Chapter 2 Rational Numbers

Terms to Know

• Divisible: a number is divisible by a second number if the number can be divided by the second number with a remainder of 0– 10 is divisible by 5– 5 is not divisible by 10

Terms to Know

• Factor: an integer that divides another integer with a remainder of 0– 5 is a factor of 10– 10 is not a factor of 5

– 10 is divisible by 5, so 5 is a factor of 10

Terms to Know

• Prime Number: A whole number greater then 1, with exactly 2 factors, 1 and itself

• Are the following numbers prime? (your divisibility rules can help you)– 1– 37– 20724

Terms to Know

• Composite Number: A whole number greater than 1, with more than two factors

• Are the following composite numbers?(use your divisibility rules to help you)– 1– 37– 2724

Explaining why a number is prim or composite

• Is 14582 prime? Explain.

• NO

• 14582 is not prime because it is divisible by 2. The number 14582 is composite.

Explaining why a number is prim or composite

• Is 29 prime?

• YES

• It can only be divided by 1 and 29, therefor the number 29 is prime

Prime Factorization

• Prime factorization: A composite number written as a product of prime numbers

• To find prime factorization use a factor tree.

Write the prime factorization of 54

54

2 27

3 9

3 3

Prime Factorization :2·3·3·3

OR2·33

NOTE: When solved the prime factorization equals 54, the original number

Finding the GCF

• Greatest Common Factor (GCF): The GCF of two or more numbers is the greatest number that is a factor of all of the numbers

• Two ways of doing this…– List the factors (works great for smaller numbers)– Prime Factorization (Makes the hard numbers

easier to deal with)

Finding GCF by Listing

Find the GCF of 54 and 63

54:

63:

Greatest factor both numbers share is 9, therefor 9 is the GCF

1 542 276 9

1 633 217 9

Greatest factor both numbers share is 9, therefor 9 is the GCF

Find the GCF through Prime Factorization

Find the GCF of 54 and 6354

2 27

3 9

3 3

63

9 7

3 3

54 = 2 · 3 · 3 · 363 = 3 · 3 · 7

Both P.F. share a 3 and another 3, multiply these together = 9

9 is the GCF of 54 and 63

If you struggle with these topics view your work on page 55

Rational number

Any number that can be written as a fractionAre the following rational numbers?

5

36.3

0

9 87.0

9

52

Between Fractions

10

9

440

436

40

36

40

36

410

49

10

9

Simplifying Fractions

Fraction to Decimal

3612512536125

36

The fraction bar is just a division sign

288.0000.36125

125

36

Decimal to Fraction

1000

1345

10001

1000345.1

1

345.1345.1

Don’t forget to simplify!

200

691

200

269

51000

51345

1000

1345

If you are struggling with this look at your work from page 59

Comparing Fractions

9

412

5

45 48

12

5

9

44845

Comparing Decimals

• Line up the decimal points (you need to compare the tens place with other tens places etc.)

• Digits to the left have greater value

2.3452.4351.3452.34

Using these two methods and your know how to change fractions to decimals and vise versa, you

can order larger groups of numbers

If you struggle with this subject review your work on page 64

Adding and Subtracting Rational Numbers

1) Make all mixed numbers improper fractions2) Find common denominators3) Add or subtract the numerators4) Keep the denominator5) Simplify

4

11

3

11

4

32

3

23

4

32

3

23

12

44

43

411

3

11

12

33

34

311

4

11

12

77

12

33

12

44

2

13

and 4

641.612

56

12

775

If you struggle on this topic refer to your work from page 68

Multiplying Rational Numbers

1)Make into improper fractions2) Pre-cancel if you can3) Multiply numerators4) Multiply denominators5) Simplify

7

21

5

42

7

9

5

14

7

21

5

42 1

1

9

5

7

7

9

5

14

7

1

2

5

16

1

9

5

7

3And

4

5

13

5

165

Dividing Fractions

1) Make into improper factions2) Multiply by the reciprocal3) Simplify

7

21

5

42

7

9

5

14

7

21

5

42 1

45

98

9

7

5

14

• First fraction stays the same• Division become Multiplication• Second fraction is flipped• See Multiplying Fractions

2

45

82

45

983

If you struggle with this refer to your work on page 75

Formulas

You do not need to memorize formulas refer to page 648 in the text

Formula: A rule that shows the relationship between two or more quantities

It’s an expression you are given to calculate information about certain scenarios. Just plug in and chug out.

4 steps

1. Choose the formula2. List your known and unknown variables3. Substitute values into the formula4. Solve the formula based on it’s rules

1. Choose the formula

• Make sure you know what the question is asking for and your formula can solve for it.

• If I want to find the area for a square, a formula that solves for volume is pretty useless

• Copy the formula down and give yourself plenty of room to work

2 list your variable

• Read the question carefully and list what you know and what you need to find

)(2

121 bbhA

cmb

cmb

cmh

A

6

5

4

?

2

1

NOTE: ½ is a constant. It does not change and is part of the rule of the formula

I picked random numbers to use as an example. Read carefully, your problems will tell you what goes where. If you have more then 1 unknown, you can not solve.

3 Substitute into the formula

6

5

4

?

2

1

b

b

h

A )(2

121 bbhA

)65)(4(2

1A

NOTE: Use parentheses to keep work neat and clear

NOTE: only replace the variables, keep the rules the same

5 Solve using the rules of the formula

222

)44(2

1

)11)(4(2

1

)65)(4(2

1

cmA

A

A

A

NOTE: Formulas will have units of measure associated with them. Pay attention and include them with your answer

Area will be squared unitsVolume will be cubed units

Isolating Variables

You pretend you know the other values and solve for the unknown you are asked to isolate

Many times you don’t need to do this, you can just use algebra after you substitute in.

How to isolate a variable

• Determine which variable you want to isolate• Ask yourself what operation is going on

(Adding, subtracting, multiplying, dividing, etc.)

• Do the opposite of that operation to both sides of the equal sign. This should cancel out the unwanted variables attached to the one you want to isolate

Isolate the in the following formulal

lw

Aw

wl

w

A

wlA

You are multiplying by (w), so divide each side of the equal sign by (w)

W’s cancel

If you are struggling on this topic review your work from page 83

Powers and Exponents

42Exponent

Base

Exponents

• Be careful with you signs!

625)5555(5

625)5)(5)(5)(5()5(

4

4

If you are struggling with this section refer to your work on page 88

Scientific Notation

• Makes large numbers smaller and easier to work with

• Also some calculators can only show so many numbers and they will make use of this with EE instead of a X10

Scientific to Standard Notation

• You are only moving decimal points and filling in empty stops with zeros

• Positive powers of 10 move to the right• Negative powers of 10 move to the left

Power of 4, move decimal 4 places to the right

.246001046.2 4

000246.01046.2 4

Power of -4, move decimal 4 places to the left

Standard to Scientific Notation

• First number is always between 1 and 10!• Count the number of places you need to move

the decimal to make a number between 1 and 10

• The number of places you moved the decimal is your exponent

• You are always multiplying by a power of 10

1310353.100001353000000 Needed to move the decimal 13 places to the left

so you multiply by 1013

1110353.113530000000000.0 Needed to move the decimal 11 places to the

right so you multiply by 10-11

NOTE: If your number is less then 1 in standard notation, you will have a negative exponent in scientific notation

If you are struggling on this topic refer to your work on page 94