You probably know that, when doing arithmetic, the minus sign, -, means that you are to subtract, and the plus sign, +, means that you are to add. Likewise, on your calculator, the key with the + sign and the key with the - sign mean add and subtract. But we also use the signs + and - for another purpose: to indicate positive and negative numbers. This is not exactly the same as add and subtract!

When we talk about a number, 5 for example, we mean +5. On the 'number line', below, that means 5 notches above 0. Which notch is called zero is either a random choice, or is chosen for a particular reason. If we want the fifth notch below the zero notch, we call it -5. These are called SIGNED NUMBERS.

 If you have been in the Opticianry field for very long, you have probably become familiar with a number line that is vertical, instead of the horizontal number line that is usually used in math books. It is the power drum on your lensometer.

The notch that is called zero is the point on the power drum that indicates no lens power. For now, we will just consider what happens when traveling from one point of the number line to another point.

On the number line above, we consider travel from lower to higher to be positive travel.  So, when we wish to add, we will travel in a positive direction. We will consider travel from higher to lower to be negative travel.  So when we wish to subtract, we will travel in a negative direction.

In a math course, any number that does not have a sign is assumed to be positive. In optics, when dealing with lens power, we do NOT make that assumption: lens powers should have either a plus sign or a minus sign. In this discussion of arithmetic with signed numbers, all of the numbers will have either a plus sign or a minus sign.

Suppose we wish to ADD a +3 to a +2. Locate the +2 on the number line.

Since we are adding a positive number, we will travel in a positive direction: up. We are adding +3, so we will travel three notches up. The notch that we end out on is +5. So, (+2) + (+3) = +5. You already knew that, didn't you?

Or, start with -2 and add +3. Locate the -2 on the number line and then move in the positive direction 3 notches. The result is +1. So, (-2) + (+3) = +1.

OK. Now, lets pretend that I have nothing in my pocket, and I want to buy a $3.00 chocolate sundae, and you have a $10.00 bill in your pocket, and you want one too. What happens? Well, you buy us both a chocolate sundae! (What? You would rather have strawberry? Well, OK, you are buying, after all!) What is the result for me? I started with 0, and I added a debt of $3.00, which is less than 0, so I ADDED -3 to 0, and ended out with -3. So, 0 + (-3) = -3. What if I actually had $1, and only needed to borrow $2? Then (+1) + (-3) = -2.

How about if I had already borrowed $2 from you, so my pocket already has -2 in it, and I want that chocolate sundae anyway? Then (-2) + (-3) = -5.

We have demonstrated the first rule of arithmetic for signed numbers.
 
 

1. When ADDING two numbers with the same sign, add the amounts and give the result the common sign.
• (+6) + (+3) = +9
• (+1) + (+10.50) = +11.50
• (-4.50) + (-2.50) = -7.00
• (-18) + (-32) = -50
2. When adding two numbers with unlike signs, subtract the amounts and give the result the sign of the larger number.
• (+6) + (-2) = +4
• (-1) + (+10.50) = +9.50
• (-4.50) + (+2.50) = -2.00
• (+48) + (-32) = +16

• (+6) - (+3) = (+6) + (-3) = +3
• (+1) - (+10.50) = (+1) + (-10.50) = -9.50
• (-4.50) - (-2.50) = (-4.50) + (+2.50) = -2.00
• (-18) - (-32) = (-18) + (+32) = +14
• (+6) - (-2) = (+6) + (+2) = +8
• (-1) - (+10.50) = (-1) + (-10.50) = -11.50
• (-4.50) - (+2.50) = (-4.50) + (-2.50) = -7.00
• (+48) - (-32) = (+48) + (+32) = +80

PS:  Don't use your calculator this time.  ABO does not allow calculators.  Some of the exercises I'll tell you to use your calculator on.  You SHOULD be able to do these with paper and pencil.

;-)  <-- that's a smile, by the way.

a. +44
b. -4.50
c. -54.3879
d. +4.25
e. -8.25
f. -9.12

If I want to enter (-1) - (-2) into the calculator without doing any calculations on paper or in my head [such as changing the problem to (-1) + (+2)], I need a way to change the sign of the number.

On your calculator there is a key marked +/-. This key means: "change the sign of the number shown in the display". It does not result in addition or subtraction. It changes the sign of  the number in the display [either from + to - or from - to +]. So, for the problem (-1) - (-2), we will enter 1, press the +/- key, [the display now shows -1] then the - key [meaning subtract], then 2, press the +/- key, [the display now shows -2], then press = or "enter". The answer 1 [really meaning +1] should show up on the screen.

Try this one for yourself: just press in digits in the order that they are shown. When you come to a subtraction or an addition, use the - key or the + key. When you come to a parenthesis, enter the digits, then the +/- if the number is negative, then go right on to the next operation.

(+12) + (-13) - (+15) + (+10) + (-6) - (-5) =

 Do it first, write down your answer, then follow my steps.

• enter 12
• press +
• enter 13
• press +/-
• press -
• enter 15
• press +
• enter 10
• press +
• enter 6
• press +/-
• press -
• enter 5
• press +/-
• press = or "enter", whichever your calculator says.

Do these with your calculator:
 

g. +47.5
h. -50.1379

This brings us to the rules for multiplying signed numbers:

1. When multiplying two numbers with THE SAME sign, multiply the amounts and give the result a plus sign.
• (+6)(+3) = +18
• (-1)(-10.50) = +10.50
• (-4.50)(-2.50) = +11.25
• (+18)(+32) = +576
[Remember: sets of parenthesis with nothing between them mean multiply.]
 
2. When multiplying two numbers with UNLIKE signs, multiply the amounts and give the result a minus sign.
• (+6)(-2) = -12
• (-1)(+10.50) = -10.50
• (-4.50)(+2.50) = -11.25
• (+48)(-32) = -1,536

 What about using the calculator? You will use the +/- key the same way we did in the last section: for the problem above, you enter:

• enter 3
• press +/-
• press x
• enter 5
• press +/-
• press x
• enter 2
• press x
• enter 1
• press +/-
• press = or "enter", whichever your calculator says.

One final item about multiplication: what happens when you square a number? Squaring a number is the same as multiplying it times itself., Cubing a number is multiplying it by itself three times. Thus,

  (-3) = (-3)(-3) = +9
  (-5) = (-5)(-5)(-5)(-5)(-5) = -3,125

Notice that, since a minus times a minus is plus, any number squared will result in a positive number: (-1)(-1) = +1. A number to the 4th power will also always be positive: (-1)(-1)(-1)(-1) = +1. A negative number to an EVEN numbered power will be positive. A negative number to an ODD numbered power will be negative.

[When multiplying you can ignore the signs until the multiplication is complete, and then count the minus signs. If there were an even number of minuses, the result is plus. If there were an odd number of minuses, the result is minus.]

Try these:
 

i. (+15)(+29)(-4)(-0.5) =

j. (+2.00)(-1.50) =

i. +870
j. -13.5

RIGHT
It is POSITIVE!

1. When dividing two numbers with THE SAME sign, divide the amounts and give the result a plus sign.
• (+6) / (+3) = +2
• (-1) / (-10.50) = +0.095238...
• (-4.50) / (-2) = +2.25
• (+18) / (+36) = +0.5

 
2. When dividing two numbers with UNLIKE signs, divide the amounts and give the result a minus sign.
• (+6) / (-2) = -3
• (-1) / (+10.50) = -0.095238...
• (-4.50) / (+2) = -2.25
• (+18) / (-36) = -0.5

k. +0.6
l. -0.4

If you had trouble with any of this, please go to your friendly local public library and get a basic math textbook and do the exercises.