1. HOW TO READ AND WRITE
WHOLE NUMBERS
THE POWERS OF 10
In this Lesson, we will answer the following:
1.What are the ten digits?
2.Which numbers are the powers of 10?
3.What are the names of the classes?
4.How do we read a whole number, however large?
Section 2: Place value. Positional numeration.
To which place does each digit belong?
What does it mean to write a number in expanded form?
What is the relationship between the units of adjacent place value?
How do we round off, or approximate, a whole number to a given place?
2. What are the ten digits?
The ten symbols: 0 1 2 3 4 5 6 7 8 9
Examples.
105 is a three-digit number. The digits are 1, 0, and 5.
28 ends in the digit 8.
$364 has the same digits as $3.64.
Those ten marks are also known as the because it was
the Arab mathematicians who introduced them into Europe from India,
where their forms evolved.
364' is a , which is a symbol for a number. A number is the actual
collection of .
3. The number we call One Thousand is a collection of ten One Hundreds.
The powers of 10
Number Ten is a collection of
ten Ones
One Hundred is a collection of
ten Tens.
The number we call One Thousand a collection of ten One Hundreds.
4. The Powers of 10
Which numbers are the powers of 10?
They are the numbers produced when, starting with One, we repeatedly
collect them into groups of 10.
10 Ones. 10 Tens. 10 Hundreds. 10 Thousands. And so on.
Here are their names and numerals.
Class of One 1
Ones Ten 10
Hundred 100
Class of One thousand 1,000
Thousands Ten thousand 10,000
Hundred thousand 100,000
Class of One million 1,000,000
Millions Ten million 10,000,000
Hundred million 100,000,000
Class of One billion 1,000,000,000
Billions Ten billion 10,000,000,000
Hundred billion 100,000,000,000
Each power is composed of ten of the one above.
(The metric system is the system of measurement
based on the powers of 10; see )
Strictly, 1 is not a power of 10. The first power of 10 is
10 itself. Its numeral is a 1 followed by one 0.
The second power of 10 is 100; it has two 0's. The
third power has three 0's. And so on.
Notice how the names fall into groups of three:
One thousand, Ten thousand, Hundred thousand.
One million, Ten million, Hundred million.
Each group of three -- Ones, Tens, Hundreds
is called a class.
5. Starting with Billions (bi for two), each class has a Latin prefix. To read a
number more easily, we separate each class -- each group of three digits -- by
commas.
Note that each class is 1000 times the previous class; the Thousands are 1000
times the Ones; the Millions are 1000 times the Thousands; and so on.
In previous lesson we showed how to read and write any number from
1 to 999, which are the numbers in the class of Ones. Together with
knowing the sequence of class names, that is all that is necessary to be able
to name or read any whole number.
Example 1. Read this number:
256,312,785,649,408,163
Answer. Starting from the left, 256, read each three-digit group. Then say the name of the class.
Say:
"256 Quadrillion, 312 Trillion, 785 Billion, 649
Million, 408 Thousand, 163."
Do not say the class name "Ones."
Example 2. To distinguish the classes, place commas in this number: 8792456
Answer. Starting from the right, place commas every three digits: 8,792,456
Read the number:
"8 million, 792 thousand, 456."
6. Example 3. Read this number: 7,000,020,002
Answer. "Seven billion , twenty thousand, two."
When a class is absent, we do not say its name; we do not say, "Seven billion, no
million, ..."
Also, every class has three digits and so we must distinguish the following:
002 "Two"
020 "Twenty"
200 "Two hundred"
As for "and," in speech it is common to say "Six hundred and
nine," but in writing we should reserve "and" for the decimal
point.
(For example, we should write $609.50 as "Six hundred nine
dollars and fifty cents." Not "Six hundred and nine dollars.")
7. Example 4. Write in numerals:
Four hundred eight million, twenty-nine thousand, three
hundred fifty-six.
Answer. Pick out the classes: "million", "thousand". Each
class (except perhaps the first class on the left) has exactly
three digits:
8. Example 5. Write in numerals:
Five billion, sixteen thousand, nine.
Answer. After the billions, we expect the millions, but it is absent. Therefore write
5,000,016,009
Again, we must write "sixteen thousand" as 016; and "nine" as 009; because
each class must have three digits. The exception is the class on the extreme
left. We may write "Five" as 5 rather than 005
.
When writing a four-digit number, such as Four thousand five
hundred, it is permissible to omit the comma and write 4500. In
fact, we often read that as "Forty-five hundred." But when a
number has more than four digits, then for the sake of clarity we
should always place the commas.
a) Two hundred
seventeen million
b) Two
hundred million seven
teen
Answers.
a) 217,000,000 b) 200,000,017
Example 6. Distinguish the following:
