Tag

INTEGERS

Browsing

Continuing on in our review of basic math, I welcome you to Day 2. The answers to Day 1 questions are at the bottom of the post — along with new questions. But first, let’s learn how to multiply and divide integers.

Let’s say you have a bank account with a service fee of $15 per month. If that amount was deducted every single month, how can you represent the yearly amount for these fees? Well, you would multiply -$15 (the fee is negative because it’s taken out of the account) by 12 (the number of months in the year). But how the heck do you multiply negative and positive numbers? Let’s find out.

Remember integers — those negative and positive numbers that aren’t fractions, decimals, square roots, etc.? I like to think of them as positive and negative whole numbers (though most real mathematicians would argue against that classification). On Wednesday, you learned how to add and subtract these little buggers. (Check out the post here, if you missed it.)  Today, we multiply and divide.

Her’s the really good news: it is way, way easier to multiply and divide integers than to add and subtract them. First, though, it’s a good idea to understand how the rules work. When you first started multiplying numbers, you did things like this:

2 x 3 = 2 + 2 + 2 = 6

In other words “2 x 3” is the same thing as adding up three 2s. Get it? And because you started working with positive numbers when smacking a girl upside the head meant you “like-liked” her, you know without a shadow of a doubt that the answer is positive.

Let’s see what happens when you multiply a negative number by a positive number:

-2 x 3 = -2 + -2 + -2 = -6

Now to understand this, you need to either pull up your mental number line and count or remember the addition rules from Wednesday’s post. When you add two numbers with the same sign, add the numerals and then take the sign. So -3 + -3 is -6.

But what about multiplying two negative numbers? Admittedly, this is a little trickier to explain. It helps to look for a pattern using a number line. Let’s try it with -2 x -3.

-2 x 2 = -4
-2 x 1 = -2
-2 x 0 = 0
-2 x -1 = ?
-2 x -2 = ?

Based on the pattern shown on the number line, what is -2 x -1? What is -2 x -2? If you said 2 and 4, you are right on the money.

And now we can summarize the above with some rules. Believe me, this is one math concept that is much, much easier to remember with the rules. Still, if knowing why helps anyone get it, I’m all for pulling back the curtain.

When multiplying integers:
If the signs are the same, the answer is positive;
If the signs are different, the answer is negative.

Bonus: The same rules work for division. That’s because division is the inverse (or opposite) of multiplication.

When dividing integers:
If the signs are the same, the answer is positive;
If the signs are different, the answer is negative.

The only tricky part is this: Sometimes it seems that if you are multiplying or dividing two negative numbers, the answer should be negative. It’s a trap! (Not really, but you could think of it that way, if it helps.) The key in multiplying and dividing integers is noticing whether the signs are the same or different.

In fact, if you are doing a whole set of these kinds of problems, you can simply run through the problems and assign the signs to the answers — before even multiplying or dividing. (I tell students to do this all the time, because I think it helps them to remember the rules.)

4 x -3 → signs are different → answer is negative
-4 x -3 → signs are the same → answer is positive
-4 x 3 → signs are different → answer is negative
4 x 3 → signs are the same → answer is positive

Then all you’d need to do is the multiplication itself:

4 x -3 = -12
-4 x -3 = 12
-4 x 3 = -12
4 x 3 = 12

And like I said, division works the same way:

-24 ÷ -2 = +? = 12
24 ÷ -2 = -? = -12
24 ÷ 2 = +? = 12
-24 ÷ 2 = -? = -12

Got it? Try these examples on your own.

1. 5 x -6 = ?

2. -18 ÷ 9 = ?

3. -20 ÷ -4 = ?

4. 8 x 4 = ?

5. -2 x 7 = ?

Questions? Ask them in the comments section. Up Monday are fractions. If you can’t remember how to add, subtract, multiply or divide fractions or mixed numbers, tune in. 

Answers to Wednesday’s “homework.” (It’s not really homework, I promise.) -10, -4, 2, -15, -2. How did you do?

Welcome to Day 1 of our tour of basic math. If your New Year’s Resolution is to brush up on your math skills. You’re in the right place. 

Winter is really the perfect time to talk about integers.

But first, what are integers? It’s quite simple, really. They’re positive and negative whole numbers. These are integers: -547, 9, 783, and -1. These are not integers: 0.034, -0.034, √3, and -1/2.

You are very familiar with positive integers. For the first three years of your formal education, you probably worked exclusively with these little buggers — or as you called them, “numbers.” You learned to count them, tell time with them, add/subtract/multiply/divide them, and even write them out as words.

(Soon after, you learned about fractions and then decimals, which are not integers, but are still positive, so it was all good.)

If you’re like me, the part that completely blew your mind was when you first learned that numbers could be negative. Now that I think back, this was kind of a silly surprise in my world. I grew up in an area of the United States that gets pretty cold in the winter. This means two things: we measured temperatures with Fahrenheit and the temps got below zero. And those two things pointed to negative numbers. Duh.

Regardless, with a lot of work and determination, I finally understood integers, which included adding and subtracting negative and positive whole numbers. But before I show you how this is done, let’s take a look at the number line, which can help you visualize how this works.

The number line isn’t a real thing. It’s just a way to visualize how numbers work. And the key is the zero in the middle of the line. Notice what happens on the right — the numbers get larger, one by one, right? And what happens on the left? Yep, they get smaller.

Did you get that smaller part? If not, don’t worry. You’re just a little rusty. See, when two numbers are negative, the smaller one actually has the larger numeral. In other words -37 is smaller than -1, while 1 is smaller than 37.

(This is a good time to note something else that you may have forgotten. If a number has no sign, it is positive. The positive sign, +, is understood.)

If you can picture a number line, you can add and subtract integers, no problem. Here’s how:

-1 + 3 = ?

Start at -1 and count three places to the right. We’re counting to the right because we’re adding. What is the number on the number line? If you said 2, you’re right on target.

4 – 5 = ?

This time start at 4 and count five places to the left. That’s because we’re subtracting. What do you get? If you said -1, give yourself a gold star.

So this number line thingy is pretty cool, but it’s not all that useful if you need to find an answer pretty quickly. And what happens if the second number is negative? (Well, you change direction, actually, but that’s pretty clunky and somewhat confusing. So how about if we find another process?)

Once you understand the why of adding and subtracting integers, you can learn an algorithm that works every single time. It goes like this:

This is much easier to understand with an example:

-10 + 4 = ?

We’re adding two numbers with different signs. That means we need to ignore the signs, find the difference and take the sign of the larger numeral. But what does “find the difference” mean? It’s pretty simple, actually. Just subtract the smaller number (without the sign) from the larger number (without the sign). 10 – 4 is 6, and if we take the sign of the larger numeral, the answer is -6.

Another way to think of “difference” is the distance between the two numbers on the number line. So if you got back to the number line, it’s a matter of counting spaces between the two numbers. Then take the sign of the larger numeral. Make sense?

-10 + 4 = -6

Okay, let’s try a subtraction example.

-3 – 9 = ?

First step is to change the subtraction to addition and change the sign of the second number.

-3 + -9 = ?

Now all you need to do is follow the addition rule for numbers with the same signs. That means to ignore the signs, add, and keep the sign.

-3 + -9 = -12

So, no need to pull out a number line for these. Just practice with these rules, and you’ll have them down in no time at all. Here are a few additional examples to help you.

5 – 8 = 5 + -8 = -3

-7 – 4 = -7 + -4 = -11

3 + -3 = 0

-12 + 8 = -4

Now, try these out on your own. I’ll post the correct answers on Friday. And if you have questions, ask them in the comments section.

1. 15 – 25 = ?

2. -7 – -3 = ?

3. 10 + – 8 = ?

4. -3 – 12 = ?

5. -6 + 4 = ?

Photo courtesy of .raindrops.

Every so often, at around 7:00 p.m., I’ll get a call from someone I know.  “I don’t understand my kid’s math homework,” they’ll say.

These folks aren’t dumb or bad at math.  But almost always, they’ve hit a concept that they used to know, but don’t remember any more.  And those things can trip them up — big time. So, I thought it might be helpful to review 4 middle school math facts that may give parents trouble.

Every number has two square roots.

This is the question that prompted this blog post.  I got a call from a friend who didn’t understand this question in her daughter’s math homework: “Find both square roots of 25.”  Both?

Most adults have probably forgotten that each number has two square roots. That’s because we are typically only interested in only one of them — the positive one.

Yep, the square roots of 25 are 5 and -5.  In other words:

[pmath]sqrt{25}[/pmath] = 5 and -5

It should be pretty easy to see why this is true.  (You just have to remember that when you multiply two negative numbers, your answer is positive.)

5 · 5 = 25

-5 · -5 = 25

1 is not prime.

This question came up in my own daughter’s homework last week — a review of prime and composite numbers.  Remember, prime numbers have only two factors, 1 and the number itself.  So, 7 is prime.  And so are 13, 19 and even 3.  But what about 1?

Well, it turns out the definition of a prime number is a little more complicated than what we may assume.  And I’m not even going to get into that here.

But there is a way for less-geeky folks to remember that 1 is not prime. Let’s look at the factors of each of the prime numbers we listed above:

7: 1, 7

13: 1, 13

19: 1, 19

3: 1, 3

Now, what about the factors of 1?

1: 1

Notice the difference?  Prime numbers have two factors, 1 and the number itself.  But 1 only has one factor.

0 is an even number.

This idea seems to trip up teachers, students and parents.  That’s because we tend to depend on this definition of even: A number is even, if it is evenly divisible by 2.  How can you divide 0 into two equal parts?

It might help to think of the multiplication facts for 2:

2 x 0 = 0

2 x 1 = 2

2 x 2 = 4

2 x 3 = 6 …

All of the multiples of 2 are even, and as you can see from this list, 0 is a multiple of 2.

Anything divided by 0 is undefined.

Okay, this gets a little complex, so bear with me.  (Of course, if you want, you can just memorize this rule and be done with it.)

First, we can describe division like this:

[pmath]r={a/b}[/pmath]

Using a little bit of algebra you can get this:

r · b = a

So, what if = 0?

r · 0 = a

That only works if is also 0, and 0 ÷ 0 gives us all kinds of other problems.  (Trust me on that.  This is where things get pretty darned complicated!)

So how many of you have thought while reading this, “I will never use this stuff, so what’s the point?” You may be right.  Knowing that 0 is an even number is probably not such a big deal.  But at least your kid will think you’re extra smart, when you can help him with his math homework.

What are your math questions?  Is there anything that’s been bugging you for ages that you still can’t figure out?  Ask your questions in the comments section.  I’ll answer some here and create entire posts out of others.