dan's arithmetic lesson
©1997-2016 - all rights reserved - dan bach @dansmath
the basic operations
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.
Alternate notation: 6 * 2 = 12, 6 ÷ 2 = 3.
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 operations below.)
Exponents
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.
Notation: The exponent is usually placed to the upper right of the base:
Example: Which is bigger, 4 ^ 5 or 5 ^ 4 ?
Answer: 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 x 10^3
0.00003802 = 3.802 x 10^(-5)
Order of Operations
To do the operations in the right order, remember PEMDAS, which stands for:
Parentheses, Exponents, Multiplication-Division, Addition-Subtraction.
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 additions and subtractions from left to right.
In California we say Powerful Earthquakes May Deliver After-Shocks.
(Speaking of powerful, someone e-mailed me a long time ago and said this section gave him confidence and changed his life!)
Example: What is 2 + 3 x 4 ?
Do it in your head, and then try it on your calculator. If either answer is 20, you have a problem.
Don't do the 2 + 3 (addition) until all multiplications are done:
2 + 3 x 4 = 2 + 12 = 14 (the answer!)
Question: In 2 + 3 x 4, what's wrong with doing 2 + 3 = 5, then 5 x 4 = 20? (You tell me)
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 must been done first; 4 x 6 = 24. Answer: 3 + 24 – 2 = 27 – 2 = 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 problem (meaning 4 x 4, or 16), so by PEMDAS, do it before multiplication.
3 x 4 ^ 2 = 3 x 16 = 48.
Note: If you really want it to mean 12 ^ 2, you have to write (3 x 4) ^ 2.
Prime Numbers and Factorizations
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.
Example: With the help of exponents, we can write:
36 = 2 ^ 2 * 3 ^ 2, and
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?
Example: Let's see if 101 is prime.
We can try 2, see if it goes into 101 (it doesn't), then 3 into 101 (it doesn't go 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!
Example: 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.)
Note: There are 25 primes under 100. Can you find them all?
Does this mean there are 250 primes less than 1000?
For more advanced topics go to the Number Theory page.
Fractions
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 people think that pi is equal to 3 1/7, but it's merely close. Pi is an irrational number, which means it's a real number, but not equal to any fraction. (And 355/113 is even 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 that 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 could get my textbook: Prealgebra: Mathematics for a Variable World, Third Edition, by Bach & Leitner. (It's published by McGraw-Hill and you can order it through amazon.com and other sites.)
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 denominator)
Subtracting: (3/4) – (5/6) = (9/12) – (10/12) = (9–10) / 12 = –1/12
Percentages and Decimals (a quick tour)
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.
Decimals already stand for fractions: 0.23 means 23 / 100 , and 0.6 means 6/10 or 3/5.
Decimals (more details and examples)
Rounding decimals to a certain accuracy or number of decimal places:
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).
Square Roots (what they are and how to do 'em!)
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.)
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 once 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 of my scratchwork showing -/10.7 : You chop up the number in twos starting at the decimal point, then (under)estimate the first square root, 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?
Now you try to continue it!