Fractions, decimals and percents.Prime factorization and all factors.

Decimals, fractions and percents

There are three standard ways to represent rational numbers, fractions, decimals and percents. Many people are most comfortable with decimals because that is the way a calculator displays a number, but all three of the methods have their advantages and disadvantages.

Example: The number .735 is a decimal.  It is common to call this "point seven three five", but its proper name would be "seven hundred thirty five thousandths". That is because it has three places after the decimal, which is the thousandth place.

Easy rule: how many ever places there are after the decimal, write a fraction, "1" over "1" followed by that many "0".

One place after the decimal corresponds to 1/10 or one tenth.
Two places after the decimal corresponds to 1/100 or one hundredth.
Three places after the decimal corresponds to 1/1000 or one thousandth.


Following this rule, .735 = 735/1000.  You should be able to see that both of these numbers are divisible by 5, so this is not in lowest terms. 735/1000 = 147/200, and this is in lowest terms.  (If you have the TI-30XIIs, you can type 735 [A b/c] 1000 [enter], and the answer will be 147/200.)

The word "per" is from Latin and means "out of". It always represents division. "Percent" is "out of 100". The decimal .735 is equal to 73.5%.  Technically we multiply .735 by 100 to get 73.5, then by putting the percent sign at the back we have divided by 100, so the number is the same, only the representation has changed.  The mechanical rule we are usually taught is.

Decimal to percent:  Move the decimal point two places to the right and put a percent sign behind it.
Percent to decimal:   Move the decimal point two places to the left and remove the percent sign.

That means 33% = .33 and 3% = .03.  As fractions .33 =33/100 and .03 = 3/100.

Prime factorization and all factors of a number

A factor of a number n is a number that divides into n evenly.

Example: 4 is a factor of 12 because 12/4 = 3, which can also be stated as 4x3 = 12.
Counterexample: 5 is not a factor of 12 because 12/5 = 2.4, a decimal and not a whole number.

Every number has a prime factorization, which means we break it down to a product of prime numbers. A prime can only be divided evenly by 1 and itself. (By definition, 1 is not a prime. It is called a unit.)

Here is a list of the primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ...
There are infinitely many primes, so this list goes on forever.

Find the prime factorization of 45.
Step 1: Find two numbers other than 1 and 45 that multiply to 45.  An easy pair is 5x9.
Step 2: 5 is prime, but 9 is not. That means we can find a pair of numbers (not 1 and 9) that multiply to 9, in this case 3x3. 3 is prime, so 45 = 5x3x3 is the prime factorization of 45.

Find all factors of 45.
When making a list of all factors, it makes sense to write them in pairs. I recommend starting with 1 and the number itself.

1 and 45

Now find the next biggest number after 1 that divides evenly into 45.  2 won't work because 45/2 = 22.5.  3 does work, because 45/3 = 15.

3 and 15

4 doesn't work but 5 does.

5 and 9.

You can try 6, 7 and 8 if you like, but none of them work.  9 works, but it's already on our list, so we can stop.

All the factors of 45 in order are 1, 3, 5, 9, 15 and 45.

Practice

a) Change .09 into a percent.
b) Change .09 into a fraction in lowest terms.
c) Change 24% into a decimal.
d) Change 24% into a fraction in lowest terms.
e) Change 3/8 into a decimal.
f) Change 3/8 into a percent.

Answers in the comments.


For practice on factorization, click on this link.

Practice problems for first homework of Summer 2012

Change these numbers in Hindu-Arabic to Roman numerals.

648 = ______

15,781 = _____

Change these numbers in Roman numerals to Hindu-Arabic.
__
IXCDXLVII = ____

DCXCIV = ____

Find the approximation of 7/37 to the following number of decimal places and find the exact repeating decimal representation.

7/37 to the nearest tenth = _____
7/37 to the nearest hundredth = _____
7/37 to the nearest thousandth = ____

Repeating decimal representation = _____

Find the prime factorization of 48.

Find all the factors of 48.

Add or subtract these times, given in minutes and seconds.  If the minutes are 60 or more, change the answer to hours:minutes:seconds.

27:25 + 18:58 + 21:44 = __________

17:21 - 9:46 = __________

Answers in the comments.

Practice problems for homework 1 of Summer 2011.

Change these numbers in Hindu-Arabic to Roman numerals.

648 = ______

15,781 = _____

Change these numbers in Roman numerals to Hindu-Arabic.
__
IXCDXLVII = ____

DCXCIV = ____

Find the approximation of 7/37 to the following number of decimal places and find the exact repeating decimal representation.

7/37 to the nearest tenth = _____
7/37 to the nearest hundredth = _____
7/37 to the nearest thousandth = ____

Repeating decimal representation = _____

Find the prime factorization of 48.

Find all the factors of 48.


Answers in the comments.

Practice problems for prime factorization.

Here's the link to problems with answers. If you need more, let me know.

Representing numbers by prime factorization

I'm going to be putting previews of notes for class before the class, then fleshing in the details and possibly adding more or taking away topics from the list, depending on how far we get.


Prime numbers, composite numbers and the number 1 (unit)
prime factorization
relatively prime numbers
shortcuts for divisibility
reducing fractions
greatest common divisor (g.c.d.) and least common multiple (l.c.m.) for two numbers

These topics are covered on pages 23 through 28 in the notes.