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Below are the 13 most recent journal entries recorded in The Counter's LiveJournal:

Friday, April 26th, 2013
8:55 pm
[joe_doe_zero]
Math and Money
Math and Money
Wednesday, April 24th, 2013
12:36 am
[joe_doe_zero]
Hello
I am the Counter
Sunday, July 17th, 2005
7:10 pm
[joe_doe_zero]
General subtraction - lesson 2
If we want to subtract 566 from 1707 how would we do it?

First, let's use the lessons we learned from adding multi-digit numbers. We would turn 566 into 0566.

1707 - 0566 = ?

1 - 0 = 1
7 - 5 = 2
0 - 6 = ???
7 - 6 = 1

You might have noticed that the tens digit in the minuend is smaller than the tens digit in the subtrahend and the problem seems incomputable. But we can always "borrow" from an upper row. That is, we subtract 1 from the row we are borrowing from and add ten to the row we are borrowing to.

Let's borrow a 10 from the hundreds column in this problem.

1 - 0 = 1
7 - 5 = 2 <- borrow 1 from this column (subtract 1 from the minuend)
0 - 6 = ??? <- borrow 10 to this row (add 10 to the minuend)

7 - 6 = 1

We get:

1 - 0 = 1
6 - 5 = 1
10 - 6 = 4

7 - 6 = 1

We get 1707 - 566 = 1141

Let's try another: 1200 - 975
1 - 0 = 1
2 - 9 = ???
0 - 7 = ???
0 - 5 = ???


First we borrow a 1 from the thousands row (in the minuend, of course). We add 10 to the hundreds row.
0 - 0 = 0
12 - 9 = 3
0 - 7 = ???
0 - 5 = ???

So far, we have no definite answer but at least the 3 in the hundreds row is settled. The problem of 1200 - 975 has been reduced to 300 - 75.

3 - 0 = 3
0 - 7 = ???
0 - 5 = ???

We "borrow" from the hundreds row again. We subtract 1 from the 3 and add 10 to the number beneath it.

2 - 0 = 2
10 - 7 = 3

0 - 5 = ???

Now we have a more definite answer: 230. The problem has been reduced from 300 - 75 to 230 - 5. But there's still more "borrowing" to do. One more "loan" ought to do it.

2 - 0 = 2
3 - 0 = 3 (borrow a 1 from here)
0 - 5 = ??? (add a 10 to here)


2 - 0 = 2
2 - 0 = 2
10 - 5 = 5


Done. And done. Our final answer is 225. Thus 1200 - 975 = 225

Another one to try: 9,000 - 5,678 = ?

9 - 5 = 4
0 - 6 = ???
0 - 7 = ???
0 - 8 = ???

A shortcut to take is to do multiple borrowing. Borrow a 1 from the 9 and put in a 10 into the 0 directly beneath it. Then borrow a 1 from that zero and put a 10 into the 0 directly beneath that. (The second zero beneath the 9) And finally, borrow a 1 from the tens row and put a 10 into the ones row (the bottom row). Here's what the problem looks like after the first step
8 - 5 = 3
10 - 6 = ?
0 - 7 = ???
0 - 8 = ???

After a second loan has been made:
8 - 5 = 3
9 - 6 = ?
10 - 7 = ?
0 - 8 = ???

And after the third loan:
8 - 5 = 3
9 - 6 = ?
9 - 7 = ?
10 - 8 = ?

And it's ready to be solved!
8 - 5 = 3
9 - 6 = 3
9 - 7 = 2
10 - 8 = 2

9,000 - 5,678 = 3,322.

Thus this stumper:

10,000,000,000 - 1,234,567,891

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

After a series of borrowing:
9 - 1 = 8
9 - 2 = 7
9 - 3 = 6
9 - 4 = 5
9 - 5 = 4
9 - 6 = 3
9 - 7 = 2
9 - 8 = 1
9 - 9 = 0
10 - 1 = 9


10,000,000,000 - 1,234,567,891 = 8,765,432,109

You might have noticed that the difference in that last problem has exactly one of each kind of digit in it.
-----------------------------------------------------------------------------------------------------------

Some Life Help


If you must pay for something that costs $59.99 and you have a $100.00 bill, one way you can rapidly compute the difference is by doing "subtraction digit pairs". Pair each digit using this formula:

9 - (digit) = (its subtraction pair)
9 - 0 = 9
9 - 1 = 8
9 - 2 = 7
etc.

So 0 is paired with 9, 1 with 8, 2 with 7, 3 with 6, 4, with 5, and so on.

Then for every digit except for the ones digit, simple put in a pair. For the digits with the greatest value, simply subtract 1.

$100.00 - $59.99
10 - 5 = 4 (Think 9 - 5 = 4)
0 - 9 = 0 (0 is paired with 9)
0 - 9 = 0 (0 is paired with 9)
0 - 9 = 1 (1 + 9 = 10)
Your change is $40.01

Another: $800.00 - $567.98
8 - 5 = 2 (7 - 5 = 4)
0 - 6 = 3 (3 is paired with 6)
0 - 7 = 2 (2 is paired with 7)
0 - 9 = 0 (0 is paired with 9)
0 - 8 = 2 (2 + 8 = 10)
Your change is $$432.02
6:59 pm
[joe_doe_zero]
General subtraction - lesson 1
Subtracting two or more numbers is almost as easy as adding two or more numbers. There are a few things that need to be cleared, however.

Let's say we want to do: 9 - 5 - 3

This can mean two things: Either (9 - 5) - 3 or 9 - (5 - 3). The answers to them are different. Remember that we must do what is written in the parenthesis first.

In the first statement:

(9 - 5) - 3

9 - 5 = 4
4 - 3 = 1

(9 - 5) - 3 = 1

In the second statement:

9 - (5 - 3)

5 - 3 = 2
9 - 2 = 7

9 - (5 - 3) = 7

Very different answers. Remember that subtraction is not commutative so the rule, generally, is to do what is in the parenthesis first. Do the inner parentheses first and work your way outward.

((9 - 2) - 3) - (2 - 1)
(7 - 3) - (2 - 1)
4 - 1
3

((9 - 2) - 3) - (2 - 1) = 3

Subtracting numbers with more than one digit


Let's say we want to do 95 - 33.

95 - 33 = ?

We do it in the same way as addition but using subtraction instead of addition.

9 - 3 = 6
5 - 3 = 2

95 - 33 = 62

78 - 54

7 - 5 = 2
8 - 4 = 4

78 - 54 = 24

And now let's move on to the more fun lesson in subtraction - borrowing tens.
5:44 pm
[joe_doe_zero]
General adding - lesson 2

Adding numbers with more than one digit



We know to a good extent how to add numbers containing only one digit. But let's say we want to add numbers with more than one digit.

We want to add: 10 + 10

While we could look it in the adding chart and find the answer (it's there!), it would be too impractical to have an adding chart for every single number. Therefore we need a more general method to add numbers rather than looking it up in an adding chart.

10 + 10 = 20.

Does this problem resemble the problem 1 + 1 = 2? Let's take a closer look and compare the two problems:

1 + 1 = 2
10 + 10 = 20

Instead of adding by ones, we added by tens in the last problem. Let's try another:

10 + 20 = ?

We find [in our adding chart] that 1 + 2 = 3. Using the same type of logic we find that

10 + 20 = 30

What about 50 + 30 = ?

5 + 3 = 8
Therefore 50 + 30 = 80.

Could we use the same with bigger numbers? Yes, we can.

100 + 300 = ?

From the adding chart, 1 + 3 = 4

Therefore 100 + 300 = 400.

What if we wanted to add 100 + 30?

Can we use 1 + 3 = 4 for this problem, too? No, we cannot. 30 is a smaller number than 300. Therefore

100 + 30 is NOT equal to 100 + 300

If we cross out a zero from each numeral.

100 + 30

We get 10 + 3

10 + 3 = 13

Now add back a zero to the answer.

100 + 30 = 130

Let's try another: 20 + 7 = ?

Remember our "breaking" technique from the previous lesson? 20 can be broken into 10 + 10

10 + 10 + 7
10 + 7 = 17
17 + 10 = 27

How exactly did we add 17 and 10? This problem is actually two addition problems in one:

17 + 10 = ?

1 + 1 = 2
7 + 0 = 7

17 + 10 = 27

1 + 1 = 2 (each 1 is actually a 10)
7 + 0 = 7

Another problem: 26 + 20 = ?

2 + 2 = 4 (each 2 is actually a 20)
6 + 0 = 6

26 + 20 = 46

Let's try a more difficult one: 140 + 45 = ?

Think of the number 45 as 045. The number is not changed, you're merely adding an imaginary zero to the front of it.

So the problem becomes 140 + 045 = ?
1 + 0 = 1 (each digit is actually a hundred)
4 + 4 = 8 (each digit is actually a ten)
0 + 5 = 5 (each digit is a one)
140 + 045 = 185
140 + 45 = 185.

Let's do an even more difficult one: 132 + 66
66 becomes 066

So the problem becomes 132 + 066 = ?
1 + 0 = 1 (each digit is a hundred)
3 + 6 = 9 (each digit is a ten)
2 + 6 = 8 (each digit is a one)
132 + 66 = 198.

Another one: 133 + 256 = ?
1 + 2 = 3
3 + 5 = 8
3 + 6 = 9
133 + 256 = 389

Problems in real life are not always as easy (or as pretty) as the ones presented above.

Let's try: 356 + 289 = ?

3 + 2 = 5
5 + 8 = 13
6 + 9 = 15
But the answer is not 51315. Let's go back to an earlier problem we worked:

133 + 256 = ?
1 + 2 = 3 (each digit is a hundred)
3 + 5 = 8 (each digit is a ten)
3 + 6 = 9 (each digit is a one)

What does it mean when each digit is a hundred? When we were adding 1 + 2, in essence we were adding 100 + 200 and got 300. When we were adding 3 + 5, we were actually adding 30 + 50 to get 80. We added the 300 + 80 + 9 to get 389.

133 + 256 = 389

Now let's take a look at this problem:
3 + 2 = 5
5 + 8 = 13
6 + 9 = 15

The 5 is actually a 500; the 13 is actually a 130; the 15 is a 15. So we have

500 + 130 + 15 = ?
Turn the 15 into 015 and remember how we add more than two numbers.
5 + 1 + 0 = 6
0 + 3 + 1 = 4
0 + 0 + 5 = 5

Since the 6 is actually a 600; 4 is a 40; 5 is a 5, we get 645 as our answer.

356 + 289 = 645.

Another, just for more practice: 781 + 998

7 + 9 = 16
8 + 9 = 17
1 + 8 = 9

16 is really 1600; 17 is really 170; 9 is really a 9.

1600 + 170 + 9
Translate: 1600 + 0170 + 0009

1 + 0 + 0 = 1
6 + 1 + 0 = 7
0 + 7 + 0 = 7
0 + 0 + 9 = 9

781 + 998 = 1779.

The technique we just outlined is called "carrying the 1". If we have a sum in a column that is greater than ten (usually between 10 and 19, inclusive) we only count the lowest digit and add a one to the column on the left.

Say we want to add: 25 + 37

We simply add 5 + 7 to get 12 and then "carry the 1" from the 12 and then add the 1 to 2 + 3.

The ones digit will be 2 and the tens digit will be computed by 1 + 2 + 3 = 6

Thus giving us an answer of 62.

Finally, let's go for something that is sure to get our minds going:

34,768 + 8,051*

34768 + 08051

3 + 0 = 3
4 + 8 = 12
7 + 0 = 7
6 + 5 = 11
8 + 1 = 9

3 = 30,000; 12 = 12,000; 7 = 700; 11 = 110; 9 = 9.

30,000 + 12,000 = 42,000
700 + 110 = 810
9 = 9
42,000 + 810 + 9

810 + 9 = 819
42,000 + 819 = 42,819

34,768 + 8,051 = 42,819

And now you know how to add all whole numbers.

*When numerals have more than three digits a comma is placed between every three digits. Thus 10000000000 would be written as 10,000,000,000.
5:22 pm
[joe_doe_zero]
General adding - lesson 1

Adding more than two numbers



It is possible to add more than two numbers. Let's say we want to add 3, 5, and 6

3 + 5 + 6 = ?

There is a hidden beauty here. We can add the numbers in any order we desire. Let's first do (3 + 5) + 6

3 + 5 = 8
8 + 6 = 14

If that was too inconvinient for you, let's try in a different order: 3 + (5 + 6)
5 + 6 = 11
11 + 3 = 14

We arrive at the same answer!

Now, let's try a third order. (3 + 6) + 5
3 + 6 = 9
9 + 5 = 14

Another way:
5 + 3 = 8
8 + 6 = 14

There are two more ways left but we will not go into them since we have arrived at the same answer consistently.

Let's try another: 2 + 6 + 8 = ?

8 and 2 can be added to get 10 so
8 + 2 = 10
10 + 6 = 16

Another way:
2 + 6 = 8
8 + 8 = 16

This is often helpful because sometimes we can add certain numbers more easily than other numbers. The number 10 is easy to add by. Let's try:

3 + 9 + 1 + 7 = ?

3 + (9 + 1) + 7 = ?
3 + 10 + 7 = ?
3 + 7 + 10 = ?
(3 + 7) + 10 = ?
10 + 10 = ?
10 + 10 = 20

We simply paired up the 9 and 1 togather and paired 3 and 7 togather. Both add up to 10 and we added the tens to get 20.

Let's try 7 + 1 + 2 + 5

Adding by 1 is easy so 2 + 1 = 3
We're left with 7 + 3 + 5

7 and 3 make 10 leaving us with

10 + 5 = ?
10 + 5 = 15

It often helps to "pair" or "lump" numbers togather to make computation easier. Let's say we want to add: 6 + 6 + 2 + 4 + 2

Pair one 6 with a 4
Lump the other 6 with 2 and 2

6 + 4 = 10
6 + 2 + 2 = 10

10 + 10 = 20.

Thus 6 + 6 + 2 + 4 + 2 = 20

The following "number pairs might be helpful:

Pair 1 with 9
Pair 2 with 8
Pair 3 with 7
Pair 4 with 6
Pair 5 with 5 (itself)
Pair 6 with 4
Pair 7 with 3
Pair 8 with 2
Pair 9 with 1

Actually, you only need five pairs: (1 and 9), (2 and 8), (3 and 7), (4 and 6), (5 and itself).

As for the number 0, just disregard it. If you're ever faced with a problem such as

6 + 0 + 9 = ?

Think of it as simply:

6 + 9 = ?

Just like numbers can be paired, they can also be "broken". For example, 4 can be "broken" into 3 and 1 r it can be broken into 2 and 2.

4 can be broken into 3 + 1
4 can be broken into 2 + 2

Why is this helpful?

Say you want to add 7 + 8. You know how to add 7 + 3. So "break" 8 into 5 + 3. "Pair" 7 and 3 to get 10 and add the remaining 5 to get 15.

Try this one: 9 + 9 + 3

"Break" one 9 into 7 + 2 and add the 7 to 3 to get a 10. We now have

9 + 10 + 2

9 + 2 = 11
10 + 11 = 21

The beauty of these problems is that you can often do the same problem in many different ways. You can take as many steps as you'd prefer to solve a problem. Some problems can be easily solved in three steps, while others may take six steps. Choose the method that works best for you. And do not be afraid to use your creativity; have fun in the process! :)
5:09 pm
[joe_doe_zero]
Connections Between Addition and Subtraction
Addition and subtraction are almost totally linked. Let's take three numbers in which one is the sum of the other two: 3, 5, and 8

3 + 5 = 8

5 + 3 = 8
8 = 5 + 3
8 = 3 + 5

Now let's do it with subtraction.

8 - 5 = 3
8 - 3 = 5
5 = 8 - 3
5 = 8 - 5

Ignoring the fact that 5 + 3 = 8 and 8 = 5 + 3 are the same thing, let's only focus on the addition and subtraction. Let's take three more numbers: 4, 2, and 6. We can construct four equations.

4 + 2 = 6
2 + 4 = 6
6 - 2 = 4
6 - 4 = 2

Let sum be represented by an "S".

General pattern:
A + B = S
B + A = S
S - B = A
S - A = B

Using the Adding Chart and what we just learned we can construct an entire system of adding and subtracting numbers. (Doing so would be an exercise in tedium so please don't do so unless you need a hand workout.)

You also should have noticed that

A + 0 = A
0 + A = A
A - 0 = A
A - A = 0

Addition is commutative while subtraction is not commutative. That is

A + B is equal to B + A
A - B is NOT equal to B - A

If A and B are equal then A - B = B - A but the statement is true only for that particular case. In order for an equation or a statement to be held as being true in mathematics, it must hold true for every case and not just specific cases.
4:13 pm
[joe_doe_zero]
Subtraction in Detail
Before learning this lesson, make sure you have the adding chart memorized. If you haven't memorized it, make sure you at least know how to count and some basic adding concepts. Let's do a quick review.

Concept 1 The counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc.
Concept 2 The Adding Chart
Concept 3 The addition patterns

The addition patterns:
(first number) + (second number) = sum
(second number) + (first number) = sum
(first number) + 0 = (first number)
(second number) + 0 = (second number)
(first number) + (second number) = (second number) + (first number)

To put it in a more concise way, let "A" stand for the first number and "B" stand for the second number.

A + B = sum
B + A = sum
A + 0 = A
B + 0 = B
A + B = B + A

The last statement is known as the Law of Commutativity

Addition is known as a commutative operation because it makes no difference which order the numbers are added in.

For example:

A + B + C = A + C + B = B + C + A = B + A + C = C + B + A = C + A + B

We shall go on to learn subtraction. Subtraction is adding in reverse. In former lessons you had to solve problems like:

1 + 1 = ?
3 + 4 = ?
5 + 2 = ?
9 + 6 = ?
2 + 5 = ?

In the previous problems, you were given the first number and the second number and had to compute the sum. That is, you were given the addends and had to compute the sum. In the next few problems we will work with problems where the sum is given but one of the addends is missing. We must compute the addends.

Let's take the problem: 3 + ? = 7

The problem is asking: What, when added to 3, gives us the sum of 7?

Taking a quick look at the addition chart we will find that 3 + 4 = 7. Therefore, when 4 is added to 3, it gives us 7. How about this one:

? + 5 = 8

What, when added to 5 gives us the sum of 8?

Again, a quick look at the adding chart should give us the answer: 3 + 5 = 8.

There is a more direct way to solve these problems without the use of an adding chart. The process is called subtraction.

Instead of doing 3 + ? = 7, we subtract and thus do

7 - 3 = ?

Doing this visually, let's take seven objects

7 objects:
(object #1)
(object #2)
(object #3)
(object #4)
(object #5)
(object #6)
(object #7)

And now delete three of the seven objects. (Any three, it makes little difference.)

Delete 3 objects:
(object #1)
(object #2)
(object #3)
(object #4)
(object #5)
(object #6)
(object #7)


Objects remaining:
(object #1)
(object #2)
(object #3)
(object #4)

Now count the remaining objects: 1... 2... 3... 4.

We have four remaining. Thus

7 - 3 = 4

Let's try another example:

8 - 6 = ?

8 objects:
(object #1)
(object #2)
(object #3)
(object #4)
(object #5)
(object #6)
(object #7)
(object #8)

Delete 6 objects:
(object #1)
(object #2)
(object #3)

(object #4)
(object #5)
(object #6)

(object #7)
(object #8)

Objects remaining:
(object #4)
(object #7)

Count the remaining objects: 1... 2.

Thus 8 - 6 = 2.

The number 8 is known as the minuend of the equation; the number 6 is known as the subtrahend of the equation; the number 2 is known is the difference of the equation. 2 is the difference of and 8 and 6.

Another one: 5 - 5 = ?

5 objects:
(object #1)
(object #2)
(object #3)
(object #4)
(object #5)

Delete 5 objects:
(object #1)
(object #2)
(object #3)
(object #4)
(object #5)


Objects remaining:
(none)

Count the remaining objects: (none)

Thus 5 - 5 = 0.

Try 3 - 0 = ?

3 objects:
(object #1)
(object #2)
(object #3)

Delete 0 objects:
(object #1)
(object #2)
(object #3)

Objects remaining:
(object #1)
(object #2)
(object #3)

Thus 3 - 0 = 3.

Try 3 - 5 = ?

This cannot be done (unless we invent a new type of number). We are asking to remove five objects from a group that does not contain that many objects. Five is a bigger number than three. Five is greater than three. Mathematically written, it is

5 > 3

The concept of "greater than" can wait for another lesson. For now, let's focus on some properties of subtraction.

General formula: (Minuend) - (Subtrahend) = (Difference)
General pattern: (number) - 0 = (number)

3 - 5 is not the same as 5 - 3

5 - 3 = 2 while 3 - 5 cannot be done in our elementary system of arithmetic.

Subtraction property 1: A - B is not equal to B - A.
Subtraction property 2: A - 0 = A
Subtraction property 3: A - A = 0
4:11 pm
[joe_doe_zero]
The Adding Table
First make sure that you have the lesson "Concepts in Addition" down pat before taking on this lesson.

Counting numbers review: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and so on.

In the previous lesson we learned to add by one. Adding by one was the same as counting to the next number.

3 + 1 = 4
4 + 1 = 5
etc.

Now let's learn to add by the number 2. Eventuall we will construct an entire adding table.

1 + 2 = 3
2 + 2 = 4
3 + 2 = 5
4 + 2 = 6
5 + 2 = 7
6 + 2 = 8
7 + 2 = 9
8 + 2 = 10
9 + 2 = 11
10 + 2 = 12

Adding by two is just like adding by one twice.

2 + 2 = 4
(2 + 1) + 1 = 4
2 + (1 + 1) = 4

(The parenthesis means to do this operation first.)

It helps if you can memorize them in terms of evens and odds.

Odds
1 + 2 = 3
3 + 2 = 5
5 + 2 = 7
7 + 2 = 9

Evens
2 + 2 = 4
4 + 2 = 6
6 + 2 = 8
8 + 2 = 10

Finally, I would like to introduce you to one concept and number that is conspicuously absent from our adding lessons. The number 0. It is perhaps the easiest number to add.

1 + 0 = 1
2 + 0 = 2

A general pattern: number + 0 = number

Now let's construct an adding table: Adding Chart
3:25 pm
[joe_doe_zero]
Concepts in Addition
If you know how to count, then you're on your way to being an adding whiz! Even if all you can do in your head is addition, then you're still considered a calculating whiz.

For those of you who don't know counting, just memorize these numbers in this particular order: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...

Those are known as the counting numbers. (If you're a mathematics major, you might better know them as the natural numbers.)

Now, let's add by 1.

1 + 1 = 2
2 + 1 = 3
3 + 1 = 4
4 + 1 = 5
5 + 1 = 6
6 + 1 = 7
7 + 1 = 8
8 + 1 = 9
9 + 1 = 10

Simple? See if you can memorize it. Better yet, see if you can find an alternative to memorization. You should know that if 1 is added to a number then the result is the next counting number.

A general pattern: (counting number) + 1 = (next counting number)

So if 13 is the next counting number after 12, then 12 + 1 = 13

Let's take the number 6.

6 + 1 = 7

We can reverse the order the numbers are added and still get the same result.

6 + 1 = 7
1 + 6 = 7

6 + 1 is the same thing as 1 + 6.

Let's do a quick review of the counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and so on...

1 and 6 are known as addends and the number 7 is known as the sum.

A general pattern: addend + addend = sum

If you're starting to think of mathematics as a bunch of general patterns, then you're already thinking like a mathematician.

But the general pattern can be put into a more general pattern.

Sum = addend + addend

Let's say with the numbers 6, 1, and 7, we can formulate four equations.

6 + 1 = 7
1 + 6 = 7
7 = 1 + 6
7 = 6 + 1

Let's take another case, say 3, 1, and 4.

3 + 1 = 4
1 + 3 = 4
4 = 1 + 3
4 = 3 + 1

Got it? Good. Now make equations out of these numbers.

Problem 1: 1, 2, and 3
Problem 2: 6, 1, and 7 (already done)
Problem 3: 8, 1, and 9
Problem 4: 9, 1, and 10
Problem 5: 10, 1, and 9 (see problem 4 but re-do it)
Problem 6: 11, 12, and 1
Problem 7: 10, 1, and 11
Problem 8: 88, 89, and 1
Problem 9: 1, 100, and 99
Problem 10: 566, 1, and 567
Sunday, January 2nd, 2005
1:02 am
[joe_doe_zero]
Subtraction

Subtraction



If you have mastered simple addition then subtraction should come easy. Subtraction is "taking away" something. If I have five apples and I take away some apples then I will have fewer apples. The subtraction sign is the minus sign (-) just as the addition sign is plus sign (+).

For example let's say we have five apples and we take away two. How many remain?

Problem: 5 - 2

Let's remember our number line.

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

We start with five (5) because that is our starting quantity. Now we take away two (2). We move to the left instead of to the right.

0--1--2--3**4**5--6--7--8--9--

That leaves us at three (3). Therefore:

5 - 2 = 3

or...

Five minus two equals three.

That is how to do simple subtraction.
Saturday, January 1st, 2005
11:02 am
[joe_doe_zero]
Addition

Addition



Now that you know how to count, next we can learn how to do it faster. Let's say we have the number one (1) and we want to add one to it two times. We can do it just by adding two. We have:

1 + 1 + 1 = 1 + 2

1 + 1 = 2 (1 time)
2 + 1 = 3 (2 times)

or

1 + 2 = 3

Let's say we want to add 3 to 5. We have:

3 + 1 = 4 (1 time)
4 + 1 = 5 (2 times)
5 + 1 = 6 (3 times)
6 + 1 = 7 (4 times)
7 + 1 = 8 (5 times)

or

3 + 5 = 8

If we want to add 8 to 5, we have

5 + 1 = 6 (1 time)
6 + 1 = 7 (2 times)
7 + 1 = 8 (3 times)
8 + 1 = 9 (4 times)
9 + 1 = 10 (5 times)
10 + 1 = 11 (6 times)
11 + 1 = 12 (7 times)
12 + 1 = 13 (8 times)

or

5 + 8 = 13

To do this a faster way draw a line and write numerals on it such as this:

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

Now to add, we start from the starting quantity and move the number of spaces that we are adding by.

To do 3 + 5 we go

0--1--2--3**4**5**6**7**8--9--

To get eight (8).

That is how to do simple addition.
10:50 pm
[joe_doe_zero]
Our first lesson...

Lesson 1 - Numbers and Numerals



For this lesson I will assume you do not know the concept of numbers or numerals. First, what is a number? It is a quantity. Let's say you are thirsty and need some glasses of water. How much water do you need? At times you might need only a glass of water. At other times you might want a glass followed another glass. And yet at other times you might need a glass followed by another glass which is still followed by another glass.

If you only need a glass then you need one glass of water. The quantity of glasses of water is one.
If you need a glass followed by another glass then you need two glasses of water. The quantity of glasses of water is two.
If you need a glass followed by another glass which is followed by yet another glass then you need three glasses of water. The quantity of glasses of water in this case is three.

In everyday usage we assign numerals to the quantities.
The number one is represented by the numeral 1.
The number two is represented by the numeral 2.
The number three is represented by the numeral 3.

Adding and Counting
Let's say we have one glass of water and we wish to make it two. In this case we add one glass of water to what is already there. If we add a glass of water then we denote it by + 1. The + tells us that we added and the numeral after it tells us how many we added. If we wish to have three glasses of water and we only have two then we add one glass of water to it. In this case we have 2 + 1 = 3. The 2 tells us how many we had before. The + tells us that we are adding. The 3 tells us that we now have three glasses of water. The = is the equals sign. It tells us that the quantities on both sides of the equals sign are the same in quantity.

2 + 1 = 3 could be read in everyday language as "two plus one equals three".

Other numbers and numerals
To start out counting in everyday usage, we first start with the number one (1). To continue counting we add one to it (+1) to get two (2). To get to the next counting number we add one to it again and get three (3). Thus far we have:

1 + 1 = 2
2 + 1 = 3

We can continue to get another quantity yet higher than three which is four written as a 4.

3 + 1 = 4

Then a quantity called five (5) follows.

4 + 1 = 5

Then six (6), seven (7), eight (8), nine (9), and then ten (10).

5 + 1 = 6
6 + 1 = 7
7 + 1 = 8
8 + 1 = 9
9 + 1 = 10

You might be curious as to why ten is represented by two numerals (namely 1 and 0) instead of a numeral of its own. In this case the 1 stands for the quantity ten. Think of the numbers we just covered as this:

1 is 01
2 is 02
3 is 03
4 is 04
5 is 05
6 is 06
7 is 07
8 is 08
9 is 09
10 is 10

Since we have exhausted the everyday numerals, we then write the 1 in the next column. We can continue counting if we so wish.

10 + 1 = 11
11 + 1 = 12
19 + 1 = 20

And so on. In the case of 12, the 2 is in the ones' place and the 1 is in the tens' place.
There is another quantity which we have not yet covered. This is the quantity of zero also written as (0). Zero is such a quantity that

0 + 1 = 1.

Zero plus one equals one. Now practice counting: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11...
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