Building Simple Words: Multiplication

Multiplication is the process of multiple additions. Multiplication is shorthand for multiple addition operations. Like addition, multiplication is a binary operation, meaning there are two terms. The first number is called the multiplicand and the second is called the multiplier. The result is called the product. The numbers that make up a product are also called the factors.

Multiplication is the process of adding multiple times. For example,
2 × 3 would be the same as 2+2+2=6.

Multiplication can be performed on the number line by starting at 0 and skipping to the right to the answer. Multiplying 2 by 3 would require starting at 0 then skipping to the right by 2's three times. The first skip lands on 2, the second skip lands on 4 and the third skip lands on 6.

0--1--2--3--4--5--6--7--8--9--10->

Just as we can develop a table for addition, we can develop a table for multiplication.

×	0	1	2	3	4	5	6	7	8	9	10
0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10
2	0	2	4	6	8	10	12	14	16	18	20
3	0	3	6	9	12	15	18	21	24	27	30
4	0	4	8	12	16	20	24	28	32	36	40
5	0	5	10	15	20	25	30	35	40	45	50
6	0	6	12	18	24	30	36	42	48	54	60
7	0	7	14	21	28	35	42	49	56	63	70
8	0	8	16	24	32	40	48	56	64	72	80
9	0	9	18	27	36	45	54	63	72	81	90
10	0	10	20	30	40	50	60	70	80	90	100

To find the product of two numbers, we start with the multiplicand at the top and read down the column to the row starting with the multiplier. Where the two meet is the product.

It is useful to memorize the multiplication table, just as it is useful to memorize the addition table.

Multiplying large numbers requires multiplication by each column and then addition of those multiples.

	1	3	5	7
×		2	4	6

We start by multiplying 1357 by 6. Any multiple that is greater than 9 carries the tens digit to the next column to the left as an add after the multiplication of that column. The colored number will be the carry.
7 × 6=42
We keep the ones value and carry the tens value to the next column.
5 × 6=30+4=34
We keep the ones value and carry the tens value to the next column.
3 × 6=18+3=21
We keep the ones value and carry the tens value to the next column.
1 × 6=6+2=8
So our first row of multiplication is 8142.

	1	3	5	7
×		2	4	6
	8	1	4	2

Now we multiply 1357 by 40. Since we are multiplying by a multiple of 10, we place a 0 in the ones column to hold the position open, then simply multiply by 4.
7 × 4=28
We keep the ones value and carry the tens to the next column.
5 × 4=20+2=22
We keep the ones value and carry the tens value to the next column.
3 × 4=12+2=14
We keep the ones value and carry the tens value to the next column.
1 × 4=4+1=5
So our second row of multiplication is 54280. Notice the 0 holding the ones position open.

	1	3	5	7
×		2	4	6
	8	1	4	2
5	4	2	8	0

Now we multiply 1357 by 200. Since we are multiplying by a multiple of 100, we place a 0 in both the ones and the tens column to hold the position open and multiply by 2.
7 × 2=14
We keep the ones value and carry the tens to the next column.
5 × 2=10+1=11
We keep the ones value and carry the tens to the next column.
3 × 2=6+1=7
We don't have a carry this time.
1 × 2=2
So our third row of multiplication is 271400. Notice the zeros holding the tens and ones positions open.

		1	3	5	7
	×		2	4	6
		8	1	4	2
	5	4	2	8	0
2	7	1	4	0	0

Now we add up the three rows of multiples.

			8	1	4	2
		5	4	2	8	0
+	2	7	1	4	0	0
	3	3	3	8	2	2

So 1357 × 246=333822. That's 1357 added to itself 246 times. We had to make three multiplications and one addition to get the answer. That's a lot faster than doing 246 additions.

Just like with addition, multiplication is commutative; 3 × 2=2 × 3. And, just like with addition, multiplication is also associative;(3 × 2)× 4=3 ×(2 × 4). And, just like with addition, there is also an identity element in multiplication; it is 1. Multiplying a number by 1 doesn't change its identity. This will be an important tool later on. Also, notice that multiplying any number by 0 gives 0. There is an important rule to remember in mathematics that if the product of two numbers is 0, one of them must have been 0 to begin with.

Addition and multiplication can be mixed in a sentence. When we see addition and multiplication together, there is a particular order that we do the operations in. In English, "cats eat birds" means something completely different if we change the order to "birds eat cats." In math, the order of the operations can change the meaning of a mathematical sentence. Look at the sentence 2+3 × 4 for example.What if we add 2 and 3 then multiply by 4. We get 5 × 4=20. But what if we multiply 3 and 4 first, then add 2. We get 2+12=14. In math, multiplication is always done before addition. So 2+3 × 4 is 14. To avoid confusion, we use parenthesis around operations. Operations inside parenthesis are done first. So it would be clear that 2+(3 × 4) is 14 and (2+3)× 4=20.

When mixing multiplication and addition, there is a special property called distribution, when we have multiplication on the outside of a parenthesis containing an addition. In something like
2×(3+4), the multiplication distributes to each term of the addition inside the parenthesis.
2×(3+4) is the same as 2 × 3+2 × 4. It is clear that 2 × 7=14 and 6+8=14. We will use this distributive property a lot in the future.

There is a special multiple that will become important later on. It is called the least common multiple or lcm. It is the smallest number that is a common factor of two numbers. You start with the prime factorization of two numbers, then you count the each factor at least once.

Some rules for factoring numbers are as follows. If the number is even, 2 is a factor. If the number ends in 5, then 5 is a factor, If the sum if the digits of a number add up to a multiple of 3, then 3 is a factor. There are other rules that can be found at Wikipedia http://en.wikipedia.org/wiki/Divisibility_rule

Let's find the least common multiple of 48 and 52. If we just look at the product, 48×52=2496. Now let's factor. The factors of 48 are 2,2,2,2 and 3. The factors of 52 are 2,2, and 13. The lcm of 48 and 52 would be the product of four 2's, a 3 and a 13. 2×2×2×2×3×13=624. The reason the lcm is less than the straight product of 48 and 52 is that we don't have to multiply by the extra 2's in 52; we already counted them in the factors of 48. The smallest number that is a factor of both 48 and 52 is 48×13=624 and 52×12=624.

Building Simple Words: Addition

Because there is an order to the natural numbers, we can lay them out on a number line. It looks like this:

0---1---2---3---4---5---6---7---8---9---10--->

Starting with a number on the number line, we can get to another number on the number line by stepping to the right on the line. This process is called addition. We take two numbers and add them together to form another number. For this reason, addition is called a binary (meaning two) operation. The two numbers are called the addends and the result of the operation is called the sum.

We work our way from left to right along the number line. For example, let's add 3 + 2.

Start on 3 on the number line and step to the right 2 times.

0---1---2---3---4---5---6---7---8---9---10--->

We stop on 5. So 3 + 2 = 5.

We can build a table of the results of adding the numbers 0 to 9 and memorize the results

+	0	1	2	3	4	5	6	7	8	9
0	0	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9	10
2	2	3	4	5	6	7	8	9	10	11
3	3	4	5	6	7	8	9	10	11	12
4	4	5	6	7	8	9	10	11	12	13
5	5	6	7	8	9	10	11	12	13	14
6	6	7	8	9	10	11	12	13	14	15
7	7	8	9	10	11	12	13	14	15	16
8	8	9	10	11	12	13	14	15	16	17
9	9	10	11	12	13	14	15	16	17	18

To add two numbers, we find the first number on the top row and follow the column down to the row with the second number on the left. Where the column and row meet is the sum of the two numbers. For example, 7 + 6 = 13. Start with 7 on the top and follow the column down to the row that starts with 6. Where the column and row meet is 13.

Adding larger numbers requires lining up the positions in columns. All of the ones line up under each other; all of the tens line up under each other; all of the hundreds line up under each other and so on. Then you add down the columns starting with the ones column and move to the left. If the sum of the column is more than 9, you keep the ones value from the sum and add 1 to the column on the left. The number that is "carried" to the column on the left is called the carry.

Let's add 1357 + 246.
Line the numbers up in columns

1	3	5	7
+	2	4	6

Starting with the ones column adding down the column: 7+6=13. Keep the 3 in the ones column and add 1 to the tens column. The carry is in color.

		1
1	3	5	7
+	2	4	6
			3

Now add down the tens column: 1+5+4=10. Keep the 0 in the tens column and add 1 to the hundreds.

	1
1	3	5	7
+	2	4	6
		0	3

Now add down the hundreds column: 1+3+2=6.

1	3	5	7
+	2	4	6
	6	0	3

Since there is no number in the thousands column in the second number, we fill in the position with a 0. Now add down the thousands column: 1+0=1

1	3	5	7
+	2	4	6
1	6	0	3

Adding more than two numbers at a time, you still line up the numbers in columns and add down the columns starting with the ones column. You can only add two numbers at a time, but you can keep a running total as you add your way down the column. When you get to the bottom, keep the ones value and carry the rest of the number to the columns on the left. If your carry is larger than 9, the carry overflows to the columns on the left, carrying to more than one column if you have to.

Addition is the simplest way to form larger numbers. As you can see from the addition table, there are several ways to get to other numbers: 3+2=5 and 4+1=5. You might also notice that changing the order of the numbers doesn't make a difference: 3+2=2+3. This is called commutativity and is a basic property of the natural numbers.

When adding more than tow numbers, we use parenthesis around numbers to indicate the order we want to add. Numbers inside of parenthesis get added first, then that sum gets added to the numbers on the outside of the parenthesis. 3+(4+2) means add 4+2 then add 3 to that sum.
3+(4+2)=
3+ 6 = 9.
This way we can group addition together.

Another basic property of the natural numbers is associativity. This means the grouping doesn't matter. If we add more than two numbers at a time, it doesn't matter how we group the numbers together:

3+(4+2)=(3+4)+2
3+ 6 = 7 +2
9 = 9

These basic properties will follow us all the way through the alphabet. As we add more types numbers to our alphabet, those numbers will also be commutative and associative. Remember, commutative means order and associative means group. An easy way to remember it is that when a governor commutes a sentence he changes the court's order and an association is a group of people. This ability of numbers to commute and associate is a very powerful tool when we start looking at much more complicated forms of math.

In addition, 0 is a very special number. As you can see, anytime we add 0 to a number, the number doesn't change. 0 is called the identity element, because addition by 0 doesn't change a number's identity. This will become an important tool to use later on. The ability to change a number's look without changing its value is very useful.

The Alphabet of Math: The counting numbers

Just like any language, math has an alphabet. We will build the alphabet in steps. The basic alphabet is the natural or counting numbers. The natural numbers occur in nature. They are also called the counting numbers because they are numbers used to count things. Just like the English alphabet, there is an order. In order, the natural numbers are 1,2,3,4,5,6,7,8,9.

According to The History of Numbers, when math started out, there were only three numbers; one, two and many. There are so called "primitive" cultures alive in the world today where those are still the only numbers used. One, two and many isn't very precise. So many has to be quantified.

According to The History of Numbers, people naturally began counting on fingers. That was good for five or ten things, but what if you had more things to count? Then you could count two more things by counting arms, two more by counting legs, two more by counting eyes or ears. In some primitive cultures, it is possible to count a large number of items by ticking off the fingers and various other body parts. To know if you have lost any sheep during the day, just remember what part of the body you counted up to in the morning and make sure you get that far at the end of the day.

The most commonly developed scheme for keeping track of numbers is using marks on a piece of wood; one mark for each item. Grouping marks, say into fives or tens, makes it easy to see at a glance how many items you have. With practice, you learn to recognize multiples of five or ten. The Sumerians were the first to use symbols for numbers, but they were really only sophisticated groups of tick marks. Counting on body parts is good for a relatively small number of items. The Sumerians were able to represent truly large numbers in the millions to keep track of grain harvests and distribution of resources and even do some astronomy which calls for REALLY big numbers.

The numbers that we use today come from India through Arabia. With the addition of the concept of zero, which we also get from the Indians, we can represent any number. We start counting at 0, denoting an absence of items and count from 1 to 9. To represent larger numbers, we borrow a concept from the Sumerians called positional notation. As we go above 9, we add numbers to the left. Each position to the left represents a multiple of 10. Each position to the left gets a new name. From right to left they go ones, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions and so on. So 25 represents 2 tens and 5. The number 347 is 3 hundreds, 4 tens and 7.

With the alphabet from 1 to 9 and positional notation, it is possible to represent any number we can count, no matter how large. With this alphabet, we are ready to learn how to combine numbers together to form larger numbers. We'll discuss that in my next blog on basic arithmetic.

Introduction to Math Speak

Mathematics is a language, just like English, French or Russian. It has an alphabet and rules of grammar. In this blog, I will take you through the process of learning the language of mathematics from counting to high school algebra. Feel free to post for homework help.