The Alphabet of Math: The integers and subtraction

We can do some simple forms of math using only the natural numbers and zero. The natural numbers and zero are called the whole numbers. Right now we haven't developed the notion of pieces of numbers, but since we call the natural numbers and zero the whole numbers, you can guess that we are going to eventually talk about numbers that aren't whole. We will, but one step at a time.

All of our numbers, so far, have fit on a number line that starts at zero and moves to the right. But what happens if you want to start at zero and move to the left? The number line can be extended to the left from zero using the concept of negative numbers. We simply number from zero, counting up just like we do on the right, but we preface each number with a negative sign to indicate it is to the left of zero.

-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6

This extended set of numbers (the natural numbers, zero and the negative numbers) are called the integers. The integers with a negative sign are called negative numbers and those without a negative sign are called positive. We can do addition and multiplication on positive numbers just as before. But, since negative numbers are the opposite of positive numbers, when we encounter them in addition or multiplication, we change our direction and move to the left. If an addend is positive, move to the right. If an addend is negative, move to the left.

Let's look at some examples of addition using negative numbers.

6+ -2. Start at 6 and move to the left by 2.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6
6+ -2 =4

-6+ 2. Start at -6 and move to the right by 2.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6
-6+ 2 =-4

6+ -8. Start at 6 and move to the left by 8.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6
6+ -8=-2

8+ -6. Start at 8 and move to the left by 6.
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8
8+ -6=2

Notice the similarity between 8+ -6 and 6+ -8. They both equal 2 but the signs are different. Notice how in 6+ -8 the answer has the sign of the number with the higher magnitude. The same thing happened with 6+ -2 and -6+2. The answer is still 4, but with the sign is the same as the number with the higher magnitude.

This process of adding by moving to the left instead of the right is called subtraction. Subtraction is the opposite of addition. In addition, you start at a number and move to the right. In subtraction, you start at a number and move to the left. Negative signs tend to have more power than a positive sign, so when you see them side by side, the negative sign takes over. Because of this, 6+ -8 is written as 6-8 and 6+ -2 is written as 6-2.

What happens if we start with a negative number and subtract?
-4-2 = -6
-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6

When the signs are the same, you add the numbers and keep the sign. When the signs are different, you subtract the numbers and keep the sign of the larger number.

The integers inherit the commutativity and associativity of the natural numbers. Zero is still the identity of addition. With the integers, addition gets another special number called an inverse. The inverse is a number that you add by to get the identity. (Multiplication also has an inverse, but we need another type of number that will come later.) So the inverse of 4 is -4 because 4-4=0. The inverse of -4 is 4 because -4+4=0.

How about multiplication of negative numbers? We need to add a step to our multiplication process to account for negative multiplicands. If the multiplicand (the number on the left) is positive, we face to the right. If the multiplicand is negative, we face to the left. If the multiplier is positive we jump forward. If the multiplier is negative we jump backward.

3×-3
Start facing to the right and jump backwards.
························································>
-9···-8···-7···-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8···9
3×-3=-9

-3×3
Start facing to the left and jump forwards.
························································<
-9···-8···-7···-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8···9
-3×3=-9

-3×-3
Start facing to the left and jump backwards.
························································<
-9···-8···-7···-6···-5···-4···-3···-2···-1···0···1···2···3···4···5···6···7···8···9
-3×-3=9

Notice when the signs are different, the answer is negative and when the signs are the same the answer is positive. If you are multiplying a string of numbers and the number of negative signs is odd the answer is always negative.

So far we have been looking at what happens when we add or subtract two numbers. We can also compare two numbers. We look at the relative position of the two numbers on the number line and determine which one is further to the left or right. If a number is further to the right than the other, it is said to be greater than the other. If a number is further to the left than the other, it is said to be less than the other. If two numbers are at the same place on the number line, then they are said to be equal to each other. There is a law in mathematics called the trichotomy law. It says that a number can compare to another number in only one of three ways (tri is Greek for three). It is either greater than, less than or equal to the other number. Greater than is denoted by < and less than is denoted by >. The arrow always points to the smaller number.

3 < 6 and -3 > -6.

It gets confusing with negative numbers so you have to mentally place the numbers on the number line and look at which number is further to the right. That number is the greater number. On a number line, -3 is further to the right than -6, so -3 is greater than -6.

Comparisons are not usually done on individual numbers. You are more likely to see them in reference to two equations. You simplify each equation separately, then compare the resulting numbers.
3+6 < 3×4
9 < 12

4×4 = 12+4
16 = 16

3×1-2 > 3+4×-2
3-2 > 3-8
1 > -5

Now that we have the operation of subtraction, let's revisit division in the next blog.