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Arithmetic (The basic operations and what order to do them)
Prealgebra (Introduction to symbols and expressions)
Beginning Algebra (Simplifying, solving, and graphing)

Arithmetic [ top of page ] Click & choose a topic or just scroll & learn!

The Basic Operations [ top of page ] . . . 8/97 , revised 7/98

There are four basic operations:

+ (addition) . . . . . .as in 6 + 2 = 8; the sum of 6 and 2 is 8.
- (subtraction) . . . .as in 6 - 2 = 4; the difference of 6 and 2 is 4.
x (multiplication) . as in 6 x 2 = 12; the product of 6 and 2 is 12.
/ (division) . . . . . .as in 6 / 2 = 3; the quotient of 6 and 2 is 3.

Also you use parentheses ( ) for grouping and sometimes multiplication.
Examples: 20 - 12 - 7 = 8 - 7 = 1 . . while 20 - (12 - 7) = 20 - 5 = 15
. . . . also , (3 + 4)(6 - 2) = (7)(4) = 7 x 4 = 28 ,
. . . . while 3 + (4)(6) - 2 = 3 + 24 - 2 = 27 - 2 = 25 . (See order of oper's below.)

Exponents [ top of page ] . . . 8/97

Oh, there's one more: ^ (exponentiation), so make that five operations.

This last operation would give 6 ^ 2. This means "6 to the power 2", or in other words:

6 ^ 2 = 6 x 6 = 36. (Remember, it's not just 6 x 2.)
The 6 is the base and the 2 is the exponent.
This is also called "6 squared" ; it's the area of a square of side 6. (See square roots.)

Another example would be 3 ^ 4 = 3 x 3 x 3 x 3 = 9 x 9 = 81.

Example: Which is bigger, 4 ^ 5 or 5 ^ 4 ?
4 ^ 5 = 4 x 4 x 4 x 4 x 4 = 16 x 16 x 4 = 256 x 4 = 1024, while
5 ^ 4 = 5 x 5 x 5 x 5 = 25 x 25 = 625.
To answer the question, 4 ^ 5 is bigger.
As a rule, the smaller number to the bigger power often (but not always) comes out bigger.

One important use of exponents is to express really large (or really small) numbers:
This is called scientific notation and uses powers of ten:

Example: 4560 = 4.56 * 10^3 and 0.00003802 = 3.802 * 10^(-5)

[ arithmetic | top of page ]

Order of Operations [ top of page ] . . . 8/97 , revised 7/98

To do the operations in the right order, remember PEMDAS, which stands for:

P . . . .Start by working inside parentheses, innermost first.
E . . . .Simplify any exponent expression next.
M-D . Then work all mults and divs, from left to right, as they appear.
A-S . . Finally work all add'ns and subtr's from left to right.

In California we say Powerful Earthquakes May Deliver After-Shocks.
(Speaking of powerful, someone e-mailed me and said this section gave him confidence and changed his life!)

Example: What is 2 + 3 x 4 ?
If either answer is 20, then think again (and get a better calculator!).
Don't do the 2 + 3 (addition) until all mults are done:
2 + 3 x 4 = 2 + 12 = 14 (the answer!)
Question: What's wrong with doing 2 + 3 = 5, then 5 x 4 = 20?
Some calculators will give you 20; throw them out or send them back to the factory, then get a "scientific" calulator (about \$9).

Example: What about PEMDAS in 3 + 4 x 6 ­ 2 from the previous section?
Well, again the multiplication has been done first; 4 x 6 = 24. Answer = 25.

Example: Ok, what about 3 x 4 ^ 2 ? Is that the same as 12 ^ 2 ? (^2 means squared, or times itself.)
Answer: 4 ^ 2 is an exponent prob (meaning 4 x 4, or 16), so by PEMDAS, do it before multip.
3 x 4 ^ 2 = 3 * 16 = 48. But if you want to do 12 ^ 2 , you have to write (3 x 4) ^ 2 .

Prime Numbers and Factorizations [ top of page ] . . . (8/97)

Primes are a lot of fun for me; they're the "building blocks" of the natural numbers!
First things first: 1 (one) is NOT a prime. Well then, what is?

Definition: A prime number is a natural number with exactly two divisors.
This excludes 1, since it has only itself as a divisor.

Every whole number starting with 2 can be written as a product of primes; for example 10 = 2 x 5 or 2 * 5, while 36 = 2 * 2 * 3 * 3. This is called the prime factorization of the number, and is what gives each number its own individual character, or "DNA sequence," if you will.

With the help of exponents, we can write: 36 = 2^2 * 3^2, or
1 million = 10^6 = (2 * 5)^6 = 2^6 * 5^6. Are you still with me?

How can you tell if a number is prime? . . . (11/97)

Let's try 101. We can try 2, see if it goes into 101 (it doesn't), then 3 (it doesn't either), etc. We don't need to try 4 because 2 didn't go in, so we only need to try dividing in primes. We quickly see that 2, 3, 5, and 7 don't go into 101. The next prime is 11; but 11 x 11 = 121, more than 101. If anything goes into 101, it must be less than 11. But we tried all that stuff before. So 101 is prime!

What about 1001? Try 2, 3, and 5; no dice.
But 7 goes in; 1001 / 7 = 143.
Well, 7 won't go into 143, but 11 will: 143 / 11 = 13.
So 1001 = 7 x 11 x 13. (The prime factorization.)

There are 25 primes under 100. Can you find them all? Does this mean there are 250 primes less than 1000? (Actually there are only 168.)

For more advanced topics go to the Number Theory page.

[ arithmetic | top of page ]

Fractions -- by request! (7/00)

Fractions are another way of expressing division. The expression 12/3 is equal to 4 because 12 divided by 3 is 4. If you don't believe me, check that 3 * 4 = 12; that's an equivalent statement.

Now you have your proper fractions, where the numerator (top) is smaller than the denominator (bottom), like 5/12. These give numbers that are less than 1.

On the other hand, there are improper fractions like 19/8. Think of pizzas each cut into eight slices; each slice is one eighth, or 1/8 of a pizza. Then 19 slices would be the same as 16 slices and 3 more slices; making 2 pizzas and 3 extra slices.
Therefore we get 19 / 8 = 2 + 3/8 , which is written as 2 3/8 , called a mixed number.

To convert 19/8 you do division; 8 into 19 goes 2 times with 3 left over; so 2 3/8.

If you have a mixed number like 3 1/7 , you do the reverse: 3 * 7 + 1 = 22 ; so 22/7.

Some misguided people (not you!) think that pi is equal to 3 1/7; it's merely close. Pi is an irrational number , which means it's a real number, but not equal to any fraction. (But 355/113 is closer to pi.)

Fractions can be put into lowest terms, meaning you cancel out common factors of the top & bottom. For example, 6 / 8 = (2*3) / (2*4) = 3 / 4 ; six eighths equals three fourths.
(Think of eating six slices of the aforementioned pizza! Hungry yet?)

Now I spoze you want to know all about doing operations like multiplication and addition on any fractions. I'll give you a couple of examples but for the full treatment you should get my textbook: Prealgebra: Mathematics for a Variable World, by Bach & Leitner. (It's published by Houghton Mifflin and you can order it through amazon.com and such sites.)

Examples of fraction operations:
Multiplying: (3/4) * (5/6) = (3*5) / (4*6) = 15 / 24 = 5/8 (in lowest terms.)
Dividing . . : (3/4) / (5/6) = (3/4) * (6/5) = 18 / 20 = 9/10 (invert and multiply)
Adding . . . : (3/4) + (5/6) = (9/12) + (10/12) = (9+10)/12 = 19/12 = 1 7/12 (common denom)

[ arithmetic | top of page ]

Percentages and Decimals - a quick tour -- (10/99)

The % is a percent sign, meaning divided by 100.
So 25% means 25/100, or 1/4.
To convert a percentage to a decimal, divide by 100.
So 25% is 25/100, or 0.25.
To convert a decimal to a percentage, multiply by 100 (just move the decimal
point 2 places to the right). For example, 0.065 = 6.5% and 3.75 = 375%.
To find a percentage of a number, say 30% of 40, just multiply:
(30/100)(40) = 0.3 x 40 = 12.
To find what percent a number is of another, divide 'em:
3/4 = 0.75 = 75%, so 3 is 75% of 4.
To make a fraction into a decimal, you divide:
3/4 = 0.75 = 75%, to recycle a recent example.
0.23 means 23 / 100 , and 0.6 means 6/10 or 3/5.

Decimals - more details and examples -- (3/01)

Rounding decimals to a certain accuracy or number of decimal places:
For example, 5.1837 to the nearest hundredth would be 5.18 (round down),
while to the nearest 3 places would be 5.184 (round up because of the 7)

Order matters when calculating and rounding (vs. rounding then calculating):
3.7 + 2.6 --> 4 + 3 --> 7 rounding first to nearest whole number then adding
3.7 + 2.6 --> 6.3 --> 6 adding first and then rounding at the end. Which is
correct? The second one really, but the first one is quicker for rough work!

Significant digits measure overall relative accuracy of a value: for example
the approx number 3.85 has 3 sig digs, while 0.00034 has only two. In this
case we would consider 18.40 as more accurate than 18.4 (4 sig digs to 3).

[ arithmetic | top of page ]

Square Roots - what they are and how to do 'em! -- (3/02)
What do you have to multiply by itself to get an answer of 25?
You could start with 5 , then do 5 * 5 = 5^2 = 25. (Remember the exponent notation from before.)
So we call 5 the "square root" of 25 , and write 25 = 5. (Put the 25 inside the .)
For example, 81 = 9 because 9^2 = 81, while 10.7 = 3.2710854 . . . (decimal goes on forever)
Note: You could round to the nearest hundredth : 10.7 ~ 3.27 but this is not exact ; 3.27^2 = 10.6929.
 I had a request to show the old "pencil and paper square root method" I learned it in pre-calc (meaning before calculators!); here's a picture showing10.7 ; you chop the number in twos starting at the decimal point, then (under)estimate the first sq rt, then subtract, bring down a pair, double what's on top, and see how many times that goes in, sort of. The blanks match up; we figure 62*2 is closer to 170 than 61*1, and 63*3 is too big. See? Try to continue it!

If you want the (Newton's) Divide and Average method, that's a different story: (6/03)
1) Guess the square root of a number , 2) divide the guess into the original number,
3) Take the average of the guess and quotient, 4) Repeat with this average as new guess.
"Guess, Divide, Average, Repeat." (I guess GDAR isn't a great way to remember that...)

Example : 10.7 = ? Guess: sqrt[10.7] is about 3 ; Divide 3 ) 10.7 and get
10.7 / 3 = 3.566666 ;
Average (3 + 3.566666) / 2 = 3.283333 ; Repeat:
10.7 / 3.283333 = 3.258884 ; (3.283333 + 3.258884) / 2 = 3.271108
close to real answer of 3.2710854...

[ arithmetic | top of page ]

Prealgebra [ top of page ] Click & choose a topic or just scroll & learn! . . . 9/97

A friend of mine once said, "I had algebra, it's that 'a + b = c' stuff." When I asked, the friend couldn't explain what a, b, and c were, what they might have meant, or why you would even add letters in the first place. Patricia Leitner and I decided to write a book about this very thing:
Prealgebra, Mathematics for a Variable World, by Daniel Bach and Patricia Leitner, 2nd edition, Houghton Mifflin Company. (sorry, another shameless plug)

Variables [ top of page ] . . . 9/97

The main idea is that a variable represents a number whose value might vary; hence the name!

Example: My sister Emily is 4 years older than me, so:

When I was 10, she was 10 + 4 = 14 .

When I was 17, she was 17 + 4 = 21 .

When I was (Dan's age), she was (Dan's age) + 4 .

We can say (Emily's age) = (Dan's age) + 4 ,

or simply E = D + 4 , where E = Emily's age, and D = Dan's age .

The quantities "Dan's age", "Emily's age", "D", and "E" are variables because they can represent many different numbers.

[ prealgebra | top of page ]

Laws of Arithmetic [ top of page ] . . . 9/97

You know 3 + 5 = 8. Does it matter what order you add?
No; 5 + 3 = 8 too. So 3 + 5 = 5 + 3. (thanks to daniel kenner for fixing my typo.)
Also (20) + (-3) = (-3) + (20); both are 17.
So we can pretty much say that

a + b = b + a , for any numbers a and b.

This is the commutative law of addition. (when you commute, you go back and forth.)

Which other operations commute? Try subtraction, multiplication, division; even
exponentiation. Think about it; I'll wait.
Ok, not subtraction: 7 ­ 3 = 4 , while 3 ­ 7 = - 4 .
Multiplication is good: 6 x 8 = 8 x 6 = 48, so we can say

a x b = b x a.

But this x might look like a variable, so let's write a * b = b * a , or even better,

a b = b a . (the commutative law of multiplication.)

Which do you add first in 3 + 4 + 5? Well,
(3 + 4) + 5 = (7) + 5 = 12 , and
3 + (4 + 5) = 3 + (9) = 12 . It works here, and in general:

(a + b) + c = a + (b + c) . (the associative law of addition.)

This ought to work for multiplication, but we'd better check:

(5*4)*6 = 20*6 = 120 and 5*(4*6) = 5*24 = 120 ; in general

(a * b) * c = a * (b * c) . (the associative law of multiplication.)
For simplicity we write (ab)c = a(bc) ; both are the same as just abc.

Let's see what happens if we do 4(7 + 3):
4(7 + 3) = 4(10) = 40 . This follows the order of PEMDAS.

But . . . 4(7) + 4(3) = 28 + 12 = 40 , too, as seen in this picture:

```... .......  This is either a
... .......  4 x 10 rectangle of
... .......  dots, or a 4 x 3
... .......  and a 4 x 7.```
Let's write this as 4(7 + 3) = 4(7) + 4(3) .
We say we "distribute" the 4 to the terms inside.
This translates to the general case:

a(b + c) = a b + a c (called the distributive law.)

[ prealgebra | top of page ]

Like and Unlike Terms [ top of page ] . . . 10/97

Ever hear the phrase "That's like comparing apples to oranges!"? Well, that's a lot like the problem you'll have if you try to add or subtract "unlike" terms like 3x and 4y. Huh? Read on!

First, what are "terms"? Then we'll see what "like terms" are.

A term is a product of a number and some variables, like 3xy or -4x^2. If the letter part is the same in two term's, they're called like terms: 3x and 5x are like terms but 3x and 5y are unlike terms; so are 5ab and 7ab^2. (that's b squared, as in "b there or b^2".)

If two terms are "like" then you can add or subtract them. Notice the use of the "distributive law" (see above):

3x + 5x = (3 + 5)x = 8x.

We can keep the same letter part and add the coefficients (the numbers in front of the variables).

But unlike terms can't be added: (The symbol /= means "not equal to.")

3x + 4y /= 7x or 7y or 7xy, and

6 + 5x /= 11x (multiply before adding).

Well, that's it for now. Check back often for new stuff!
Click below for other topics, or visit the ask dan page!
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