Gauss-Jordan Elimination

Chapter

If you'll just look over to your left, you will see that we have a rather colorful place-value table there.

Notice also that the chart is divided into four different sections.

Three of the sections, which we call periods, are labeled at the top.

On the right, we have the ones period. Note that the ones period is subdivided into 3 individual places: the ones place, the tens place, and the hundreds place.

Then we have the thousands period, and within the thousands period, again, we have three places: the one thousands place, the ten thousands place, and the hundrend thousands place. (Did you notice the repetition of one, ten, and a hundred?)

And finally, we have the millions period. And once again, within the millions period, we have the one millions place, the ten millions place, and the hundred millions place.

As you can see, the periods are separated from one another by commas.

When we read such numbers, we read this set of three digits that come between the commas as we normally would, but we replace the commas with the names of the periods. Or in other words, every time you see a comma, you're going to say the name of the period that preceded it.

7,824,315 reads seven million, eight hundred twenty four thousand, three hundred, fifteen.

Caution: Do not read the word “and” when reading whole numbers. “And” indicates the placement of a decimal point. (e.g. 8.4 is “eight and four tenths.”)

Stress use of commas as an aid in reading whole numbers and the use of the decimal point in separating whole numbers and decimals.

Spelling-correct use the –th at the end of words indicating decimals: tenths, hundredths, etc.

NUMBER SENSE AND NUMERATION

There are three different parts to the place-value chart. You should keep studying the chart to the right until you are able to draw it by memory!

The Periods

The Places

The Digits

Place Value Table

For example, the standard form of the number we read as

two hundred six thousand, fifty-four is written this way:

206,054

Each group of three digits, separated by a comma, is called a period. When writing numbers in standard form, always remember to use commas to separate the periods.

Note the way that the standard form of the number was written in the place-value chart above. Now it’s your turn!

Directions: Write each number in words.

5. 80, 062 _____________________________________

6. 10,800 ______________________________________

Directions: Write each number in standard form.

7. eight thousand, six hundred nine ______________

8. two million, fifty-six thousand, three ______________

Directions: Place a comma in each of the numerals below. Then rewrite each numeral in the place-value chart located to the left.

1. 37200005

2. 36200

3. 1903702

4. 5053

Name: __________________________________________

Date: _____/_____/_____

Place-Value Chart

Place-Value Prowl !

GAME TIME

Once your teacher feels you are ready, you may begin the Place-Value Prowl activity.

DIRECTIONS:

This game is best played in pairs.

Two copies of the place-value chart.

You will need a die, pencil, and possibly scratch paper.

The first player rolls the die.

S/he then chooses a place to write the digit that was rolled. (Should s/he write it in the ones place, tens place, hundreds place, etc? The placement of the digit will effect its value, and ultimately, the value of the final number.)

The goal of the game is to create the largest number possible.

After the first player has finished rolling their number and writing it in their place-value chart, the second player does the same.

Continue taking turns rolling the die and writing the digit that comes up in the place out your table until all the boxes in the digit section of both place-value charts our filled.

Then compare the two numbers and see who has the higher number.

NUMBER SENSE AND NUMERATION

Comparing

Numbers

NUMBER SENSE AND NUMERATION

38,509

38,590

One of the skills you are required to master this year is the ability to order and compare whole numbers. To compare two or more numbers, follow these steps:

First, line up the numbers using place value. For example, to compare 38,509 and 38,590 you begin in this way:

Next, begin at th greatest place value, and then move right until you find the first place value where the digits are different:

Compare those digits. Write > , < , or = .

Because 9 > 0, you can say that 38,590 > 38,509.

Because 0 < 9, you can also say that 38,509 < 38,590.

38,509

38,590

↓

↑

Directions: Study the number 890,536. Write the digit that is in the

1. hundreds place ___

2. ten thousands place ___

3. tens place ___

4. thousands place ___

5. ones place ___

6. hundred thousands place ___

Number Sense

Place-value Table

(By the way, the symbols we use for the digits are called Arabic numerals. However, the Arabs did not invent these symbols, but learned them from the people of India.)

→

In the fourth grade you will need to be able to tell which digit is in what place value and give the value of that digit.

The first step toward reaching this goal is being able to write numbers in a place value chart like the one below.

Number Sense

The Places

The Periods

The Digits

Ones

Turning the first (algebraic) expression into a verbal expression (into words) we have, "3 × m, divided by 2."

Translating the second expression into words we have, "3 × n, minus 6."

Millions

Evaluate if x = 15 and y = 7

2.31 – x + y

SPECIAL NOTATION

8 × m > 8m, 8 ∙ m

5 × x × y >

10 - q >

Evaluate if x = 4 , y = 3 and z = 2

1.4x + 3y – z2. – (z + x)

16 + 9 – 2

25 – 2 = 23

EXAMPLES

Standard Form: 764,341,237

n ,y ,x ,a

Place Value Through the Hundred Millions Place:

THE PERIODS

Thousands

Hundreds l Tens l Ones

7 6 4, 3 4 1, 2 3 7

Word Form: seven hundred sixty-four million,three hundred forty-one thousand two hundred thirty-seven

Expanded Form: 700,000,000 + 60,000,000 + 4,000,000 + 300,000 + 40,000 + 1,000 + 200 + 30 + 7

Which is farther: 3609 miles or 3690 miles?

Which is more: 10,478 units or 10,748 units?

Which is highest?

978,000,534 feet

978,000,354 feet

978,000,435 feet

Which is warmer: 280176 or 820176?

Which is more: 421 gallons or 412 gallons?

Which is heaviest?

3740 pounds

3790 pounds

3709 pounds

Which is more: $905 or $950?

Which is farther: 1,470 miles or 1,407 miles?

Which is heaviest?

784 pounds

874 pounds

748 pounds

Which is farther: 36.9 miles or 39.6 miles?

Which is more: 14.7 units or 17.4 units?

Which is highest?

534.5 feet

354.5 feet

435.5 feet

Which is warmer: 60.7 degrees or 67.0 degrees?

Which is more: 421.5 gallons or 412.5 gallons?

Which is heaviest?

3,740.9 pounds

3,790.4 pounds

3,709.4 pounds

Which is more: $103.95 or $139.50?

Which is farther: 1470.89 miles or 1407.89 miles?

Which is heaviest?

9,784.50 pounds

9,785.40 pounds

9,784.05 pounds

Hello everyone! My name is Mr. Will and today

I will be teaching a lesson overview on the base-ten number system, which is the Hindu-Arabic number system (or sequence of numbers) that we use today.

One of the things that made this system unique is the fact that there is a symbol for zero.

Moreover, in this system all of the numbers are constructed from ten numerals, which we call digits. The ten digits (or numerals) are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. They can be used to represent any number. You should memorize them!

We say that base-ten is a "positional" numeral system because the value of any particular digit as determined, at least in part, by its position within the number in which it appears.

Since the value of a digit depends on the place where it is located, we often speak of place value which, in turn, is based on groupings of ten. This is from where we get the term: base ten.

Now, in order to identify each place's value, different places are given different names, such as the ones place, the tens place, the hundreds place, the thousands place, etc., and each name identifies a different unit, and stands for a different multiple of ten.

(A unit is a basic amount that is used to count or measure.)

Since every unit has its own name and value based on position, each unit's "place" never changes. And as we said before, each place is worth a different multiple of ten, which is to be multiplied times whatever digit happens to appear in that particular place.

Finally, note that each position has a value ten times greater than the value of the position to its immediate right.

All of this information, taken together, comprises (or forms) the concept we know as place value. You will need to develop a solid understanding of base ten and place value to to do well in math.

Another way to present all of this information is to display it in what is known as the Place-Value Chart

REVIEW: Place value is the value of a digit as determined by its position in a number.

PLACE-VALUE CHART

THOUSANDS

MILLIONS

ONES

INSTRUCTOR: Mr. Will

CLASS: Our Number System

When it comes to writing numbers, there are three different ways that it can be done.

Standard form is the manner in which numbers are normally written. That's when the numbers are written out using digits.

Then there is word form. Of course, that's when numbers are written using letters (and words). Notice how you write the name of the period at the end, followed by a comma.

So, you read only the numbers between the commas, ending you see the name of the period. The knee read the next three numbers, and then say the name of the period in which they appear. Etc.

Expanded form is writing the value of each digit and adding the values together. So you have the seven, and the value of the seven is 700 million. Etc.

So you write 700 million plus 60 million plus 4 million etc.

A good way to find the value of each individual digit is to write the name of the place in which the digit appears in standard form. Then multiplythe digit times what you wrote.

Now you try it out. Writes this number into other forms. You have 62,200,043. You haven't written out in word for. You try to write it in standard form and in expanded form.

I like to draw a chart so that I can see where everything falls.

It says 63, and then it says million there, so you know you're going to have a comma there. There is no hundred, so you just write 43.

Now that you wrote it in the place by you charge, you can see that you have...

Now, let's find the value of the red digit. Remember the place all you chart, and remember the periods.

If you look at the place values, you see that it lands in the millions period and in the 10 Millions Place, so you multiply the 4 times 10 million to get 40 million. as the value of that digit.

Another type of problem is one in which you have to find the missing number.

Is when you're given a group of numbers, and you have to figure out what the missing number is.

A good way of doing this is examined the digits place by place in order to find out which place has the digits that do not match.

You have 67,321,417. So, it's here in the thousands period that the digits are different.

Specifically, it's the two and the three in the 10 thousand period. So, known that the 2 and the 3 changed. Right? Four hundred seventeen stays the same.

Then we have 67,351,417.

Here, the five is different. So, once again, we see the 10 Thousands Place changing. It goes from 20,000, to 30, 000, to the missing number, to 50,000. So, we have 2, 3, ___, and five.

Since four comes between three and five, it's clear that for is the missing digit. So now we can write the missing number with afforded 10 Thousands Place, and everything else th same.

We can save the 67 million is the same, followed by the blank number,...

So when you're finding the missing number, find the digit that is changing, and then figure out how it is changing, and then write the missing number.

Decimal Numbers

A number that uses place value and a decimal point to show numbers that include tents, hundreds, thousands, and so on is a decimal number.

The part of the decimal number that represents the amount that is less than one is called the decimal part. The decimal part of the decimal number is the part that is to the right of the decimal point.

A decimal point is the period that is used in decimal numbers to separate the whole number part from the decimal part.

Numerals are written symbols that are used to represent quantities or numbers. The numerals we use today are called Hindu Arabic numerals. The term digit is a synonym for numeral.

Whole numbers are the set of natural numbers and zero, which can be represented using the digits.

The Base Ten System

The Hindu-Arabic system that we use today where:

All numbers are constructed from the ten digits.

The system uses place a value based on groupings of ten.

There is a symbol for zero.

Decimal Numbers

A number that uses place value and a decimal point to show numbers that include tents, hundreds, thousands, and so on is a decimal number.

A decimal point is the period that is used in decimal numbers to separate the whole number part from the decimal part.

Numerals are written symbols that are used to represent quantities or numbers. The numerals we use today are called Hindu Arabic numerals. The term digit is a synonym for numeral.

Place Value Chart

Place Value:

Hundred Millions