# I am going crazy

Today I was sitting in the library. I had gotten some studying done, but was not planning to leave. I planned to stay for at least 3 more hours. But I came across a problem, realized something, felt I had lost my mind, and so came back home.

I learned the formula for the area of a square in like 5th grade. Now I'm an undergraduate freshman, weeks form finishing my first year.
I have known the formula for the area of a square for the past 8 years or so.

Today I was going through a physics problems. Like (what I assumed to have done a million times before), I converted the dimensions of a rectangle from centimeters to meters, and went on to find the magnetic flux.

I needed to find the area of the rectangle for that. The dimensions were 15 cm and 10 cm. And I got .15 m and .10m. I multiplied the two dimensions, and realized that I couldn't do that because the area was smaller than each dimension.

I thought I had lost my mind. I couldn't believe this. How could I have not realized this before. I've been doing this for so long, I've done this so many times. How can it be that I didn't realize this until now?

How come when I was taught the formula for finding the area of a circle, I wasn't told the domain was restricted to be greater than or equal to +1.

What is going on? How can I be so far behind? What else don't I know?

How do I find the area of a square with side 0.1m?

## Answers and Replies

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I agree, your life is f***ed (1).

I had these doubts when I was just around your age, but its not a problem if you think about it more carefully.

How do I find the area of a square with side 0.1m?
It's 0.01m^2. You can't compare m^2 to m. If you convert to cm, ft, etc everything works. There is no problem!

How come when I was taught the formula for finding the area of a circle, I wasn't told the domain was restricted to be greater than or equal to +1.
A square that is 0.5 cm by 0.5 cm has 0.25 cm^2 area. You can't compare units of length to units of area.

How do you picture something like that.
How can the magnitude of the area be smaller.

Pengwuino
Gold Member
Well, no pun intended, but you're comparing apples to oranges. Think about it for a second, let's say you have 2 sticks, both of 0.5m in length. Let's also say you have 2 more sticks both of 1m length. If you make the 2 sides of a square using the 1m sticks, you obviously have a 1m^2 square almost outlined. Now if you make the 2 sides of a square using 0.5m sticks, its obvious that it's a whole 1/4 the size of the square using 1m sticks! That is, it's 0.25m^2.

Also, why do you think the domain of the area of a circle is restricted to being greater then 1? Those formulas work for every value, period, end of statement.

lisab
Staff Emeritus
Gold Member
What others have said is correct. Comparing cm to cm2 is an apples-to-oranges comparison (no pun intended, apples! ).

Edit: Ahahaha Pengwiuino, I didn't see your post until after I posted...:rofl:

How do you picture something like that.
How can the magnitude of the area be smaller.
y=x is greater than y = x^2 from 0 to 1.

if x = 1/10
y = x: y = 1/10
y = x^2 = 1/10 * 1/10 = 1/100
y = x^3 = 1/1000

....

y = x^inf = 0

But seriously, you don't need to get stuck on these things. Something of more concern would be if you are getting enough sleep or active enough.

"What is going on? How can I be so far behind? What else don't I know?"

A strong coffee would might help.

You need only 10 little 0.1 inch line segments to fill up a 1 inch line, whereas you need 100 little 0.1x0.1 squares to fill up a 1x1 square.

Therefore the area ratio of 0.1x0.1 square to the unit square is much smaller than the length ratio of a 0.1 inch line segment to the unit line segment.

Hmm, I think I understand now.
I guess even though I've converted from cm to meters and m^2 to cm^2 numerous times, I had kinda assumed that the area is always bigger than the sides. Maybe because I mostly dealt with values greater than 1, not in decimals. Or if I dealt with decimals, I never thought about them.

I secretly always compared m and m^2 and made some sort of reasoning in my mind.

Yeah, now I don't understand what I was confused about, and how I got confused.

Pengwuino
Gold Member
Yah NEVER confuse or try to compare different units. In all honesty, it's all arbitrary. You could measure small things in meters and be using decimals and have this thing with magnitudes being smaller or bigger then the magnitudes of the side or you could change the way you measure. If something is 0.5m and you measure with a meter stick, well it's obviously 0.5m. What if you measured with toothpicks? Well then its... i don't know, 30 toothpicks long! A square would be 0.25m^2 while in the toothpick measurement world, it would be 900 toothpicks^2!

lisab
Staff Emeritus
Gold Member
Here's a visual:

The red line is half of the whole side, but the blue square is just a quarter of the whole square.

Did you just type:

Like (what I assumed to have done a million times before), I converted the dimensions of a rectangle from centimeters to meters, and went on to find the magnetic flux.
Like, LIKE, LIKE!?. I can understanding say the word like as a filler in conversation by accident, but typing the word like??? Good god, never say that word around here again please.

Pengwuino
Gold Member
Don't mind cyrus, no one LIKES him :rofl:

California, baby.

California, baby.
That doesn't excuse anything. Honestly, I'd expect a 16 year old girl to use that when they talk, but to write like that - ughhHHgghHHghGHh.

Yeah, now I don't understand what I was confused about, and how I got confused.
I think you just had one of those out of body experiences we all get from time to time. They happen more as you get older. So watch out!

Usually I experience this when I look at a word I've seen a million times. Suddenly this one word looks totally foreign. I just stare at it and swear I've never seen it before and the spelling seems wrong even though I know it's right.

The mind works in strange ways sometimes.

EnumaElish
Homework Helper
There are some math puzzles that are based on comparing numbers squared (say, dollars squared) with numbers themselves (say, dollars). I cannot think of an example right now... Another related mistake is to compare a variance (squared measure) to an average (linear measure).

It's happened to other people.

Here's a visual:

The red line is half of the whole side, but the blue square is just a quarter of the whole square.
Excellent example, lisab.

I have to admit, the OP had me going for a few seconds. I thought, "Wait a minute..." Then I bonked myself in the forehead and all was better once again.

I always get confused about LCMs where

1/A + 1/B and you need to do (..)/A assuming A = aB

So far simplicity, I just do A+B / AB and then factor out numerator and reduce the denominator.

or converting between sin and cos (so have to draw graph each time I go from sin to cos)

Yeah, now I don't understand what I was confused about, and how I got confused.
That happens to me occasionally. Usually when I over think a problem. Suddenly something so basic, which I know I understand, seems wrong and I spend several minutes trying to figure out what magic trick my brain just played on me.