And why do negative numbers exist?
EDIT: thank you all so much for answering. I actually understand now. I wish they explained it this simply in school 😭
In: 4
Division is simply multiplication of the reciprocal i.e. divided by 3 is just times 1/3. So a decimal like .44 is 44/100. So if you divide you are multiplying by 100/44 or 25/11.
Negative numbers exist because it is important to have a concept of negatives when calculating things like debts. If you owe money your income from that source is negative.
When you’re dividing, you’re really just asking, “how many of one number can fit into another number.” 4÷2 just means, “how many 2s can fit into a 4”. The answer is 2.
When you divide by a fraction, it’s the same logic. 4 ÷ (1/2) is the same as asking, “how many halves can fit into 4”. 8 halves can fit into a 4. So, 4 ÷ (1/2) = 8. Same as 4 × 2 = 8.
EDIT: And a very non-mathematician answer to the negative numbers thing is that negative and positive numbers are opposite sides of the same coin. They just represent movement on a number line in one direction or another. Positive numbers go up, negative numbers go down.
Because devidng by a less than whole number makes more of whatever.
An illustration makes it easier to understand. You have a pizza with eight slices. You devide each one on half (0.5). You now have sixteen slices.
If someone gave you a five-dollar bill and asked for change in quarters, how many should you give them?
This is the same as asking “how many times does 0.25 go into 5?” or “what is 5 ÷ 0.25?”
The answer is 20, because 20 x 0.25 = 5.
1) Dividing by numbers larger than 1 makes the number smaller. (1/2, 1/4, 1/8) Dividing a number by 1 doesn’t change the number at all (1/1, 2/1) so it makes sense that dividing by numbers smaller than 1 makes the number larger (1/0.5, 1/0.25). Same thing goes for multiplication but in the other way: multiplying by numbers larger than 1 makes the number bigger, but multiplying them by a number smaller than 1 makes the number smaller.
2) Because they’re extremely useful. You can’t have “negative sheep” in a pen, but when you’re keeping track of something that can flip between two states (debit/credit, positive/negative, above sea level / below sea level) then it’s far simpler to just note one half of the scale as negative and keep the math simple. That way, you can easily draft more money out of your bank account than it contains if you have to, and you know exactly how much you have to pay back until you’re back at even zero.
How many times can you fit 1 inside of 1? Just once.
How many times can you fit .1 into 1? 10 times. See how the resulting number got bigger?
If you don’t like working with the decimal, you can also change it to a fraction.
(.1 = 1/10, because it is in the tenths decimal slot, where as .01 = 1/100, .001= 1/1000, and so on.)
Negative numbers exist because you can set a limit in the real world and still go below that limit. Think about it.
People go into debt to pay for things they don’t have the money for.
If you go cave diving but we measure altitude by height above sea level, how would you describe where you are if you go a certain depth below the surface?
There’s a real world use for Negative numbers, so they have to exist.
For the negative numbers part of the question:
Let’s say you have a train, and you’re on a very long, straight line of track. Somewhere on the track there is a checkered flag planted next to it that we’ll call the starting flag.
Your train only goes forwards. Say you drive it for 10 kilometers. Then your phone rings. It’s a friend of yours. They’re at the starting flag, and they want to know where you are. So you tell them, “10 km away”. And with that information alone, they will be able to find you. There’s only one place you could be.
But what if your train could also go in *reverse*? There’s track running *behind* the start flag, too. What if you chose to go in reverse for 10 km up the track?
Your friend calls, again asking you where you are. You again answer, “10 km away.” Which is true. But this time, your friend is confused. With only that info, there’s *two* places you could be. There was not enough information in that answer to tell you where you precisely are. You also need to tell them which side of the start flag you are on, either one way or the other.
This is what negative numbers are for. They are just normal numbers, like any other, but they have a little marker next to them — that minus sign — that indicates that, somewhere in the context of the problem, there is an arbitrary line where knowing which side of it you’re on is important, and that this number happens to be “behind” it instead of “in front of” it. In this example, it happened to be tracking the position of a train on a track relative to a flag. If you say you’re “-10 km away”, that can tell someone that you are 10 km away but *behind* the flag.
They didn’t teach you why dividing is the same as multiplying by the reciprocal? 😳
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