Errors in Sums and Differences

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. You take two readings: ten point two, plus or minus zero point one, and nine point eight, plus or minus zero point one. Each reading is only about one percent uncertain. Very good readings, right? Now you subtract them, and the answer is zero point four. So tell me, how uncertain is this answer? Fifty percent! Wait, how did one percent become fifty percent? Today we will see what happens to errors when you add or subtract measured quantities. And look, in JEE, error questions use this idea again and again, and the trap is always the minus sign. So, let's begin.

0:45Sum or difference: the errors add

So, here is our setup. Two measured quantities, A and B, each with its own error. The little triangle on screen is delta. Delta A is the absolute error in quantity A, and delta B is the absolute error in quantity B, okay? And plus or minus delta A means the true value can be that much above, or that much below. Clear? First, a sum: Z equals A plus B, okay? Now think. Quantity A can be off by delta A, and quantity B can be off by delta B, right? In the worst case, how much can Z be off? Yes, by both together! So, delta Z equals delta A plus delta B, as simple as that. If both readings are off in the same direction, the total is off by both errors. Simple, isn't it? Alright, now a difference: Z equals A minus B, okay? The values get subtracted, so the errors also get subtracted. Delta A minus delta B, right? No, no, careful! This is the JEE trap. Even for a difference, delta Z equals delta A plus delta B, with a plus sign! Why plus? Because we always quote the maximum error, the worst case. Think of it this way: quantity A is read a little too high, and quantity B a little too low. Then the gap between them is off by both errors together. With me? So, write this one line in your notebook. For a sum or a difference, the maximum absolute errors always add. You subtract the values, but you never subtract the errors. Clear?

2:27The trap: subtracting nearly equal numbers

Now, back to our puzzle. Ten point two plus or minus zero point one, minus nine point eight plus or minus zero point one. Use the rule we just learnt. Find the value, and its error. Got it? The value is ten point two minus nine point eight, that is zero point four. And the error? The errors add: zero point one plus zero point one, zero point two. So the result is zero point four, plus or minus zero point two. Now the percentage error, from our last lesson: the error divided by the value, times hundred. Zero point two divided by zero point four, times hundred. Fifty percent! And each reading on its own? Zero point one out of about ten. Only about one percent. Look, mystery solved. The error grew only a little, to zero point two. But the answer itself shrank to just zero point four. So, compared with the answer, the error is huge. So here is the lesson for the lab. Avoid designing experiments that subtract two nearly equal numbers. Remember this one; it is a very common trap.

3:42Topper's trick: two columns, value and error

Now let me share how toppers handle these questions with zero confusion. They work in two columns. Column one, the value: add or subtract, exactly as the question says. Column two, the error: here there is no minus sign at all. Add the absolute errors, always. Plus in the value means plus in the error, and minus in the value also means plus in the error. And step three, only if the question asks: percentage error equals delta Z divided by Z, times hundred, okay? Always divide by the final answer, not by the readings. That's it, three steps.

4:23Example 1: error in two resistors joined in series

Now, let's practise. Two resistors, one hundred ohm plus or minus three ohm, and two hundred ohm plus or minus four ohm, are joined in series. Find the equivalent resistance, with its error. Pause the video, and try it first. Done? Okay, let's check. In series, the resistances simply add. So R equals one hundred plus two hundred, that is three hundred ohm. The omega symbol on screen stands for ohm. Now the error. It is a sum, so the errors add. Delta R equals three plus four, seven ohm. So the equivalent resistance is three hundred, plus or minus seven ohm. Nice and simple. Notice, the error grew from three and four to seven. The errors never cancel.

5:15Example 2: error in a temperature difference

Now one more, and this one has a hidden trap. The temperatures of two bodies are twenty point zero, plus or minus zero point five degrees Celsius, and fifty point zero, plus or minus zero point five degrees Celsius. Find their difference, with its error, and its percentage error. Pause the video and try it. Hint: watch the minus sign. Done? Okay, let's check. The difference: fifty point zero minus twenty point zero, that is thirty point zero degrees Celsius. Now the error. Both readings have an error of zero point five. It is a difference, so subtract: zero point five minus zero point five, zero. That is a perfect answer, with no error at all, right? No, no, careful! What did we just learn? Yes, the errors always add! Zero point five plus zero point five, that is one point zero. So the difference is thirty point zero, plus or minus one point zero degrees Celsius. The double arrow on screen just means, which gives. Now the percentage error. Divide by the final answer: one point zero divided by thirty point zero, times hundred. About three point three percent. The wavy equals sign means approximately. Alright, now compare with the readings themselves. Zero point five out of twenty point zero: two point five percent. Zero point five out of fifty point zero: just one percent. See, the difference has a larger percentage error than either reading. Same story as our puzzle, right? Subtracting made the answer smaller, but the errors still added. So, compared with the answer, the error grew. Got it?

7:03Quick revision: in a sum or difference, errors add

So, quick revision. For a sum or a difference, the absolute errors add: delta Z equals delta A plus delta B, always. You subtract the values, but never the errors. And when you subtract two nearly equal numbers, the percentage error becomes huge, so avoid that in experiments. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!