Greatest Common Factor 5-5
If we list the factors of 30 and 42, we notice
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
We notice that 1, 2, 3, and 6 are all COMMON factors of these two numbers. The number 6 is the greatest of these and therefore is called the
GREATEST COMMON FACTOR of the two numbers. We write
GCF(30,42) = 6
Although listing the factors of two numbers and then comparing their common factors is one way to determine the greatest common factor, using prime factorization is another easy way to find the GCF
Find GCF(54, 72)
54 = 2 ⋅ 3 ⋅ 3 ⋅ 3
72 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3
Find the greatest power of 2 that occurs IN BOTH prime factorization. The greatest power of 2 that occurs in both is just 2 1
Find the greatest power of 3 that occurs IN BOTH prime factorizations. The greatest power of 3 that occurs in both is 32
Therefore
GCF(54, 72) = 2 ⋅ 32 = 18
In class we circled the common factors and realized that
GCF(54, 72) = 2 ⋅ 3 ⋅ 3 = 18
Fin the GCF( 45, 60)
45 = 3 ⋅ 3⋅ 5
60 = 2⋅ 2⋅ 3⋅ 5
Since 2 is NOT a factor of 45-- there is NO greatest power of 2 that occurs in both prime factorizations.
The greatest power of 3 is just 31
and the greatest power of 5 is just 51
Therefore,
GCF(45,60) = 3⋅ 5 = 15
The number 1 is a common factor of any two whole numbers!! If 1 is the GCF , then the two numbers are said to be RELATIVELY PRIME. Two numbers can be relatively prime even if one or both of them are composite.
Show that 15 and 16 are relatively prime
List the factors of each number
FACTORS of 15: 1, 3, 5, 15
FACTORS of 16: 1, 2, 4, 8, 16
Since the GCF(15,16) = 1. The two numbers are relatively prime!!
Friday, December 2, 2011
Wednesday, November 30, 2011
Algebra Honors (Period 6 & 7)
Multiplying Fractions 6-2
You know from previous years that
ac/bd = a/b ⋅ c/d
and you know the converse is also true
a/b ⋅ c/d = ac/bd
That means you could solve
8/9⋅3/10 by either multiplying first and then simplify or you could simplify first and then multiply.
I find it works so much better to simplify first
8/9⋅3/10 = 4/15
6x/y3⋅y2/15 = 2x/5y where y ≠0
Which simplifies to
This textbook wants us to keep the factored form as our answers--> so let's continue to do that. In addition it states, " ...from now on, assume that the domains of the variables do not include values for which any denominator is ZERO. Therefore it will NOT be necessary to show the excluded [or restrictions] values of the variables."
I know everyone is jumping for joy!!
Rule of Exponents for a Power of a Quotient
(a/b)m = am/bm
(x/3)3 = x3/27
(-c/2)2⋅4/3c
you must do the exponent portion first!!
c2/4⋅(4/3c)
c/3
Find the volume of a cube if each edge has length 6n/7 in
You just need to cube each factor
(6n/7)3 = 216n3/343 inches cubed
If you traveled for 7t/60 hours at 80r/9 mi/h, how far have you gone?
Just multiply
7t/60⋅80r/9
but simplify first and you get
28rt/27 miles
You know from previous years that
ac/bd = a/b ⋅ c/d
and you know the converse is also true
a/b ⋅ c/d = ac/bd
That means you could solve
8/9⋅3/10 by either multiplying first and then simplify or you could simplify first and then multiply.
I find it works so much better to simplify first
8/9⋅3/10 = 4/15
6x/y3⋅y2/15 = 2x/5y where y ≠0
Which simplifies to
This textbook wants us to keep the factored form as our answers--> so let's continue to do that. In addition it states, " ...from now on, assume that the domains of the variables do not include values for which any denominator is ZERO. Therefore it will NOT be necessary to show the excluded [or restrictions] values of the variables."
I know everyone is jumping for joy!!
Rule of Exponents for a Power of a Quotient
(a/b)m = am/bm
(x/3)3 = x3/27
(-c/2)2⋅4/3c
you must do the exponent portion first!!
c2/4⋅(4/3c)
c/3
Find the volume of a cube if each edge has length 6n/7 in
You just need to cube each factor
(6n/7)3 = 216n3/343 inches cubed
If you traveled for 7t/60 hours at 80r/9 mi/h, how far have you gone?
Just multiply
7t/60⋅80r/9
but simplify first and you get
28rt/27 miles
Math 6 Honors ( Periods 1, 2, & 3)
Prime Numbers & Composite Numbers 5-4
A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...
Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Sieve of Eratosthenes - We did it!! :)
Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.
Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7
When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.
Two is the only even prime number because all the other even numbers have two as a factor.
Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.
List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.
Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.
A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...
Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!
Sieve of Eratosthenes - We did it!! :)
Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.
Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7
When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.
Two is the only even prime number because all the other even numbers have two as a factor.
Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.
List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.
Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.
Math 6 Honors ( Periods 1, 2, & 3)
Prime Numbers and Composite Numbers 5-4
A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers
A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.
Since 1 has exactly 1 factor, it is neither prime nor composite.
About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.
You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree
Both were shown in class.
Could you start the factor tree differently? If so, would you end up with the same answer?
The prime factors of 42 are the same in either factor tree, except for their order.
Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.
When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42
Notice the order in which prime factorization is written.
Let’s try finding the prime factorization of 60
The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 22∙ 3∙ 5
A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers
A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.
Since 1 has exactly 1 factor, it is neither prime nor composite.
About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.
You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree
Both were shown in class.
Could you start the factor tree differently? If so, would you end up with the same answer?
The prime factors of 42 are the same in either factor tree, except for their order.
Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.
When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42
Notice the order in which prime factorization is written.
Let’s try finding the prime factorization of 60
The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 22∙ 3∙ 5
Tuesday, November 29, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Square Numbers and Square Roots 5-3
Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.
One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.
Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol
so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.
Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)
Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40
Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...
In this case, 9 + 16 = 25... hmmm... what's 25???
Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.
One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.
Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol
so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.
Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)
Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40
Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...
In this case, 9 + 16 = 25... hmmm... what's 25???
Wednesday, November 16, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Tests for Divisibility 5-2
It is important to learn the following divisibility rules:
A number is divisibility by:
2 ... if the ones digit of the number is even
3 ... if the sum of the digits is divisible by three ( add the digits together)
4 ... if the number formed by the last two digits is divisible by by four ( Just LOOK at the last two numbers-- DON"T ADD them!!)
5 ... if the ones digits of the number is a 5 or a 0
6 ... if the number is divisible by both 2 and 3... (or if it is even and divisible by 3)
8 ... if the number formed by the last three digits is divisible by 8. (Like FOUR, just look at the last three digits-- divide them by 8)
9 ... if the sum of the digits is divisible by 9
10 ... if the ones digits of the number is a 0.
You will not need to know the divisibility rules for 7 or 11 but they are interesting...
You can test for divisibility by 7
Let's start with a number 959
Step 1: drop the one's digit so we have 95
Step 2: Subtract twice the ones' digit ( that you dropped) in this case we dropped a 9
so we double that and subtract 18 from 95
or 95-18 = 77. If the results, in the case, 77, is divisible by 7 --- so is the original number 959.
Step 3: If the number you get is still to big.. continue the process until you can determine if your number is divisible by 7.
To test for divisibility by 11
add the alternative digits beginning with the first
so let's try the following
4,378,396
Step 1: Add the alternate digits beginning with the 1st 4 + 7+ 3 + 6 = 20
Step 2: Add alternate digits beginning with the 2nd 3 + 8 + 9 = 20
Step 3: If the difference of the sums is divisible by 11 so is the original number.
In this case, 20-20 = 0 and 0/11= 0 so
4,378,396 is divisible by 11.
A good test for divisibility by 25 would be if the last two digits represent a multiple of 25.
A perfect number is one that is the SUM of all its factors except itself. The smallest perfect number is 6, since 6 = 1 + 2+ 3
The next perfect number is 28 since
28 = 1 + 2 + 4 + 7 + 14
What is the next perfect number?
It is important to learn the following divisibility rules:
A number is divisibility by:
2 ... if the ones digit of the number is even
3 ... if the sum of the digits is divisible by three ( add the digits together)
4 ... if the number formed by the last two digits is divisible by by four ( Just LOOK at the last two numbers-- DON"T ADD them!!)
5 ... if the ones digits of the number is a 5 or a 0
6 ... if the number is divisible by both 2 and 3... (or if it is even and divisible by 3)
8 ... if the number formed by the last three digits is divisible by 8. (Like FOUR, just look at the last three digits-- divide them by 8)
9 ... if the sum of the digits is divisible by 9
10 ... if the ones digits of the number is a 0.
You will not need to know the divisibility rules for 7 or 11 but they are interesting...
You can test for divisibility by 7
Let's start with a number 959
Step 1: drop the one's digit so we have 95
Step 2: Subtract twice the ones' digit ( that you dropped) in this case we dropped a 9
so we double that and subtract 18 from 95
or 95-18 = 77. If the results, in the case, 77, is divisible by 7 --- so is the original number 959.
Step 3: If the number you get is still to big.. continue the process until you can determine if your number is divisible by 7.
To test for divisibility by 11
add the alternative digits beginning with the first
so let's try the following
4,378,396
Step 1: Add the alternate digits beginning with the 1st 4 + 7+ 3 + 6 = 20
Step 2: Add alternate digits beginning with the 2nd 3 + 8 + 9 = 20
Step 3: If the difference of the sums is divisible by 11 so is the original number.
In this case, 20-20 = 0 and 0/11= 0 so
4,378,396 is divisible by 11.
A good test for divisibility by 25 would be if the last two digits represent a multiple of 25.
A perfect number is one that is the SUM of all its factors except itself. The smallest perfect number is 6, since 6 = 1 + 2+ 3
The next perfect number is 28 since
28 = 1 + 2 + 4 + 7 + 14
What is the next perfect number?
Monday, November 14, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Finding Factors and Multiples 5-1
You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 60. A number is said to be divisible by its whole numbered factors.
To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.
We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6
When you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!
A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first five multiples of 7 are
0, 7, 14, 21, 28
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21; 4(7) = 28
... and put in set notation it would be
{0, 7, 14 ,21, 28}
If you were to ask for the first four NON-ZERO Multiples of 6
the answer would be 6, 12, 18, 24.. and in set notation
{ 6, 12, 18, 24}
Whereas the first four multiples of 6 ( you would need to include 0)
{0, 6, 12, 18}
Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!
Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number
What number is a factor of every number? ONE
Is every number a factor of itself? YES
What is ( are) the only multilpe(s) of 0? 0
How many numbers have 0 as a factor? only one number What number(s)? ZERO
The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.
You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 60. A number is said to be divisible by its whole numbered factors.
To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.
We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6
When you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!
A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first five multiples of 7 are
0, 7, 14, 21, 28
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21; 4(7) = 28
... and put in set notation it would be
{0, 7, 14 ,21, 28}
If you were to ask for the first four NON-ZERO Multiples of 6
the answer would be 6, 12, 18, 24.. and in set notation
{ 6, 12, 18, 24}
Whereas the first four multiples of 6 ( you would need to include 0)
{0, 6, 12, 18}
Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!
Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number
What number is a factor of every number? ONE
Is every number a factor of itself? YES
What is ( are) the only multilpe(s) of 0? 0
How many numbers have 0 as a factor? only one number What number(s)? ZERO
The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.
Wednesday, November 9, 2011
Algebra Honors (Period 6 & 7)
Solving Equations by Factoring Section 5-12
a⋅0 = 0
and if a = 0 or b = 0
then we know that ab= 0
This is an if, then statement
conversely
if ab = 0 then either a= 0 or b = 0
THis Zero Products Property helps us solve equations.
(x +2)(x -5) = 0
either x + 2 must equal zero or x - 5 must equal zero
so set each expression equal to zero and solve
x + 2 = 0
x= -2
and x-5 = 0
x = 5
{-2. 5}
5m(m-3)(m-4) = 0
now you have three expressions so set each of them to zero
5m = 0 so m = 0
m-3 = 0 so m=3
m-4 = 0 so m=4
{0,3,4}
What happens with
3x2+ x = 2
It isn't the 2 products property but the ZERO products property so set the expression equal to ZERO
3x2+ x -2 = 0
Now factor
(x+1)(3x -2) = 0
set each of these equal to zero
x + 1 = 0 x = -1
3x -2 = 0 so x = 2/3
10x3 - 15x2 = 0
factor
5x2(2x -3) = 0
again set each equal to zero
5x2 = 0 so x = 0
and
2x -3 = 0 so x = 3/2
polynomial equation named by the term of highes degree
ax + b = 0 linear equation
ax2 + bx + c = 0 quadratic equations
ax3 + bx2 + cx + d = 0 cubic equation
2x2 + 5x = 12
becomes
2x2 +5x - 12 = 0
(x + 4)(2x-3) = 0
so x = -4 and x = 3/2
{-4, 3/2}
18y3 + 8y + 24y2 = 0
Rearrange first
18y3+ 24y 2 + 8y = 0
Then factor the GCF
2y(9y2 +12y +4) = 0
WAIT--> its a trinomial SQ
2y(3y +2)2 = 0
2y = 0 so y = 0
and 3y + 2 = 0 so y = -2/3
-2/3 is a double or multiple root but you only list it once in solution set.
That is,
{-2/3, 0}
y = x2 + x - 12
solve for the roots means you set this quadratic equal to ZERO
so
x2 + x - 12 = 0
(x+4)(x -3) = 0
x = -4 and x = 3
a⋅0 = 0
and if a = 0 or b = 0
then we know that ab= 0
This is an if, then statement
conversely
if ab = 0 then either a= 0 or b = 0
THis Zero Products Property helps us solve equations.
(x +2)(x -5) = 0
either x + 2 must equal zero or x - 5 must equal zero
so set each expression equal to zero and solve
x + 2 = 0
x= -2
and x-5 = 0
x = 5
{-2. 5}
5m(m-3)(m-4) = 0
now you have three expressions so set each of them to zero
5m = 0 so m = 0
m-3 = 0 so m=3
m-4 = 0 so m=4
{0,3,4}
What happens with
3x2+ x = 2
It isn't the 2 products property but the ZERO products property so set the expression equal to ZERO
3x2+ x -2 = 0
Now factor
(x+1)(3x -2) = 0
set each of these equal to zero
x + 1 = 0 x = -1
3x -2 = 0 so x = 2/3
10x3 - 15x2 = 0
factor
5x2(2x -3) = 0
again set each equal to zero
5x2 = 0 so x = 0
and
2x -3 = 0 so x = 3/2
polynomial equation named by the term of highes degree
ax + b = 0 linear equation
ax2 + bx + c = 0 quadratic equations
ax3 + bx2 + cx + d = 0 cubic equation
2x2 + 5x = 12
becomes
2x2 +5x - 12 = 0
(x + 4)(2x-3) = 0
so x = -4 and x = 3/2
{-4, 3/2}
18y3 + 8y + 24y2 = 0
Rearrange first
18y3+ 24y 2 + 8y = 0
Then factor the GCF
2y(9y2 +12y +4) = 0
WAIT--> its a trinomial SQ
2y(3y +2)2 = 0
2y = 0 so y = 0
and 3y + 2 = 0 so y = -2/3
-2/3 is a double or multiple root but you only list it once in solution set.
That is,
{-2/3, 0}
y = x2 + x - 12
solve for the roots means you set this quadratic equal to ZERO
so
x2 + x - 12 = 0
(x+4)(x -3) = 0
x = -4 and x = 3
Tuesday, November 8, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Dividing Decimals 3-9 cont'd
For word Problems use the 5 step plan found on Page 18 of our textbook
296.06 ÷ (18.7 + 3.9)
Following Aunt Sally ( or PEMDAS... remember our singing...
we do the operation inside the hugs!! ( )using a sidebar
18.7 + 3.9 make sure to stack them lining up the decimals and you will get 22.6
296.06 ÷ 22.6
When dividing by a decimal remember the rule from yesterday, multiply the divisor ( 22.6) by a power of ten which makes it a natural number
then use that same power of ten and multiply the dividend,
WHen you divide you have
2960.6 ÷ 226
Please do that problem and your quotient should be 13.1
(47.1 - 16.9) ÷ (21.9 -6.8)
Again you need to do the operations inside the ( ) first. Using a side bar and lining up the decimals
47.1 - 16.9 = 30.2
and 21.9 - 6.8 = 15.1
Just take a look at those two numbers and you will notice a relationship!!
30.2 ÷ 15.1
BUT... practice your division skills and confirm what you can tell...
30.2 ÷ 15.1 = 2
At an average rate of 55 km/hour how long will it take to drive 225 km to the nearest tenth of an hour?
d = rt
What must we find and what are the clues? Well, how long... is usually time and the fact that we need to round to the nearest tenth of an hour indicates we are finding TIME as well.
So what is the distance? 225 km and what is the rate? 55 km/h
so plug into the formula
225= 55t
Now, how do we solve this one step problem?
divide both sides by 55
225/55 = 55t/55
do the division as a side bar
225/55 ≈ 4.09 so
t ≈ 4.1
and the answer is 4.1 hour
For word Problems use the 5 step plan found on Page 18 of our textbook
296.06 ÷ (18.7 + 3.9)
Following Aunt Sally ( or PEMDAS... remember our singing...
we do the operation inside the hugs!! ( )using a sidebar
18.7 + 3.9 make sure to stack them lining up the decimals and you will get 22.6
296.06 ÷ 22.6
When dividing by a decimal remember the rule from yesterday, multiply the divisor ( 22.6) by a power of ten which makes it a natural number
then use that same power of ten and multiply the dividend,
WHen you divide you have
2960.6 ÷ 226
Please do that problem and your quotient should be 13.1
(47.1 - 16.9) ÷ (21.9 -6.8)
Again you need to do the operations inside the ( ) first. Using a side bar and lining up the decimals
47.1 - 16.9 = 30.2
and 21.9 - 6.8 = 15.1
Just take a look at those two numbers and you will notice a relationship!!
30.2 ÷ 15.1
BUT... practice your division skills and confirm what you can tell...
30.2 ÷ 15.1 = 2
At an average rate of 55 km/hour how long will it take to drive 225 km to the nearest tenth of an hour?
d = rt
What must we find and what are the clues? Well, how long... is usually time and the fact that we need to round to the nearest tenth of an hour indicates we are finding TIME as well.
So what is the distance? 225 km and what is the rate? 55 km/h
so plug into the formula
225= 55t
Now, how do we solve this one step problem?
divide both sides by 55
225/55 = 55t/55
do the division as a side bar
225/55 ≈ 4.09 so
t ≈ 4.1
and the answer is 4.1 hour
Monday, November 7, 2011
Algebra Honors (Period 6 & 7)
Using Several Methods of Factoring Section 5-11
1) Always factor the GCF first
2) look for he difference of 2 SQ's
3) Look for a Perfect SQ trinomial
4) If trinomial is NOT SQ look for a pair of factors
5) If 4 or more terms-- look for a way to group the terms into pairs or into a group of 3 terms that is a perfect SQ trinomial
6) Make sure each binomial or trinomial factor is PRIME
7) check your work
-4n4 + 40 n3 -100n2
GCF
-4n2(n2 -10n + 25)
-4n2(n-5)2
What about
5a3b2 + 3a4b - 2a2b3
Again factor the GCF
a2b(5ab + 3a2-2b2)
reorder this and use either XBOX or factor pairs to solve
a2b(3a2+5ab-22)
a2b(3a2+6ab-ab-2b2)
a2b[(3a2+6ab)+(-ab-2b2)]
a2b[3a(a+2b)-b(a + 2b)]
a2b(a+2b)(3a-b)
a2bc -4bc + a2 -4b
b(a2c-4c+a2-4)
b(a2c+a2-4c-4)
b[(a2c+a2) -(4c+4)]
b[a2(c + 1) -4(c +1)]
b(c+1)(a2-4)
but we aren't finished...
b(c+1)(a+2)(a-2)
6c2+18cd+12d2
6(c2+3cd+2d2)
6(c+2d)(c+d)
3xy2-27x3
3x(y2-9x2)
3x(y+3x)(y-3x)
-n4-3n2-2n3
-n2(n2+3+2n)
-n2(n2+2n+3)
Its factored completely!!
16x2+16y -y2-64
16x2-y2+16y-64
16x2-y2+16y-64
162-(y2-16y+64)
16x2-(y-8)2
becomes the difference of two squares
(4x+y-8)(4x-y+8)
x16 -1
(x8+1)(x8-1) =
(x8+1)(x4+1)(x4-1)=
(x8+1)(x4+1)(x2+1)(x2 -1) =
(x8+1)(x4+1)(x2+1)(x +1)(x-1)
2(a +2)2 + 5(a +2) - 3
Think of this as letting a+ 2 = x
2x2 +5x -3
Factoring that is easy
(2x-1)(x +3)
so substitute in a + 2 for each x
[2(a+2) -1]{a+2 +3]
2a + 4 -1)(a +5)
(2a +3)(a +5)
1) Always factor the GCF first
2) look for he difference of 2 SQ's
3) Look for a Perfect SQ trinomial
4) If trinomial is NOT SQ look for a pair of factors
5) If 4 or more terms-- look for a way to group the terms into pairs or into a group of 3 terms that is a perfect SQ trinomial
6) Make sure each binomial or trinomial factor is PRIME
7) check your work
-4n4 + 40 n3 -100n2
GCF
-4n2(n2 -10n + 25)
-4n2(n-5)2
What about
5a3b2 + 3a4b - 2a2b3
Again factor the GCF
a2b(5ab + 3a2-2b2)
reorder this and use either XBOX or factor pairs to solve
a2b(3a2+5ab-22)
a2b(3a2+6ab-ab-2b2)
a2b[(3a2+6ab)+(-ab-2b2)]
a2b[3a(a+2b)-b(a + 2b)]
a2b(a+2b)(3a-b)
a2bc -4bc + a2 -4b
b(a2c-4c+a2-4)
b(a2c+a2-4c-4)
b[(a2c+a2) -(4c+4)]
b[a2(c + 1) -4(c +1)]
b(c+1)(a2-4)
but we aren't finished...
b(c+1)(a+2)(a-2)
6c2+18cd+12d2
6(c2+3cd+2d2)
6(c+2d)(c+d)
3xy2-27x3
3x(y2-9x2)
3x(y+3x)(y-3x)
-n4-3n2-2n3
-n2(n2+3+2n)
-n2(n2+2n+3)
Its factored completely!!
16x2+16y -y2-64
16x2-y2+16y-64
16x2-y2+16y-64
162-(y2-16y+64)
16x2-(y-8)2
becomes the difference of two squares
(4x+y-8)(4x-y+8)
x16 -1
(x8+1)(x8-1) =
(x8+1)(x4+1)(x4-1)=
(x8+1)(x4+1)(x2+1)(x2 -1) =
(x8+1)(x4+1)(x2+1)(x +1)(x-1)
2(a +2)2 + 5(a +2) - 3
Think of this as letting a+ 2 = x
2x2 +5x -3
Factoring that is easy
(2x-1)(x +3)
so substitute in a + 2 for each x
[2(a+2) -1]{a+2 +3]
2a + 4 -1)(a +5)
(2a +3)(a +5)
Math 6 Honors ( Periods 1, 2, & 3)
Dividing Decimals 3-9
According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.
Check out our textbook for some examples!!
When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.
Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!
the quotient is 0.3431 which rounds to 0.343
To divide one decimal by another
Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number
Divide the new dividend by the new divisor
Check by multiplying the quotient and the divisor.
According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.
Check out our textbook for some examples!!
When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.
Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!
the quotient is 0.3431 which rounds to 0.343
To divide one decimal by another
Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number
Divide the new dividend by the new divisor
Check by multiplying the quotient and the divisor.
Thursday, November 3, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Multiplying Decimals 3-8
According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!
You do NOT need to line up the decimal point when you are multiplying.
14.92 x 7.2 stack them but do not line up the decimals
= 107.424
11.32 X 8.73
multiply carefully and you get
11.32
8.73
98.8236
estimate and you get 11 X 9 = 99
What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)
Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1
How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50
A jet flew 820.3 km/h for 3.2 hours. How far did the jet travel?
We have a great formula distance = rate X time
or d = rt
rate means the speed
so what do we have?
distance = (820.3)(3.2)
First I think I will estimate to make sure I am in the ballpark with my actual answer
I know that 820 (3) = 2460 so I know my answer will be close to 2460 miles-- a little bit more
so when I actually carefully multiply (820.3)(3.2) I get 2624.96 km
so I know my answer is reasonable
According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!
You do NOT need to line up the decimal point when you are multiplying.
14.92 x 7.2 stack them but do not line up the decimals
= 107.424
11.32 X 8.73
multiply carefully and you get
11.32
8.73
98.8236
estimate and you get 11 X 9 = 99
What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)
Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1
How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50
A jet flew 820.3 km/h for 3.2 hours. How far did the jet travel?
We have a great formula distance = rate X time
or d = rt
rate means the speed
so what do we have?
distance = (820.3)(3.2)
First I think I will estimate to make sure I am in the ballpark with my actual answer
I know that 820 (3) = 2460 so I know my answer will be close to 2460 miles-- a little bit more
so when I actually carefully multiply (820.3)(3.2) I get 2624.96 km
so I know my answer is reasonable
Wednesday, November 2, 2011
Algebra Honors (Period 6 & 7)
Factoring Pattern for ax2 + bx + c Section 5-9
When a > 1
We used a different method than what is taught in the book. I showed you X box
2x2 + 7x -9
Multiply the 2 and the 9
put eighteen in the box
Your controllers are
2x2 and -9
THen using a T chart find the factors of 19 such that the difference is 7x
we found that +9x and -2x worked
so
2x2 +9x -2x -9
Then separate them in groups of 2
such that
(2x2 +9x) + (-2x -9)
Then realize you can factor a - from the second pair
(2x2 +9x) - (2x + 9)
Then wht is the GCF in each of the hugs( )
x(2x +9) -1(2x +9)
look they both have 2x + 9
:)
(2x +9)(x-1)
But what if you said -2x + 9x instead to make the +7x in the middle
Look what happens
(2x2 -2x) + (9x -9)
now, factor te GCF of each
2x(x -1) + 9(x -1)
now they both have x -1
(x-1)(2x +9)
SAME RESULTS!!
14x2 -17x +5
remember the second sign tells us that the numbers are the same and the first sign tells us that they are BOTH negative
create your X BOX with the product of 14 and 5 in it
70
Place your controllers on either side
14x2 and + 5
Now do your T Chart for 70
You will need two numbers whose product is 70 and whose sum is 17
that's 7 and 10
14x2 -7x -10x + 5
Now group in pairs
(14x2 -7x) + (-10x + 5)
which becomes
(14x2 -7x) - (10x - 5)
FACTOR each
7x(2x -1) - 5(2x-1)
(2x-1)(7x-5)
10 + 11x - 6x 2
sometimes its better to arrange by decreasing degree so this becomes
- 6x 2 +11x + 10
now factor out the -1 from each terms
- (6x 2 - 11x - 10)
Se up your X BOX with the product of your two controllers :)
60 We discover that +4x and -15x are the two factors
-1(6x 2 +4x - 15x - 10)
-1[(6x 2 +4x) + (- 15x - 10)]
-1[6x 2 +4x) - (15x +10)
-1[2x(3x +2) -5(3x+2)]
-(3x+2)(2x-5)
If you had worked it out as
10 + 11x -6x2 you would have ended up factoring
(5 -2x)(2 + 3x)
and we all know that
5 -2x = -(2x-5) Right ?
Next, we looked at the book and the example of
5a2 -ab - 22b2
We discussed the books instructions to test the possibilities and decided that the X BOX method was much better.... I need to check out hotmath.com... did you????
5a2 -ab - 22b2 Using X BOX method we have 110 in the box and the controllers are
5a2 and - 22b2
What two factors will multiply to 110 but have the difference -1?
Why 10 and 11
5a2 +10ab -11ab - 22b2
separate and we get
(5a2 +10ab) + (-11ab - 22b2)
( 5a2 +10ab) - (11ab + 22b2)
5a(a + 2b) -11b(a + 2b)
(a + 2b)(5a - 11b)
Factoring by Grouping 5-10
5(a -3) - 2a (3 -a)
a-3 and 3-a are OPPOSITES
so we could write 3-a as -(-3 +a) or -(a -3)
sp we have
5(a-3) -2a [-(a-3)]
which is really
5(a-3) + 2a(a-3)
wait... look... OMG they both have a-3
so
(a-3)(5 + 2a)
What about
2ab-6ac + 3b -9c
What can you combine...
some saw the following:
(2ab -6ac) + 3b -9c)
then
2a(b-3c) + 3( b-3c)
(b -3c)(2a + 3)
BUT others look at 2ab-6ac + 3b -9c and saw
2ab +3b -6ac -9c
which lead them to
(2ab + 3b) + (-6ac -9c)
b(2a +3) -3c(2a +3)
(2a +3)(b-3c)
wait that's the same!!
Hooray
What about 4p2 -4q2 +4qr -r2
First look carefully and you will see
4p2 -4q2 +4qr -r2
That's a trinomial square OMG
so isn't that
4p2 - ( 2q -r)2
BUT WAIT look at
4p2 - ( 2q -r)2 That's the
Difference of Two Squares
Which becomes
(2p + 2q -r)(2p -2q +r)
When a > 1
We used a different method than what is taught in the book. I showed you X box
2x2 + 7x -9
Multiply the 2 and the 9
put eighteen in the box
Your controllers are
2x2 and -9
THen using a T chart find the factors of 19 such that the difference is 7x
we found that +9x and -2x worked
so
2x2 +9x -2x -9
Then separate them in groups of 2
such that
(2x2 +9x) + (-2x -9)
Then realize you can factor a - from the second pair
(2x2 +9x) - (2x + 9)
Then wht is the GCF in each of the hugs( )
x(2x +9) -1(2x +9)
look they both have 2x + 9
:)
(2x +9)(x-1)
But what if you said -2x + 9x instead to make the +7x in the middle
Look what happens
(2x2 -2x) + (9x -9)
now, factor te GCF of each
2x(x -1) + 9(x -1)
now they both have x -1
(x-1)(2x +9)
SAME RESULTS!!
14x2 -17x +5
remember the second sign tells us that the numbers are the same and the first sign tells us that they are BOTH negative
create your X BOX with the product of 14 and 5 in it
70
Place your controllers on either side
14x2 and + 5
Now do your T Chart for 70
You will need two numbers whose product is 70 and whose sum is 17
that's 7 and 10
14x2 -7x -10x + 5
Now group in pairs
(14x2 -7x) + (-10x + 5)
which becomes
(14x2 -7x) - (10x - 5)
FACTOR each
7x(2x -1) - 5(2x-1)
(2x-1)(7x-5)
10 + 11x - 6x 2
sometimes its better to arrange by decreasing degree so this becomes
- 6x 2 +11x + 10
now factor out the -1 from each terms
- (6x 2 - 11x - 10)
Se up your X BOX with the product of your two controllers :)
60 We discover that +4x and -15x are the two factors
-1(6x 2 +4x - 15x - 10)
-1[(6x 2 +4x) + (- 15x - 10)]
-1[6x 2 +4x) - (15x +10)
-1[2x(3x +2) -5(3x+2)]
-(3x+2)(2x-5)
If you had worked it out as
10 + 11x -6x2 you would have ended up factoring
(5 -2x)(2 + 3x)
and we all know that
5 -2x = -(2x-5) Right ?
Next, we looked at the book and the example of
5a2 -ab - 22b2
We discussed the books instructions to test the possibilities and decided that the X BOX method was much better.... I need to check out hotmath.com... did you????
5a2 -ab - 22b2 Using X BOX method we have 110 in the box and the controllers are
5a2 and - 22b2
What two factors will multiply to 110 but have the difference -1?
Why 10 and 11
5a2 +10ab -11ab - 22b2
separate and we get
(5a2 +10ab) + (-11ab - 22b2)
( 5a2 +10ab) - (11ab + 22b2)
5a(a + 2b) -11b(a + 2b)
(a + 2b)(5a - 11b)
Factoring by Grouping 5-10
5(a -3) - 2a (3 -a)
a-3 and 3-a are OPPOSITES
so we could write 3-a as -(-3 +a) or -(a -3)
sp we have
5(a-3) -2a [-(a-3)]
which is really
5(a-3) + 2a(a-3)
wait... look... OMG they both have a-3
so
(a-3)(5 + 2a)
What about
2ab-6ac + 3b -9c
What can you combine...
some saw the following:
(2ab -6ac) + 3b -9c)
then
2a(b-3c) + 3( b-3c)
(b -3c)(2a + 3)
BUT others look at 2ab-6ac + 3b -9c and saw
2ab +3b -6ac -9c
which lead them to
(2ab + 3b) + (-6ac -9c)
b(2a +3) -3c(2a +3)
(2a +3)(b-3c)
wait that's the same!!
Hooray
What about 4p2 -4q2 +4qr -r2
First look carefully and you will see
4p2 -4q2 +4qr -r2
That's a trinomial square OMG
so isn't that
4p2 - ( 2q -r)2
BUT WAIT look at
4p2 - ( 2q -r)2 That's the
Difference of Two Squares
Which becomes
(2p + 2q -r)(2p -2q +r)
Math 6 Honors ( Periods 1, 2, & 3)
Multiplying or Dividing by a Power of Ten 3-7
We have learned that in a decimal or a whole number each place value is ten times the place value to its right.
10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000
10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01
Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.
Multiplying by ten moves the decimal point one place to the right
10 ∙ 762 = 7620
762 X 10 = 7620
4931 X 10 = 49,310
104 = 10⋅10⋅10⋅10 = 10,000
2.63874 X 104 = 26,387.4
To multiply a number by the nth power of ten--> move the decimal n places to the right.
0.0047 multiply by 100 = 0.47
0.0047 multiply by 1000 = 4.7
3.1 ÷ 104 = 0.00031
10 ∙ 4.931 = 49.31
At the beginning of this chapter you learned about powers of ten
104 = 10 ∙10 ∙ 10 ∙10 = 10,000
We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.
2.64874 ∙104 = 26,387.4
Notice that we have moved the decimal point four places to the right.
Rule
To multiply a number by the nth power of ten, move the decimal point n places to the right.
When we move a decimal point to the left, we are actually dividing by a power of ten.
Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros
Rule
To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.
2386 ÷ 103 = 2.386
Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors
(1) a number greater than or equal to 1, but less than 10,
AND
(2) a power of ten
are said to be written in scientific notation.
We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 109 to express the number in scientific notation
4.592 x 109 (Yes, you get to use the × symbol for multiplication .. but only for this!!
Way to write very large numbers AND very small numbers
Numbers expressed as products of a number greater than or equal to 1 BUT less than 10, AND a power of ten are called Scientific Notation.
Two Factors
91) 1≤ n < 10 (2) Power of 10 4,592,000,000 becomes 4.592 X 109
moved the decimal 9 places so we must multiply our number by a power of 109
98,000,000 = 9.8 X 107
320,000 = 3.2 X 105
What if I give you 7.04 X 108 and ask you to put it back into STANDARD NOTATION:
704,000,000.
0.0031 = 3.1 X 10-3
It isn't a negative number its just a very tiny number
1≤ n < 10 0.16 becomes 1.6 x 10 -1
We have learned that in a decimal or a whole number each place value is ten times the place value to its right.
10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000
10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01
Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.
Multiplying by ten moves the decimal point one place to the right
10 ∙ 762 = 7620
762 X 10 = 7620
4931 X 10 = 49,310
104 = 10⋅10⋅10⋅10 = 10,000
2.63874 X 104 = 26,387.4
To multiply a number by the nth power of ten--> move the decimal n places to the right.
0.0047 multiply by 100 = 0.47
0.0047 multiply by 1000 = 4.7
3.1 ÷ 104 = 0.00031
10 ∙ 4.931 = 49.31
At the beginning of this chapter you learned about powers of ten
104 = 10 ∙10 ∙ 10 ∙10 = 10,000
We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.
2.64874 ∙104 = 26,387.4
Notice that we have moved the decimal point four places to the right.
Rule
To multiply a number by the nth power of ten, move the decimal point n places to the right.
When we move a decimal point to the left, we are actually dividing by a power of ten.
Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros
Rule
To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.
2386 ÷ 103 = 2.386
Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors
(1) a number greater than or equal to 1, but less than 10,
AND
(2) a power of ten
are said to be written in scientific notation.
We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 109 to express the number in scientific notation
4.592 x 109 (Yes, you get to use the × symbol for multiplication .. but only for this!!
Way to write very large numbers AND very small numbers
Numbers expressed as products of a number greater than or equal to 1 BUT less than 10, AND a power of ten are called Scientific Notation.
Two Factors
91) 1≤ n < 10 (2) Power of 10 4,592,000,000 becomes 4.592 X 109
moved the decimal 9 places so we must multiply our number by a power of 109
98,000,000 = 9.8 X 107
320,000 = 3.2 X 105
What if I give you 7.04 X 108 and ask you to put it back into STANDARD NOTATION:
704,000,000.
0.0031 = 3.1 X 10-3
It isn't a negative number its just a very tiny number
1≤ n < 10 0.16 becomes 1.6 x 10 -1
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