**dan's arithmetic lesson**

**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:

**P**arentheses,** E**xponents, **M**ultiplication-**D**ivision, **A**ddition-**S**ubtraction.

**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 **P**owerful **E**arthquakes **M**ay **D**eliver **A**fter-**S**hocks.

(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 6**2 *** **2** is closer to 170 than 6**1 *** **1**, and 6**3 *** **3** is too big. See?

Now you try to continue it!