## S k i l l

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A R I T H M E T I C

### THE POWERS OF 10

In this Lesson, we will answer the following:

1. | What are the ten digits? |

The ten symbols: 0 1 2 3 4 5 6 7 8 9 |

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 Arabic numerals, because it was the Arab mathematicians who introduced them into Europe from India, where their forms evolved.

We saw in Lesson 1 that ‘364’ is a numeral, which is a symbol for a number. A number is the actual collection of units.

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 is a collection of ten One Hundreds.

Ten One Thousands are called Ten Thousand.

The numbers in that sequence are called the powers of 10 .

2. | Which numbers are the powers of 10? |

They are the numbers produced when, starting with One, we continue collecting them into groups of 10. | |

10 Ones. 10 Tens. 10 Hundreds. 10 Thousands. And so on. |

Here are their names and numerals.

The Powers of 10

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 Lesson 4.)

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 .

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 Lesson 1 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.

4. | How do we read a whole number, however large? |

256,312,785,649,408,163 | |

Starting from the left, read each three-digit group; then say the name of its class. |

Example 1. Read this number:

Answer . Starting from the left, 256, read each three-digit group. Then say the name of the class.

“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:

Answer . Starting from the right, place commas every three digits:

Read the number:

“8 million , 792 thousand , 456.”

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:

00 2 | “Two” |

0 2 0 | “Twenty” |

2 00 | “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, as we will see in the next Lesson. (For example, we should write $609.50 as “Six hundred nine dollars and fifty cents.” Not “Six hundred and nine dollars.”)

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:

Example 5. Write in numerals:

Five billion , sixteen thousand , nine.

Answer . After the billions , we expect the millions , but it is absent. Therefore write

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.

Example 6. Distinguish the following:

a) Two hundred seventeen million | b) Two hundred million seventeen |

Answers . | |

a) 217,000,000 | b) 200,000,017 |

At this point, please “turn” the page and do some Problems.

How to read a whole number. How to write whole numbers. Place value. Which numbers are the powers of 10? Names of large numbers.

## Place Value Chart

In place value chart, the digits are grouped in the threes in a big number. The number is read from left to right as вЂ¦вЂ¦вЂ¦. billion вЂ¦вЂ¦вЂ¦. million вЂ¦вЂ¦вЂ¦.. thousands вЂ¦вЂ¦вЂ¦.. ones.

The place value chart of the **International System** is given below:

100,000 = 100 thousand

1,000,000 = 1 million

10,000,000 = 10 millions

100,000,000 = 100 millions

Whole numbers can be represented on the place-value chart.

The numbers 702845 and 360400295 are represented on the place-value chart.

**We can expand these numbers as:**

702845 = 7 Г— 100000 + 2 Г— 1000 + 8 Г— 100 + 4 Г— 10 + 5 Г— 1

360400295 = 3 Г— 100000000 + 6 Г— 10000000 + 4 Г— 100000 + 2 Г— 100 + 9 Г— 10 + 5 Г— 1

Place value of 7 is 7 Г— 100000 = 700000 and the place is hundred thousand.

Place value of 2 is 2 Г— 1000 = 2000 and the place is thousands.

Place value of 8 is 8 Г— 100 = 800 and the place is hundreds.

Place value of 4 is 4 Г— 10 = 40 and the place is tens.

Place value of 5 is 5 Г— 1 = 5 and the place is ones.

Place value of 3 is 3 Г— 100000000 = 300000000 and the place is hundred million.

Place value of 6 is 6 Г— 10000000 = 60000000 and the place is ten million.

Place value of 4 is 4 Г— 100000 = 400000 and the place is hundred thousand.

Place value of 2 is 2 Г— 100 = 200 and the place is hundred.

Place value of 9 is 9 Г— 10 = 90 and the place is tens.

Place value of 5 is 5 Г— 1 = 5 and the place is ones.

The place value of digit 0 at any place is 0.

*LetвЂ™s read the numbers by observing the place value chart;*

**218705** вЂ“ The number is read as 218 thousands 705.

It is written as 218,705 (using comma).

**42156** вЂ“ The number is read as 42 thousands 156.

It is written as 42,156 (using comma).

**25374821** вЂ“ The number is read as 25 million 374 thousands 821.

It is written as 25,374,821 (using comma).

**Indian Place-Value Chart**

We know that in the Indian place-value chart, we move from right to left. Each place-value represents 10 times the place-value to its immediate right. The following place-value chart shows nine places. Starting from the right the nine places are grouped into four periods namely ones, thousands, lakhs and crores. Each period is separated by a comma.

To read a large number we say, the name for the number followed by the name of the period from left to right. To write a large number from word form to the standard form we substitute digits corresponding to the number for each period from left to right. We use commas in place of periods names.

When a period contains zero, we do not name that period in the word form.

Solved Examples on Indian Place-Value Chart:

**1. Insert commas and express the following numbers in figures.**

Seventy-four crore, three lakhs, forty-one thousand six hundred four.

Starting from left we enter the digits from crores, lakhs, thousands and ones in the following place-value chart.

Thus, the above number is read as 33,03,41,504

**2. Write the number 6,51,90,949,in words.**

We enter the digits in the places-value chart starting from right to left.

The number is written as six crore, fifty-one lakh, ninety thousand, nine hundred forty-nine.

A place that has zero in the standard form of a number is not represented in its word form.

In place value chart, the digits are grouped in the threes in a big number. The number is read from left to right as вЂ¦ billion вЂ¦million вЂ¦. thousands вЂ¦ones. The place value chart of the International System is given below: Place Value Chart 100,000 = 100 thousand 1,000,000