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

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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 a 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 dramatically.

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.

The decimal point separates the whole-number part from the fractional part. Places to the left use positive powers of $10$ ; places to the right use negative powers. Each place corresponds to a specific power:

Ones: $1 0^{0} = 1$
Tens: $1 0^{1} = 10$ , Hundreds: $1 0^{2} = 100$ , Thousands: $1 0^{3} = 1000$
Tenths: $1 0^{- 1} = 0.1$ , Hundredths: $1 0^{- 2} = 0.01$ , Thousandths: $1 0^{- 3} = 0.001$

Example: Place value across the decimal point

Consider the number $324.056$ . Each digit sits in a specific place:

$3$ 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: $0 \times 1 0^{- 1} = 0$ (placeholder - contributes nothing to the sum)

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

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

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

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.

Common pitfall: It’s easy to shift the decimal point in the wrong direction. Multiplying makes a number larger, so the decimal moves right; dividing makes it smaller, so the decimal moves left. If you find yourself moving it the opposite way, you’ve swapped multiply and divide.

Example: Multiply and divide by powers of $10$

$23.4 \times 100$ : $100 = 1 0^{2}$ , so move the decimal two places to the right. Since $23.4$ has only one digit after the decimal, append a zero as a placeholder when shifting the second place: $23.4 \to 234. \to 2340$ .

$2456 \div 1000$ : $1000 = 1 0^{3}$ , so move the decimal three places to the left. $2456 \to 2.456$ .

Answer: $23.4 \times 100 = 2340$ and $2456 \div 1000 = 2.456$

Naming decimal numbers

You name decimals the same way you name whole numbers, but you identify the decimal part by its place value. When a number has both a whole-number part and a decimal part, use the word “and” only at the decimal point - not inside the whole-number part. For example, $125$ is read “one hundred twenty-five,” not “one hundred and twenty-five.”

To choose the correct place-value name, look at the last digit on the right. Note that trailing zeros affect the name even when they don’t change the value: $0.5$ is “five tenths,” while $0.50$ is “fifty 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.”

$305.25$ is “three hundred five and twenty-five hundredths.” (The whole-number part, $305$ , is “three hundred five” - no “and” within it. The “and” falls only at the decimal point.)

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 with mixed lengths

Order $0.5$ , $0.45$ , $0.500$ , and $0.4999$ from least to greatest.

First, note that $0.5$ and $0.500$ are equal values (trailing zeros don’t change a decimal’s value), so they occupy the same position.

Pad the remaining distinct values to four decimal places for comparison: $0.5000$ , $0.4500$ , $0.4999$ .

Compare tenths: $0.4500$ and $0.4999$ both have $4$ tenths, while $0.5000$ has $5$ tenths - so both $0.4$ -something values are less than $0.5$ .

Among $0.4500$ and $0.4999$ : hundredths are $5$ vs. $9$ , so $0.4500 < 0.4999$ .

Answer: $0.45 < 0.4999 < 0.5 = 0.500$

Ordering negative decimals

The same place-value strategy applies when comparing negative numbers, but the direction reverses: more negative means further left on the number line, so a number with a larger absolute value is actually smaller. Compare the positive versions first, then flip the order.

Example: Ordering negative decimals

Order $- 0.3$ , $- 0.25$ , and $- 0.56$ from least to greatest.

As positive values: $0.25 < 0.3 < 0.56$ .

Applying negatives reverses the order: $- 0.56 < - 0.3 < - 0.25$ .

Answer: $- 0.56 < - 0.3 < - 0.25$

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 a 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 dramatically.

Ones: $1 0^{0} = 1$
Tens: $1 0^{1} = 10$ , Hundreds: $1 0^{2} = 100$ , Thousands: $1 0^{3} = 1000$
Tenths: $1 0^{- 1} = 0.1$ , Hundredths: $1 0^{- 2} = 0.01$ , Thousandths: $1 0^{- 3} = 0.001$

Example: Place value across the decimal point

Consider the number $324.056$ . Each digit sits in a specific place:

$3$ 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: $0 \times 1 0^{- 1} = 0$ (placeholder - contributes nothing to the sum)

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

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

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

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 and divide by powers of $10$

$23.4 \times 100$ : $100 = 1 0^{2}$ , so move the decimal two places to the right. Since $23.4$ has only one digit after the decimal, append a zero as a placeholder when shifting the second place: $23.4 \to 234. \to 2340$ .

$2456 \div 1000$ : $1000 = 1 0^{3}$ , so move the decimal three places to the left. $2456 \to 2.456$ .

Answer: $23.4 \times 100 = 2340$ and $2456 \div 1000 = 2.456$

Naming decimal numbers

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.”

$305.25$ is “three hundred five and twenty-five hundredths.” (The whole-number part, $305$ , is “three hundred five” - no “and” within it. The “and” falls only at the decimal point.)

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 with mixed lengths

Order $0.5$ , $0.45$ , $0.500$ , and $0.4999$ from least to greatest.

First, note that $0.5$ and $0.500$ are equal values (trailing zeros don’t change a decimal’s value), so they occupy the same position.

Pad the remaining distinct values to four decimal places for comparison: $0.5000$ , $0.4500$ , $0.4999$ .

Compare tenths: $0.4500$ and $0.4999$ both have $4$ tenths, while $0.5000$ has $5$ tenths - so both $0.4$ -something values are less than $0.5$ .

Among $0.4500$ and $0.4999$ : hundredths are $5$ vs. $9$ , so $0.4500 < 0.4999$ .

Answer: $0.45 < 0.4999 < 0.5 = 0.500$

Ordering negative decimals

Example: Ordering negative decimals

Order $- 0.3$ , $- 0.25$ , and $- 0.56$ from least to greatest.

As positive values: $0.25 < 0.3 < 0.56$ .

Applying negatives reverses the order: $- 0.56 < - 0.3 < - 0.25$ .

Answer: $- 0.56 < - 0.3 < - 0.25$

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

5 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 a 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 dramatically.

Ones: $1 0^{0} = 1$
Tens: $1 0^{1} = 10$ , Hundreds: $1 0^{2} = 100$ , Thousands: $1 0^{3} = 1000$
Tenths: $1 0^{- 1} = 0.1$ , Hundredths: $1 0^{- 2} = 0.01$ , Thousandths: $1 0^{- 3} = 0.001$

Example: Place value across the decimal point

Consider the number $324.056$ . Each digit sits in a specific place:

$3$ 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: $0 \times 1 0^{- 1} = 0$ (placeholder - contributes nothing to the sum)

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

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

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

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 and divide by powers of $10$

$23.4 \times 100$ : $100 = 1 0^{2}$ , so move the decimal two places to the right. Since $23.4$ has only one digit after the decimal, append a zero as a placeholder when shifting the second place: $23.4 \to 234. \to 2340$ .

$2456 \div 1000$ : $1000 = 1 0^{3}$ , so move the decimal three places to the left. $2456 \to 2.456$ .

Answer: $23.4 \times 100 = 2340$ and $2456 \div 1000 = 2.456$

Naming decimal numbers

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.”

$305.25$ is “three hundred five and twenty-five hundredths.” (The whole-number part, $305$ , is “three hundred five” - no “and” within it. The “and” falls only at the decimal point.)

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 with mixed lengths

Order $0.5$ , $0.45$ , $0.500$ , and $0.4999$ from least to greatest.

First, note that $0.5$ and $0.500$ are equal values (trailing zeros don’t change a decimal’s value), so they occupy the same position.

Pad the remaining distinct values to four decimal places for comparison: $0.5000$ , $0.4500$ , $0.4999$ .

Compare tenths: $0.4500$ and $0.4999$ both have $4$ tenths, while $0.5000$ has $5$ tenths - so both $0.4$ -something values are less than $0.5$ .

Among $0.4500$ and $0.4999$ : hundredths are $5$ vs. $9$ , so $0.4500 < 0.4999$ .

Answer: $0.45 < 0.4999 < 0.5 = 0.500$

Ordering negative decimals

Example: Ordering negative decimals

Order $- 0.3$ , $- 0.25$ , and $- 0.56$ from least to greatest.

As positive values: $0.25 < 0.3 < 0.56$ .

Applying negatives reverses the order: $- 0.56 < - 0.3 < - 0.25$ .

Answer: $- 0.56 < - 0.3 < - 0.25$

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 a 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 dramatically.

Ones: $1 0^{0} = 1$
Tens: $1 0^{1} = 10$ , Hundreds: $1 0^{2} = 100$ , Thousands: $1 0^{3} = 1000$
Tenths: $1 0^{- 1} = 0.1$ , Hundredths: $1 0^{- 2} = 0.01$ , Thousandths: $1 0^{- 3} = 0.001$

Example: Place value across the decimal point

Consider the number $324.056$ . Each digit sits in a specific place:

$3$ 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: $0 \times 1 0^{- 1} = 0$ (placeholder - contributes nothing to the sum)

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

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

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

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 and divide by powers of $10$

$23.4 \times 100$ : $100 = 1 0^{2}$ , so move the decimal two places to the right. Since $23.4$ has only one digit after the decimal, append a zero as a placeholder when shifting the second place: $23.4 \to 234. \to 2340$ .

$2456 \div 1000$ : $1000 = 1 0^{3}$ , so move the decimal three places to the left. $2456 \to 2.456$ .

Answer: $23.4 \times 100 = 2340$ and $2456 \div 1000 = 2.456$

Naming decimal numbers

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.”

$305.25$ is “three hundred five and twenty-five hundredths.” (The whole-number part, $305$ , is “three hundred five” - no “and” within it. The “and” falls only at the decimal point.)

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 with mixed lengths

Order $0.5$ , $0.45$ , $0.500$ , and $0.4999$ from least to greatest.

First, note that $0.5$ and $0.500$ are equal values (trailing zeros don’t change a decimal’s value), so they occupy the same position.

Pad the remaining distinct values to four decimal places for comparison: $0.5000$ , $0.4500$ , $0.4999$ .

Compare tenths: $0.4500$ and $0.4999$ both have $4$ tenths, while $0.5000$ has $5$ tenths - so both $0.4$ -something values are less than $0.5$ .

Among $0.4500$ and $0.4999$ : hundredths are $5$ vs. $9$ , so $0.4500 < 0.4999$ .

Answer: $0.45 < 0.4999 < 0.5 = 0.500$

Ordering negative decimals

Example: Ordering negative decimals

Order $- 0.3$ , $- 0.25$ , and $- 0.56$ from least to greatest.

As positive values: $0.25 < 0.3 < 0.56$ .

Applying negatives reverses the order: $- 0.56 < - 0.3 < - 0.25$ .

Answer: $- 0.56 < - 0.3 < - 0.25$

Place value and decimal representation

Place value and powers of 10

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Ordering negative decimals

Place value and decimal representation

Place value and powers of 10

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Ordering negative decimals

Place value and decimal representation

Place value and powers of 10

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Ordering negative decimals

Place value and decimal representation

Place value and powers of 10

Multiplying and dividing by powers of 10

Naming decimal numbers

Ordering numbers

Ordering negative decimals

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 $10$