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Naming and ordering numbers | Lesson

How are numbers named?

The numbers we use everyday are in the base- $10$ ‍ system. Each place represents a power of

10

and each digit is a number from

0

9

(

10

possibilities).

Below are the common names we use for powers of

10

$10^{0} = 1$ ‍: one
$10^{1} = 10$ ‍: ten
$10^{2} = 100$ ‍: one hundred
$10^{3} = 1,000$ ‍: one thousand
$10^{6} = 1,000,000$ ‍: one million
$10^{9} = 1,000,000,000$ ‍: one billion

After

10^{3}

, there is a new descriptor for every third power of

10

. The powers between descriptors represent ten and one hundred of the last power. For example:

$10^{4} = 10,000$ ‍: ten thousand
$10^{5} = 100,000$ ‍: one hundred thousand

What skills are tested?

Matching a description to its numerical representation
Determining the value of a digit in a number
Identifying a number with a digit in a specific place value
Comparing values of fractions, decimals, and mixed numbers
Ordering a list of integers, fractions, and decimals with positive and negative values

How do we rewrite the description of an integer as a numeral?

For integers, the furthest digit to the right represents

10^{0} = 1

, the ones place. The second digit from the right represents

10^{1} = 10

, the tens place. The pattern continues with each digit to the left representing the next power of

10

A place value chart is helpful for tracking the value of each digit.

Millions	Hundred thousands	Ten thousands	Thousands	Hundreds	Tens	Ones
$10^{6}$ ‍	$10^{5}$ ‍	$10^{4}$ ‍	$10^{3}$ ‍	$10^{2}$ ‍	$10^{1}$ ‍	$10^{0}$ ‍

If it is unclear what number is represented by a description, we can use the following steps to create its numeric representation:

Write the number as a sum of powers of $10$ ‍.
Map the digits onto a place value chart using $0$ ‍s for any empty places.
Add commas after every group of $3$ ‍ digits (starting from the ones place).

What number is represented by four hundred thirty-four thousand and two?

We can rewrite the description as a sum of powers of

10

Four hundred thirty-four thousand and two can be written as:

\begin{aligned} 4 00,000 + 3 0,000 + 4, 000 + 2 \\ = & 4 \times 100,000 + 3 \times 10,000 + 4 \times 1,000 + 2 \times 1 \end{aligned}

Then, we can map the leading digits onto a place value chart.

Hundred thousands	Ten thousands	Thousands	Hundreds	Tens	Ones
$4$ ‍	$3$ ‍	$4$ ‍	$0$ ‍	$0$ ‍	$2$ ‍

The number represented by four hundred thirty-four thousand and two is

434,002

How do we rewrite the description of an decimal as a numeral?

For numbers with decimals, the powers of

10

pattern continues to the right of the decimal place. Since the ones place is

10^{0}

, the first place to the right of the decimal point is

10^{- 1} = \frac{1}{10}

, the tenths place. The second place to the right of the decimal is

10^{- 2} = \frac{1}{10^{2}} = \frac{1}{100}

, the hundredths place. This pattern continues for as many digits as the decimal has.

Ones	.	Tenths	Hundredths	Thousandths	Ten thousandths
$10^{0}$ ‍	.	$10^{- 1}$ ‍	$10^{- 2}$ ‍	$10^{- 3}$ ‍	$10^{- 4}$ ‍

If it is unclear what number is represented by a description, we can use the following steps to create its numeric representation:

Write the number as a sum of powers of $10$ ‍.
Map the digits onto a place value chart using $0$ ‍s for any empty places.
Add commas after every group of $3$ ‍ digits to the left of the decimal point (starting from the ones place).

What number is represented by six hundred five and thirty-four thousandths?

We can rewrite the description as a sum of powers of

10

Six hundred five and thirty-four thousandths can be written as:

600 + 5 + \frac{34}{1,000}

Since we cannot write two digits in the thousandths place, we rewrite

\frac{34}{1,000}

\frac{30}{1,000} + \frac{4}{1,000} = \frac{3}{100} + \frac{4}{1,000}

Our final sum as powers of

10

is:

6 \times 100 + 5 \times 1 + 3 \times \frac{1}{100} + 4 \times \frac{1}{1,000}

Then, we map out the leading digits on a place value chart using

0

s in any empty places.

Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths
$6$ ‍	$0$ ‍	$5$ ‍	.	$0$ ‍	$3$ ‍	$4$ ‍

The number represented by six hundred five and thirty-four thousandths is $605.034$ ‍.

How are numbers ordered?

Ordering a list of numbers means stating the numbers from least to greatest or greatest to least.

A number line is a useful tool for comparing and ordering numbers. We read a number line from left to right with numbers further to the left having lower values.

When ordering a list of numbers:

All negative numbers are less than positive numbers.
Negative numbers with larger magnitudes (the value without the sign) are smaller.
Fractions can be compared by writing them with a common denominator or
.
Decimals can be rewritten with $0$ ‍s at the end to make comparisons easier.
Fractions and decimals are more easily compared when all are expressed as decimals.

Order the following list of numbers from least to greatest:

\frac{1}{8}

- 0.3

\frac{2}{5}

- \frac{3}{2}

0.75

First, we can rewrite all numbers as decimals, either by recognizing equivalents of common fractions or using a calculator:

\begin{aligned} \frac{1}{8} & = 0.125 \\ - 0.3 & = - 0.3 \\ \frac{2}{5} & = 0.4 \\ - \frac{3}{2} & = - 1.5 \\ 0.75 & = 0.75 \end{aligned}

Then, we can order the numbers least to greatest. The negative numbers come first, in order of decreasing magnitude:

- 1.5 = - \frac{3}{2}

- 0.3

0.125 = \frac{1}{8}

0.4 = \frac{2}{5}

0.75

We can confirm this order using a number line:

The list of numbers ordered from least to greatest is:

- \frac{3}{2}

- 0.3

\frac{1}{8}

\frac{2}{5}

0.75

Things to remember

A place value chart can be used to determine the value of each digit in a number.

For integers, the chart starts with the ones place:

Millions	Hundred thousands	Ten thousands	Thousands	Hundreds	Tens	Ones
$10^{6}$ ‍	$10^{5}$ ‍	$10^{4}$ ‍	$10^{3}$ ‍	$10^{2}$ ‍	$10^{1}$ ‍	$10^{0}$ ‍

For decimals, the chart adds a decimal after the ones place and continues to the right:

Ones	.	Tenths	Hundredths	Thousandths	Ten thousandths
$10^{0}$ ‍	.	$10^{- 1}$ ‍	$10^{- 2}$ ‍	$10^{- 3}$ ‍	$10^{- 4}$ ‍

If it is unclear what number is represented by a description, we can use the following steps to create its numeric representation:

Write the number as a sum of powers of $10$ ‍.
Map the digits onto a place value chart using $0$ ‍s for any empty places.
Add commas after every group of $3$ ‍ digits to the left of the decimal (starting from the ones place).

When ordering a list of numbers:

All negative numbers are less than positive numbers.
Negative numbers with larger magnitudes (the value without the sign) are smaller.
Fractions can be compared by writing them with a common denominator or converting them to decimals.
Decimals can be rewritten with $0$ ‍s in the end to make comparisons easier.
Fractions and decimals are more easily compared if all are expressed as decimals.

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DreamGirl
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can I just leave this e@rth?
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LN
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Please add a comment in the lesson about naming or symbol conventions that may differ depending on language or location. Lycee Francais when I was young taught comma as the decimal mark and point as the place marker. Also,"billions" may mean 10^9 or 10^12, depending where one is. Differences in convention are a terrible source of confusion and should be flagged for learners! In general, fact and convention should be distinguished for better reasoning.
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robsanzone
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Just a question on grammer: when a number has a decimal, the 4th digit to the left of the decimal it is in the position of "Thousands". When a number has a decimal, the 3rd digit to the right of the decimal it is in the position of "Thousandths" with a "ths". Is this correct?
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- laskint1
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Pitsi Lamola
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how is the ten in the value of the thousands
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- Angelica Martinez
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  thousandTHS not thousands
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Maria Vera
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When you have fractions in which the numerator is one and the denominator is greater than 1, do the ascendant numbers indicate that that number is greater of less. Ex: 1/8, 1/9, 1/10. The one is divide more times, so the number is smaller, right?They go further in the number line to the right or to the left?
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no sense for some exercises
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Course: Praxis Core Math > Unit 1

Naming and ordering numbers | Lesson

How are numbers named?

What skills are tested?

How do we rewrite the description of an integer as a numeral?

How do we rewrite the description of an decimal as a numeral?

How are numbers ordered?

Your turn!

Things to remember

Want to join the conversation?