply or divide by powers of $10$ , the digits stay the same, but the decimal point shifts.

Multiplying by $10$ , $100$ , or $1000$ moves the decimal point to the right.
Dividing by $10$ , $100$ , or $1000$ moves the decimal point to the left.

This happens because each factor of $10$ changes every place value by one step.

Example: Multiply by $100$ Find the value of $23.4 \times 100$ .

$100 = 1 0^{2}$ , so we move the decimal point two places to the right

$23.4 \to 2340$

Answer: $23.4 \times 100 = 2340$

Example: Divide by $1000$ Find the value of $2456 \div 1000$ .

(spoiler)

$1000 = 1 0^{3}$ , so we move the decimal point three places to the left

$2456 \to 2.456$

Answer: $2456 \div 1000 = 2.456$

Example: More practice with powers of $10$ Multiply or divide each expression by the given power of $10$ .

$0.75 \times 100$ $0.004 \times 100$ $34, 500 \div 1000$ $5.6 \div 1000$

(spoiler)

$0.75 \times 100 = 75$
$0.004 \times 100 = 0.4$
$34, 500 \div 1000 = 34.5$
$5.6 \div 1000 = 0.0056$

Answer: $75, 0.4, 34.5, 0.0056$

Naming decimal numbers

To choose the correct place-value name, look at the last digit on the right. For example:

If the last digit is in the thousandths place, you use “thousandths.”
If the last digit is in the hundredths place, you use “hundredths.”

Example: Reading decimal numbers Read each decimal in words.

$324.056$ is “three hundred twenty-four and fifty-six thousandths.”

$0.056$ is “fifty-six thousandths.”

$0.5$ is “five tenths.”

$100.25$ is “one hundred and twenty-five hundredths.”

Example: Reading a specific decimal Write $0.375$ in words.

(spoiler)

The last digit is in the thousandths place, so the number is read as “three hundred seventy-five thousandths.”

Answer: “three hundred seventy-five thousandths”

Ordering numbers

To order numbers, compare place values starting from the leftmost digit.

For whole numbers, compare thousands, hundreds, tens, and ones.
For decimals, keep comparing to the right: tenths, hundredths, thousandths, and beyond.

Example: Ordering decimals Order $0.25$ , $0.56$ , and $0.3$ from least to greatest.

Compare tenths: $0.25$ has $2$ , $0.3$ has $3$ , and $0.56$ has $5$

Therefore, $0.25 < 0.3 < 0.56$

Answer: $0.25 < 0.3 < 0.56$

Sidenote

Comparing decimal numbers

When comparing decimal numbers, start with the largest place value and move to the right. If two numbers match in one place, compare the next digit to the right until the tie is broken.

Example: compare the following numbers

$0.562$ $0.578$ $0.489$ $0.487$

$0.562$ and $0.578$ both have $5$ in the tenths place
Compare hundredths: $6$ vs $7$ → $0.578 > 0.562$
Compare $0.489$ and $0.487$ : ties in tenths and hundredths, but $9 > 7$ in the thousandths

Final order: $0.487 < 0.489 < 0.562 < 0.578$

Expanded form

Expanded form breaks a number into a sum of its place values, often written using powers of $10$ . You can do this for both whole numbers and decimals.

Example: Expanded form of a whole number Express $4, 309$ in terms of powers of $10$ .

(spoiler)

$4, 309 = 4 \times 1 0^{3} + 3 \times 1 0^{2} + 0 \times 1 0^{1} + 9 \times 1 0^{0} = 4000 + 300 + 0 + 9$

Answer: $4, 309 = 4000 + 300 + 9$

Example: Expanded form of a decimal Express $0.402$ in terms of powers of $10$ .

(spoiler)

$0.402 = 4 \times 1 0^{- 1} + 0 \times 1 0^{- 2} + 2 \times 1 0^{- 3} = 0.4 + 0 + 0.002$

Answer: $0.402 = 0.4 + 0.002$

Ordering mixed numbers

Example: Ordering mixed numbers Order $2 \frac{3}{4}$ , $2 \frac{2}{3}$ , and $2 \frac{5}{8}$ from least to greatest.

(spoiler)

Convert fractional parts to denominator $24$ :

$2 \frac{3}{4} = 2 \frac{18}{24}$

$2 \frac{2}{3} = 2 \frac{16}{24}$

$2 \frac{5}{8} = 2 \frac{15}{24}$ So

$> 2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Answer: $2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Each digit’s value depends on its position and corresponds to a power of $10$ .
Moving left multiplies a number by $10$ , while moving right divides by $10$ .
Positive exponents represent whole-number place values, and negative exponents represent decimal place values.
Multiplying and dividing by powers of $10$ shifts the decimal point without changing the digits.
Decimals are named based on the place value of the final digit to the right.
Numbers are ordered by comparing place values from left to right, aligning decimal points when needed.
Mixed numbers are ordered by comparing whole-number parts first, then fractional parts using a common denominator.

Place value and decimal representation

The number system you use every day is built on two ideas:

Place value (a digit’s value depends on its position)
Base $10$ (also called the decimal system)

Place value and powers of $10$

Place value tells you what each digit is worth based on where it sits in the number. In base $10$ :

Each step to the left multiplies the place value by $10$ .
Each step to the right divides the place value by $10$ .

That’s why shifting digits left or right changes a number’s size so much.

Helpful ideas:

Each place value is a power of $10$
Moving left multiplies by $10$ ; moving right divides by $10$
Whole-number places use positive exponents; decimal places use negative exponents
The decimal point separates the whole-number part from the fractional part

Each place in a number corresponds to a power of $10$ . The exponent on $10$ tells how many times $10$ is used as a factor for that place.

$1 0^{0} = 1$
$1 0^{1} = 10$
$1 0^{2} = 100$
$1 0^{3} = 1000$

For decimals, negative exponents represent fractional values:

$1 0^{- 1} = \frac{1}{10} = 0.1$
$1 0^{- 2} = \frac{1}{100} = 0.01$
$1 0^{- 3} = \frac{1}{1000} = 0.001$

Example: Place value with positive powers of $10$ Write $5, 738$ in terms of powers of $10$ .

$5$ is in the thousands place: $5 \times 1 0^{3} = 5000$

$7$ is in the hundreds place: $7 \times 1 0^{2} = 700$

$3$ is in the tens place: $3 \times 1 0^{1} = 30$

$8$ is in the ones place: $8 \times 1 0^{0} = 8$

$> 5, 738 = 5 \times 1 0^{3} + 7 \times 1 0^{2} + 3 \times 1 0^{1} + 8 \times 1 0^{0} = 5000 + 700 + 30 + 8$

Answer: $5, 738 = 5000 + 700 + 30 + 8$

Example: Place value with negative powers of $10$ Write $0.053$ in terms of powers of $10$ .

(spoiler)

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$3$ is in the thousandths place: $3 \times 1 0^{- 3} = 0.003$

$> 0.053 = 5 \times 1 0^{- 2} + 3 \times 1 0^{- 3} = 0.05 + 0.003$

Answer: $0.053 = 0.05 + 0.003$

Understanding the decimal point

The decimal point separates the whole-number part of a number from the fractional part.

Places to the left of the decimal point use positive powers of $10$ (ones, tens, hundreds, thousands).
Places to the right use negative powers of $10$ (tenths, hundredths, thousandths).

Key place values:

Ones: $1 0^{0}$
Tenths: $1 0^{- 1}$
Hundredths: $1 0^{- 2}$
Thousandths: $1 0^{- 3}$

Example: Combining positive and negative powers of $10$ Consider the number $324.056$ .

$3$ is in the hundreds place: $3 \times 1 0^{2} = 300$

$2$ is in the tens place: $2 \times 1 0^{1} = 20$

$4$ is in the ones place: $4 \times 1 0^{0} = 4$

$0$ is in the tenths place: $0 \times 1 0^{- 1} = 0.0$

$5$ is in the hundredths place: $5 \times 1 0^{- 2} = 0.05$

$6$ is in the thousandths place: $6 \times 1 0^{- 3} = 0.006$

$324.056 = 300 + 20 + 4 + 0.0 + 0.05 + 0.006$

Answer: $324.056 = 300 + 20 + 4 + 0.05 + 0.006$

Multiplying and dividing by powers of $10$

When you multiply or divide by powers of $10$ , the digits stay the same, but the decimal point shifts.

Multiplying by $10$ , $100$ , or $1000$ moves the decimal point to the right.
Dividing by $10$ , $100$ , or $1000$ moves the decimal point to the left.

This happens because each factor of $10$ changes every place value by one step.

Example: Multiply by $100$ Find the value of $23.4 \times 100$ .

$100 = 1 0^{2}$ , so we move the decimal point two places to the right

$23.4 \to 2340$

Answer: $23.4 \times 100 = 2340$

Example: Divide by $1000$ Find the value of $2456 \div 1000$ .

(spoiler)

$1000 = 1 0^{3}$ , so we move the decimal point three places to the left

$2456 \to 2.456$

Answer: $2456 \div 1000 = 2.456$

Example: More practice with powers of $10$ Multiply or divide each expression by the given power of $10$ .

$0.75 \times 100$ $0.004 \times 100$ $34, 500 \div 1000$ $5.6 \div 1000$

(spoiler)

$0.75 \times 100 = 75$
$0.004 \times 100 = 0.4$
$34, 500 \div 1000 = 34.5$
$5.6 \div 1000 = 0.0056$

Answer: $75, 0.4, 34.5, 0.0056$

Naming decimal numbers

To choose the correct place-value name, look at the last digit on the right. For example:

If the last digit is in the thousandths place, you use “thousandths.”
If the last digit is in the hundredths place, you use “hundredths.”

Example: Reading decimal numbers Read each decimal in words.

$324.056$ is “three hundred twenty-four and fifty-six thousandths.”

$0.056$ is “fifty-six thousandths.”

$0.5$ is “five tenths.”

$100.25$ is “one hundred and twenty-five hundredths.”

Example: Reading a specific decimal Write $0.375$ in words.

(spoiler)

The last digit is in the thousandths place, so the number is read as “three hundred seventy-five thousandths.”

Answer: “three hundred seventy-five thousandths”

Ordering numbers

To order numbers, compare place values starting from the leftmost digit.

For whole numbers, compare thousands, hundreds, tens, and ones.
For decimals, keep comparing to the right: tenths, hundredths, thousandths, and beyond.

Example: Ordering decimals Order $0.25$ , $0.56$ , and $0.3$ from least to greatest.

Compare tenths: $0.25$ has $2$ , $0.3$ has $3$ , and $0.56$ has $5$

Therefore, $0.25 < 0.3 < 0.56$

Answer: $0.25 < 0.3 < 0.56$

Sidenote

Comparing decimal numbers

When comparing decimal numbers, start with the largest place value and move to the right. If two numbers match in one place, compare the next digit to the right until the tie is broken.

Example: compare the following numbers

$0.562$ $0.578$ $0.489$ $0.487$

$0.562$ and $0.578$ both have $5$ in the tenths place
Compare hundredths: $6$ vs $7$ → $0.578 > 0.562$
Compare $0.489$ and $0.487$ : ties in tenths and hundredths, but $9 > 7$ in the thousandths

Final order: $0.487 < 0.489 < 0.562 < 0.578$

Expanded form

Expanded form breaks a number into a sum of its place values, often written using powers of $10$ . You can do this for both whole numbers and decimals.

Example: Expanded form of a whole number Express $4, 309$ in terms of powers of $10$ .

(spoiler)

$4, 309 = 4 \times 1 0^{3} + 3 \times 1 0^{2} + 0 \times 1 0^{1} + 9 \times 1 0^{0} = 4000 + 300 + 0 + 9$

Answer: $4, 309 = 4000 + 300 + 9$

Example: Expanded form of a decimal Express $0.402$ in terms of powers of $10$ .

(spoiler)

$0.402 = 4 \times 1 0^{- 1} + 0 \times 1 0^{- 2} + 2 \times 1 0^{- 3} = 0.4 + 0 + 0.002$

Answer: $0.402 = 0.4 + 0.002$

Ordering mixed numbers

Example: Ordering mixed numbers Order $2 \frac{3}{4}$ , $2 \frac{2}{3}$ , and $2 \frac{5}{8}$ from least to greatest.

(spoiler)

Convert fractional parts to denominator $24$ :

$2 \frac{3}{4} = 2 \frac{18}{24}$

$2 \frac{2}{3} = 2 \frac{16}{24}$

$2 \frac{5}{8} = 2 \frac{15}{24}$ So

$> 2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Answer: $2 \frac{5}{8} < 2 \frac{2}{3} < 2 \frac{3}{4}$

Place value and decimal representation

Place value and powers of 10

Understanding the decimal point

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Expanded form

Ordering mixed numbers

Place value and decimal representation

Place value and powers of 10

Understanding the decimal point

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Expanded form

Ordering mixed numbers

Place value and decimal representation

Place value and powers of 10

Understanding the decimal point

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Expanded form

Ordering mixed numbers

Place value and decimal representation

Place value and powers of 10

Understanding the decimal point

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Expanded form

Ordering mixed numbers

Place value and powers of $10$

Multiplying and dividing by powers of $10$

Place value and powers of $10$

Multiplying and dividing by powers of $10$

Place value and powers of $10$

Multiplying and dividing by powers of $10$

Place value and powers of $10$

Multiplying and dividing by powers of $10$