why is half multiplied by half equal a quarter?

Question

why is 4 times 4 16
and why is - 4 times -4 16

it does

0.5
0.5
------
0.25

Answer 1

This is an excellent question and shows that you want to understand why algebra works the way it does and not just take everything because some authority says so. Most people will just say "that's the way it is," just like they justified slavery, racial discrimination, poverty, and illegal wars and torture.

Aritmetic deals pretty much with the here and how -- the present or the immediate future. Algebra, by defining carefully operations on positive and negative numbers, is able to deal with gains and losses now, in the past, and in the future.

These results for multiplying signed numbers or common fractions are NOT "because that is just the way it is." The arabs who developed algebra struggled with these ideas and defined these operations the way they did so that algebra would produce useful results. But of course the reason we define it this way is so that multiplication is useful and corresponds to actual operations is the physical world.

1) Half of half of a cup of water is a quarter of a cup.
But it also true that under our rules of multiplication
that 1/2 x 1/2 = 1/4 by multiplying numerator, then denominators. So finding half of a half gives useful answers by defining it the way we do.

2) 4 boxes of candy with 4 pieces in each box comes to 16 pieces of candy. Four fours added together is 16. So we define 4+4+4+4, repeated addition, to be the result of 4 x 4 or 16.

3) -4(-4) is a little more difficult: Let -4 represent losing 4 dollars.(and +4 gaining 4 dollars). Let the second position -4 represent 4 months ago (and +4 represent 4 months in the future.

Then 4x4 means if you gain 4 dollars every month for 4 months you will have 16 dollars more in the future (after 4 months) than you do right now.

So -4(-4) means if you lost 4 dollars every month for the past 4 months then 4 months ago you had 16 dollars more than you do now so -4(-4) = 16

See if you can write a sentence to explain the meaning of
4(-4) and -4(4).

Answer 2

Sometimes the most difficult things to explain in mathematics are things that are supposed to be "easy". I bet there are plenty of people who can prove calculus theorems and who could not give an explanation for the things you are asking about :) So don't feel bad about it.

Anyway, I'll give it a try.

As for multiplying by 1/2, think about what that means for whole numbers: 1/2 times ten gives you one half OF ten, namely 5. check: 5 + 5 = 10.

Likewise 1/2 times 1/2 gives you 1/2 OF 1/2, namely 1/4.
check: 1/4 + 1/4 = 1/2.

As for why the product of 2 negatives is positive, I use a few different ways to motivate this when I teach my college course for future elementary school teachers. Here is the simplest, although not the "deepest":

Consider:

(4) x(4) = 16
(4) x(-4) = -16
(-4) x(4) = -16

so what should (-4) x (-4) = ?

There are only two plausible choices: 16 or -16.

If you say the answer should be -16, compare that to the third line in my list. You would be saying that multiplying (-4) by 4 is the same thing as myltiplying (-4) by (-4)! In other words, you would be saying that multiplication by 4 is the same thing as multiplication by (-4). But that is not consistent with the first two lines.

So, the only possible way out is to conclude that the product is
-16.

think of it this way: every time an additional negative sign is introduced into the product, the answer changes signs.

Answer 3

cake example is good, like pizza or pie
try thinking in terms of money
if I have to equally share 50 cents ($0.50), I would be giving a quarter (25 cents or $0.25) away to the other person to be equal

4 times 4 means the same as having 4 + 4 + 4 + 4, that's what is meant by 4 x 4. when you add them up, you get 16, okay?!

a negative number multiplied by a negative number gives you a positive number. this is a rule for multiplying negative numbers.

these are good questions because you are challenging the fundamentals; but remember to follow the rules, or you could go crazy. practice the exercises and it will come to you. hang in there

Answer 4

(1) When you are considering multiplying fractions a useful way to remember what to do is replace the 'multiply' by 'of'

So: 1/2 x 1/2

Becomes: 1/2 of 1/2

Now you can see that the answer is 1/4

(2) 4 x 4 is 4+4+4+4 = 16

(3) -4 x -4 has to = 16 for mathematical consistency

since -4 x 4 = -16

Answer 5

because when you multiply fractions you time the denominators (the bottom numbers) together. and 2 times 2 equals 4. so therefore 1/2 time 1/2 will equal 1/4

Answer 6

http://i5.tinypic.com/71dba8g.jpg
actually when I was learning multiplication, they taught me that 5X4 =20
that means, if we have 5 squares the we add another 5 & another 5 & another 5
(that means 5 squares 4 times) this equals 20
so suppose that we have a square then we divide it into 2 halves verically & into 2 halves horizontally, the we have 1/2 X 1/2 which equals 1/4 of this square

-4 * -4 =16

as
4* 4=16 (multiply on each isde by -1)
-4 * 4 = -16 (multiply on each isde by -1)
-4 * -4 = 16

Answer 7

If you add a half to a half as you've done its one.
Think of a half as 1 over 2
Multiply it by itself , you get one times one = one on the top line, and 2 times 2 = 4 on the bottom line.
Result = one over four = one quarter.
Think of your favourite pie or cake, cut it equally in to half and then cut one of the halves into two equal pieces. Do you recognise that the smallest piece is one quarter

Answer 8

Multiplying by a fraction is the same thing as dividing by the lower number (denominator) so when you multiply a half by a half you're really talking half of a half, which is a 1/4. See an earlier person's cake example.

Answer 9

think of multiplying as saying "of"

half "OF" a half is a quarter

-4 x -4 = 16

Multiplying two negatives numbers gives a positive answer.

+ and + = +
- and - = +
+ and - = -

Answer 10

(1)Because, if you change the first question to- what is 1/2 of 1/2, it becomes easier to understand.
(2)Saying times 4 or using the X sign is the same as saying-
4+4+4+4.
Because two minus signs multipled together become a plus sign. (3)But three minus signs would remain a minus, as in-
-4 x -4 x -4 =-64
I hope this helps.