Arithmetic is full of surprises! Don’t believe it? Try this question on a friend:

When does 10 + 4 = 2? He’ll probably think you are crazy, but you’re not. This question has a perfectly sensible and important answer: When it is 4 hours after ten o’clock.

Adding and subtracting on a clock doesn’t always work the same way as adding and subtracting regular numbers. Use the clock to solve the following problems. Then use the decoder and the numbers in the shaded boxes to figure out the answer to this riddle: What time did the math teacher go to the dentist?

## Calendar Math

Here is another practical arithmetic problem. When does 5 + 3 = 1? Three days after the fifth day of the week: from Thursday (5) to Sunday (1).

It doesn’t really make sense to say “Thursday + 3 = Sunday,” but if you give each day of the week a number, then you can do calendar arithmetic exactly as you did clock arithmetic. The trick is, if your result is over 7. just subtract it from the total. In our example, 3 + 5 = 8; 8 – 7 = 1. Now, what day is 12 days after a Wednesday? 4 + 12 = 16: 16 – 7 = 9; 9 – 7 = 2. This means that 12 days after Wednesday is a Monday-two weeks from then.

When Numbers Don’t Obey Why don’t hours of the day and days of the week work the same way as normal numbers? Maybe it has to do with limits. Normally, numbers go up as far as infinity, so you never have to start over. With defined terms like the day (which can never have more than twenty-four hours) or the week (which can never have more than seven days), you can’t go on forever and therefore need to start over, which messes up the calculation. Can you think of any other instances when numbers don’t behave normally?

**NUMBERS WITH DIRECTION**

recall that numbers were originally used to count things, like sheep, and it took centuries before a symbol was invented to represent zero. Are there any numbers that are less than zero?

Today, most people have heard of negative numbers, and know that they are not at all imaginary and are really quite useful. Think of where you may have heard of numbers like -5 and -10. Did you think of a thermometer? Negative numbers represent temperatures below 0 degrees (those are the cold days, whether you use a Fahrenheit or Celsius thermometer). For every “positive” number, there is its twin “negative,” and vice versa. These pairs are called “opposites”: 3 and 3; -12 and 12: 135,000,789 and 135,000.

Using a simple number substitution (A = 1, B = 2, C = 3, and so on), figure out the coded word below to get the name of a useful math tool that will help to show you negative numbers and how they work.

### Adding Signed Numbers

Why would we want negative numbers? One reason is that mathematicians don’t like problems that have no answer. Everyone knows that we can add any two numbers, say 5 + 3. Most people also know that in addition, the order of the numbers doesn’t matter. The answers to 5 + 3 and 3 + 5 are the same .

Now let’s try subtraction: 5-3 is no problem; everybody gets 2. But what if we switch the numbers? What is 3 – 5? Is it still 2? Think of what subtraction means. If you have five pieces of candy and give three to your little brother (very generous of you!), then you have two pieces left. But if you have three pieces of candy and give five to your little brother…

hey, wait a minute, you can’t do that. And if you could, you sure wouldn’t have two pieces left! So at least with candy, 3-5 doesn’t make any sense:

there is no answer.

Mathematicians really hate that. If you can do 5-3, you ought to be able to do 3-5. So what is the answer? To find out, we use a number line. Notice that if you pick any number on the number line, all the numbers to its letter are less than it while all the numbers to its right are greater.

Adding 5 + 3 on the number line is easy. You start at the 5, and then count three more spaces to the right and see where you end up (which is always on 8, since 5 + 3 = 8). Note that starting at the 3 and counting five spaces to the right gets you to the same place (3 + 5 = 8). Subtracting is just as easy. To do 5 -3, you start at the 5 and count three spaces to the left

Now, what about 3 – 5? After you pass 0. you get to -1 (one less than zero). -2 (two less than zero), 3 (three less than zero), and so on-we use a negative sign in front of the number to show numbers that are to the left of zero on the number line. If you move five spaces to the left of 3. you get to 2. So 3-5= -2. Now, what about 2-7? If you got -5.

you are on the right track!

The Bottom Line Can you come up with the rules for adding and subtracting positive and negative numbers? Here is a summary: • Add a positive number by moving right.

Add a negative number by moving left. Subtract a positive number by moving left. Subtract a negative number by moving right

**Hard to Believe!**

Even after mathematicians began thinking about negative numbers, num bers that are less than zero, many of them called such numbers “fictitious, ‘imaginary, or useless.”

MAGIC SQUARES

magic squares have been popular math puzzles for over 3,000 years, and once were thought to have mystical powers. Follow these simple rules to complete your own magic square:

1. Use each number only once. 2. Each row, column, and diagonal must a do up to the same answer.