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NUMBERS
----foundation of Mathematic
What we should do today:
 Students will be given a video which contains all the things in this section
we need to learn. While watching it, please not just laugh and play with
others, you should use information you get from the video to complete this
work sheet and hand it back at the end of this class.
 Students will learn something about numbers in chapter 3 at this class,
involving number sets, prime factors, greatest common factor(GCF), least
common factor(LCM), square root and cube root.
Number Sets
To recognize each kind of numbers and express it easily and clearly, mathematicians
make ________ _____ sets to do this job.
Numbers are divided up into several different sets:
 ________ ________ (N)
 _____________
 Often called “counting numbers” because we always use them to count.

________ ________ (W)
 _____________
 Includes all of the natural number set and has an extra Zero.

_________ (I or Z)
 _____________ or can be written as _____________
 Includes all of the natural number and whole number sets and has extra
negative numbers.

________ ________ (Q)
 _____________
 Any number that can be written as a ratio (fraction)
 Terminating decimal numbers, repeating decimal numbers
 Includes all of the natural, whole, and integer number sets. .

________ ________ (Q’)
 _____________
 Any non-terminating, non-repeating decimal number

________ ________ (R)
 All of the above number sets.
Examples for you to do:
1) Find out the irrational numbers among following:
1, 0, 1.365756, 3/5, 0.1235813…,
2) Determine the number sets of each number showing below:
a. 0
b. 3.6
c. 3/4
d. 5
e. 1.142857142857…
f. 1.23456789…
Factors , Prime, and Prime Factors
Every _______ _________ has at least one ________. The ________ means the
whole number elements of one whole number, which can _________ together to it.
Example: 3 and 4 are factors of 12 for 3*4=12
Some whole numbers greater than one have only two distinct factors, which are one
and itself, we call them _______ ________.
Example: 7 is a prime number because its only factors are 1 and 7.
Numbers greater than one that aren’t prime numbers are called __________ _______.
When the factors of a number a number are also prime, they are called _____ ______.
Example: 8 has prime factors of 2*2*2 for 2 is prime. We call this the _____
_____________ of 8.
The number 1 is not a prime number because it isn’t _________ by any whole
numbers other than itself.
The number 0 is not prime because it doesn’t have two ________ factors.
Factor trees is a ______ used to write the _____ _____________ of a whole number.
36
6
2
6
3
3
Examples:
1) Write down all the factors of 24
1, 2, 3, 4, 6, 8, 12, 24.
2) Factor the number 18 in two different ways.
2
18 = 2*9
18 = 3*6
3) Write four examples of prime numbers.
23, 31, 53
4) Factor 54 into prime factors.
54 = 2 * 3^3
Examples for you to do:
1) Write down all the factors of 72 and factor it in three different ways, where
must contain a prime factorization.
2) Write all the prime factors which are bigger than 20 and lower than 100.
GCF and LCM
The greatest common factor of two or more whole numbers is the _______ whole
number that is a ______ of two or more numbers.
Example: The GCF of 16 and 36 is 4.
Techniques to finding the GCF of different numbers:
1) List all the factors of the numbers, choose the largest factors shared by them.
2) Factor the numbers into produces of power of prime factors. The GCF is the
product of common powers with the smallest exponents associated with each
other.
3) Divide the numbers by common prime factors until all the quotients do not
have a common prime factor. In the method shown, the quotients are written
under the dividends (divided numbers).
Examples:
1) Determine the GCF of 18 and 24
18: 1, 2, 3, 6, 9, 18.
24: 1, 2, 3, 4, 6, 8, 12, 24.
So the GCF is 6.
2) Determine the GCF of 24, 36, and 64
24= 2^3 * 3
36= 2^2 * 3^2
64= 2^6
So the GCF is 2^2 equals 4.
3) Determine the GCF of 12, 18, and 27
3
12
18
27
4
6
9
So the GCF is 3.
If numbers don’t have common prime factor, the product of common prim factors
which are written in the ____ ______ is the GCF.
Examples for us to do:
1) Determine the GCF of 15 and 9, use 3 different ways.
2) Determine the GCF of 12, 8, and 36.
3) Determine the GCF of 12, 8, and 9.
The least common multiple of two or more whole number is the ________ whole
number that is a ______ of two or more whole numbers.
Example: The LCM of 6 and 8 is 24
Techniques to finding the LCM of different numbers:
1) List the multiples of numbers, until a common multiple is found.
2) Factor each number into products of powers of prime factors. The LCM is the
product of the common powers that have the largest exponents associated
with them, along with any non-common powers.
3) Use the similar division techniques as we used for the GCF, but keep dividing
until none of the numbers in a row have a common prime factor. The LCM is
the product of the left column and the button row.
Examples:
1) Determine the LCM of 4 and 6
4: 4, 8, 12, 16, 24, 36
8: 8, 16, 24, 32, 40, 48
So the LCM is 24
2) Determine the LCM of 6, 8, and 18
6= 2 * 3
8= 2^3
18= 2 * 3 ^2
So the LCM is 2^3 * 3^2 equals 72
3) Determine the LCM of 9, 16, and 18
3
2
3
9
3
3
1
16
16
8
8
18
6
3
1
So the LCM is 2*3*3*8=145
Examples for us to do:
1) Determine the LCM of 12 and 14, use 3 different ways.
2) Determine the LCM of 4, 6, and 15.
Perfect Square and Perfect Cube
For whole numbers, the word _______ in the term perfect ______ means that the
square can be written as a product of ___ __________ whole numbers.
Similarly, a perfect ____ can be written as a product of _____ __________ whole
numbers.
Also, we can think a prefect square as the ____ of a square and a perfect cube as the
______ of a cube with whole number dimensions in geometry.
We can use square tiles and small cubes to determine or show that whether a number
is a perfect square or cube or not.
Examples:
1) Determine if the number 14 is a perfect square.
Solution: 3^2=9<14, 4^2=16>14, so 14 isn’t a perfect square.
2) Deter mine if the number 64 is a perfect cube.
Solution: 3^3=27<64, 4^3=64, so 64 is a perfect cube.
Examples for us to do:
1) Determine the number 16 is a perfect square or not, use square tiles.
2) Determine the number 16 is a perfect cube or not, use small cubes.
Square Roots and Cube Roots
The ______ ____ of a number is the two _________ numbers which have the product
of it, and the ____ ____ of a number is the three ________ numbers which have the
product of it.
The symbol that we use to show the operation of taking the positive square root is __,
and the symbol for cube root is __.
The symbol __ represents the positive square root and the negative one is represented
by the symbol __. So the symbol for both positive and negative square roots is ____.
The square root of a negative number is not a ____ ______. However, cube roots have
no these problem.
Chart of some Perfect Squares and Perfect Cubes needs to know
1=1^2
25=5^2
81=9^2
8=2^3
4=2^2
36=6^2
100=10^2
27=3^3
9=3^2
49=7^2
121=11^2
125=5^3
16=4^2
64=8^2
144=12^2
1000=10^3
Examples:
1) Determine the positive square root of 576
Solution: 20^2=400, 30^2=900, so the positive square root x must be between
20 and 30, and closer to 20. Since 4^2=16, 6^2=36, so x is 24.
2) Determine the cube root of 6869
Solution: 10^3=1000, 20^3=8000, so the cube root y must be between 10 and
20, and closer to 20. Since 9^3=9, so x is 19.
Examples for us to do:
1) Determine the negative square root of 900
2) Determine the cube root of 216
3) Determine the cube root of 2744