Math 6H (Period 6 & 7)

The Decimal System 3-2

Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)

Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?

Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
Think of the comma as a 'signal' for you to mention a place name category!!

In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."

The expanded notation for 420,346 is given by

4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)

Using exponents the expanded notation may be given as
4(10⁵) + 2(10⁴)+0(10³)+3(10²)+4(10¹)+6(10⁰)

What hmmm.. how is (10⁰) = 1
any number to the zero power is 1.


From yesterday we noticed that
10² ⋅ 10³ = 10⁵
and
10⁶ ⋅ 10³ =10⁹
so we wrote a rule for any exponent values a and b
10^a ⋅ 10^b = 10^a+b
Then I asked what would be a rule for

10⁸
10⁵
and
10⁶
10⁴
Students decided that since we added the exponents when multiplying we could subtract the exponents when dividing!!

so could we write a rule for any exponent values a and b?
YES
10^a
10^b
= 10^a-b
but then
10⁵
10⁵
would be
10 ^5-5 which is 10⁰
BUT WAIT... isn't any number divided by itself equal to ONE. That is,
10⁵
10⁵
= 1
So
10⁰ = 1

279,043
is read as two hundred seventy-nine thousand, forty-three.
and in expanded notation becomes:
2(100,000) + 7(10,000) + 9(1000) + 0(100) + 4(10) + 3(1)

or with exponents:
2(10⁵) + 7(10⁴) + 9(10³) + 0(10²) + 4(10¹) + 3(10⁰)
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???

Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.

so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value

10t + 2 becomes the correct expression

What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 in front of the x for the ones' digit? No it is... invisible!!

Some of you asked that I put the Magic Number Trick that we did online... try this one on your parents...

Have your 'victim' follow these directions:
Choose any four-digit number
Write the thousand's digit
Write the thousands' and hundreds' digits
Write the thousands' hundred's and tens' digits.
Add these numbers

Multiply by 9

Find the sum of the digits of the original number
add this sum to the previous results.

Your 'victim's' answers should always be the same as their original number chosen.
Example
I used in class 1492
Write the thousand's digit-------> 1
Write the thousands' and hundreds' digits ---> 14
Write the thousands' hundred's and tens' digits--->149
Add these numbers 1 + 14 + 149 = 164
Multiply by 9 or 164(9)= 1476

Find the sum of the digits of the original number 1 + 4 + 9 + 2 = 16
add this sum to the previous results. 1476 + 16 = 1492

Voila!! It worked!!
Email me your parent's comments or post them here!!