Addition, Subtraction, and Whole Numbers

Background 

Now that we have learned about Whole Numbers, Addition, and Subtraction, let's work a problem (there's more problems in the Practice Problem section)

Question 

Find 5 + 2 - 3 - 1 + 6 - 0 + 0

Answer 

9

Analysis  

In prior entries, we've worked individual calculations, say like 5 + 2, both with and without the number line. We'll do the same here but we'll start on with the leftmost term and work our way through the different operations (the plusses and minuses).

We're evaluating 5 + 2 - 3 - 1 + 6 - 0 + 0

We can start with a number line and a red dot on the number 5:



So let's colour the 5 in our expression red: 5 + 2 - 3 - 1 + 6 - 0 + 0

Our first operation is plus 2 (+ 2). We move our point two numbers to the right, to 7 (I'll show that with blue):

5 + 2 - 3 - 1 + 6 - 0 + 0



Now starting at the blue dot at 7, we subtract 3 (- 3), or move 3 spots to the left. We'll land on the 4. I'll show that with green:

5 + 2 - 3 - 1 + 6 - 0 + 0



Starting at 4, we subtract 1 more to land on 3 (I'll show that in orange):

5 + 2 - 3 - 1 + 6 - 0 + 0



Now we add 6. Starting from the 3, we'll land on 9 (shown in purple):

5 + 2 - 3 - 1 + 6 - 0 + 0



And now we're asked to first subtract 0 and then add 0. Notice that when we add or subtract 0, the dot doesn't move! And so we leave our dot at 9.

This ability, to add and subtract 0 as many times as you want to any number and not change that number's value (like what we did with the 9 - we didn't change its value at all by adding or subtracting 0), gives 0 a special name - the additive identity.

Ok - so now let's work this same problem one more time, but this time without the number line. Here we go:

5 + 2 - 3 - 1 + 6 - 0 + 0

7 - 3 - 1 + 6 - 0 + 0

4 - 1 + 6 - 0 + 0

3 + 6 - 0 + 0

9 - 0 + 0

9

We start on the left and work our way to the right.

Vocabulary:

• Whole Numbers - The set of numbers that starts with 0 and increases by 1 (0, 1, 2, 3,...)

For more information check out these links (comment to add your favourite link): 
Where might you have come from? 

Fact-orials Index

Numbers:

• The Number 0
• Whole Numbers

Operations:

• Addition
• Subtraction

Graphing:

• Number Line

Where might we go?

Operations:

• Grouping Terms, Parentheses

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers - Practice Problems
• Multiplication, Division, and Whole Numbers
• Addition, Subtraction, and Integers

Properties:

• Associative Property

The Number 0

Background 

We've covered Counting/Natural Numbers, which is the number 1 and then the next number is plus 1, so 1, 2, 3, 4, ...

Question 

What about 0? 

Answer 

Zero took a long time to come into its own as a number. See below for a very brief history.

Analysis 

While it's pretty straightforward to recognize the existence of a thing and then count it (like with the number 1), the concept of 0 took a lot longer to come into existence. And it makes sense - how do you count absence? Non-existence? Nothing?

Consider a situation where you have 2 eggs. You then make an omelette with those 2 eggs. You had 2 eggs and now you have... well... no eggs!

The history of the Counting/Natural Numbers is lost in the mists of time across a whole host of cultures, the same can't be said for 0. For many civilizations, there was no need for 0. The Romans, for instance, had no symbol for 0 at all - the way they wrote numbers didn't require placeholders. For instance:

I = 1

II = 2

III = 3

IV = 4

V = 5

VI = 6

VII = 7

VIII = 8

IX = 9 

X = 10

and the symbols continue with ones for 50, 100, 500, and so on. No need for zeros. (See the post on Roman numerals for more).

However, if you have a number system such as our current 10 decimal system (known as base 10) where we can make all of our numbers from the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and putting them into different places (1's, 10's, 100's, and so on), where the need for a 0 to hold a place would show up. Before 0, that particular place would simply be blank. And so the numbers 11 and 101 would look like this:

11

1 1

Is that second number really 101? Or is it 11? Perhaps a 1 and another 1?

Even with that, it was only the mathematically advanced civilizations of Sumeria, India, and the Maya who used the number 0. For the Sumerians and the Maya, 0 acted only as a placeholder in larger numbers - it was never a number in and of itself.

But it was 5th century India, where the concept of "emptiness" and "emptying the mind" were coming to the fore and becoming part of religious texts, that a symbol arose to help express that emptiness. If 1 is the number for the Self, then 0 is the number for the empty mind.

Once zero was introduced as a symbol for an actual value, it started being used in mathematics. Questions such as 2 + 0 = ? made sense - and it's this ability to ask that kind of question that leads to algebra (we'll discover that in later entries).

Vocabulary:

• Counting/Natural Numbers - The set of numbers that starts with 1 and increases by 1 (1, 2, 3, 4,...)

For more information check out these links (comment to add a resource to the list!): 

Live Science article on the History of 0
http://www.newworldencyclopedia.org/entry/0_(number)

Where might you have come from? 

Fact-orials Index

Numbers:

• The Number 1
• Counting/Natural Numbers

Operations:

• Addition

Where might we go?

Numbers:

• Whole Numbers
• Prime and Composite Numbers
• Expressing Values, and Bases

Operations:

• Multiplication
• Division

Operations with different kinds of numbers:

• Addition, Subtraction, and Whole Numbers

Algebra