List of Common Misconceptions

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This page highlights common misconceptions that people have when understanding concepts. Try your best to answer these questions. If you find that you are stuck, click on the link and read on.

Arithmetic

1. Is \( 0.999 \ldots = 1?\)

2. What is \( \frac{1}{0}? \)

3. What is \( \frac{0}{0}?\)

Algebraic Expressions

1. If \( ab = ac \), is it true that \( b = c?\)

2. Is \( \frac{ \left(\frac{ a}{b}\right) } { c} = \frac{ a } { \left(\frac{b}{c}\right) }?\)

3. Is \( \frac{a}{b+c} = \frac{a}{b} + \frac{a}{c}?\) in progress (Sravanth Chebrolu)

4. Is \( \frac{a}{c} + \frac{b}{d} = \frac{a+b}{c+d}?\) in progress (Sharky)

5. Is \( \frac{a}{bc} = \frac{a}{b} \times \frac{a}{c} \)? in progress

6. If ax + by = ap + bq, must we have x = p and y = q?

Powers and Square Roots

1. What is \(0^0?\) in progress

2. How are exponent towers evaluated? Is \(a^{b^c}\) equal to \(\left(a^b \right)^c?\)

3. Does \( a^n = b^n \) imply that \( a = b?\) in progress (Sachin Vishwakarma)

4. Is \( a^2 + a^3 = a^5?\)

5. Is \((ab)^2 = a^2b^2?\) in progress

6. Do square roots multiply? Is \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}?\)

7. Is \( \sqrt{a^2+b^2} = a+b?\)

8. Is \( \sqrt{x^2} = \pm x?\)

Inequalities

1. When can we multiply inequalities without changing their directions? If \( a > b \) and \( c > d \), is it true that \( a c > b d?\) in progress

2. Does cross multiply always work for inequalities? Does \( \frac{ x+2}{x+4} > 1 \) imply \( x + 2 > x + 4?\)

3. If \(a > b\), then is \(a^2 > b^2?\)

4. Is it always true that \(ab > b ?\)

5. If \(a > b,\) is it always true that \(a^n > b^n?\) in progress

Graphing Functions

1. Can the graph of a function cross the horizontal asymptote?

Exponential and Logarithmic Functions

1. Is \( \log (a+b) = \log a + \log b?\)
   Answer: No. Logarithm is about doing multiplication in a method of addition but not a distributive law. The statement is not valid at all unless we are looking for a form of \(-\infty\) with \(a = b = 0.\)

2. Is \( \log(a×b ) = \log a \times \log b?\)
   Answer: No. \(\log (a \times b) = \log a + \log b\) on another hand is true. \(e^{\ln (a + b)} = a + b \neq e^{\ln a \times \ln b} = a^{\ln b}\) or \(b^{\ln a}.\)

See this for more information.

Complex Numbers

1. Is i<0?

2. Is 1+i>-1+i?

3. Is 1+i rational?

Linear Algebra

1. If \(A\) and \(B\) are square matrices of the same order, is \( (A+B)^2 = A^2 + 2AB + B^2?\)

Integers

1. Is 0 even, odd, or neither?

2. Is 0 a multiple of 3?

3. Is -3 a multiple of 3?

4. Is 1 a prime number?

5. Is 0 a prime number? in progress

6. Since 2 is even, it is not prime!

7. Is 0 a real number?

8. Is 0 an integer?

9. Is 0 a positive integer?

Modular Arithmetic

1. If \(a \equiv b \pmod m\) and \(a \equiv b \pmod n,\) is it true that \(a \equiv b \pmod {mn}?\)

2. If \( a \equiv b \pmod m,\) is it true that \( a = b + m?\)

3. If \( na \equiv nb \pmod {m},\) is it true that \( a \equiv b \pmod{m}?\)

4. If \( a \equiv b \pmod{m},\) is it true that \( na \equiv nb \pmod {nm}?\)

5. If \( na \equiv nb \pmod{nm},\) is it true that \( a\equiv b \pmod m?\)

• Since \(\log 0\) doesn't exist, does that mean \(\int_0^{\frac{\pi}{2}} \log(\sin x)\, dx \) doesn't converge?
  Answer: No. In fact, \(-\frac{\pi}{2} \log 2\) is its value. The infinite form at an edge of an integral does not reveal its resultant effect as its area tends to zero and it is dominated by the whole trend of sum of all others within the range.

• Do local extrema occur if and only if \(f'(x) = 0?\)

• Do global extrema occur if and only if \( f'(x) = 0?\)
  Answer: Finding the roots of \( f'(x) = 0\) will only give us the local extrema. They may or may not be the global extrema. The actual use of derivatives is to find the tangent of the curve. If the tangent has a slope equal to 0, it indicates that it is a maxima/minima. But there are many more points on the curve that have derivative equal to 0. Hence, we can conclude that \(f'(x)=0\) doesn't give the global maxima/minima always.

• A differentiable function \(f\) is strictly increasing if and only if \( f'(x) > 0 \) everywhere.

• If \(x^2 = 1 \), then we can differentiate both sides to conclude that \( 2x = 0 \). Reference explanation.

• If \( \frac{dy}{dx} = 2x \), can we say that \( dy = 2x \, dx?\) \((\)Can we multiply and divide by \( dx?)\)
  Answer: Effectively yes but traditionally no. For integration, it is \(\int \frac{dy}{dx} dx = \int 2x\, dx\) which makes the consequence. It is said that \(dy\) or \(dx\) when separated from each other or from \(\int\) cannot be meaningful as each of them would become value of indeterminate in extreme limit of an unknown destination. Only \(\delta x\) or \(\delta y\) can be written alone as expected proximity to some exact figures. However, to stay in an equation, the tradition to refuse could be improved.

• Are all continuous functions differentiable?
  Answer: No, one example is \(y=|x|\), it is continuous, but is not differentiable at \(x=0\).

• Are all differentiable functions continuous?

• True or False: Since \( \int x^2 \, dx = \frac{1}{3} x^3 + C \), hence \( \int \sin^2 x \, dx = \frac{1}{3} \sin^3 x + C \).
  Answer: False. \(d x\) ought to be \(d \sin x\) instead, to be true, which is \(d u\) as introduced usually.

• If the limit of a sequence is 0, does the series converge? (in progress)

• Can all continuous functions be drawn without lifting the pen from the paper?

• If the derivative of \(F(x)\) is \(f(x)\), is it true that \( \int_a^b f(x) \, dx = F(b) - F(a)?\)

Infinity Misconceptions

• What is \( \frac{\infty}{\infty}?\) (Aareyan)
• Why can we have \( x = x + 1?\) (Calvin)
• Are there infinitely many numbers in the interval [0,1]? (Pranshu)
• Infinity is the number at the end of the real number line. / Infinity is a real number at the end of the real number line. (Kaito)
• All figures with finite area have finite perimeter. (Zandra)
• \(1-1+1-1+1-\cdots = ?\) (Grandi's Series) (Nihar)
  

  proof text

• \( \infty + 1 > \infty \) (+ calvin's followup of 'closest to infinity') (Pranshu)
• What is \( \infty - \infty?\) OR \( \lim{x\rightarrow \infty} \frac{f(x)}{g(x)} = 1 , \lim_{x\rightarrow \infty} f(x) - g(x) = \infty\) (Vighnesh)
• What is \( \frac{1}{0}?\) (in progress)
• Is \(\infty + \infty > \infty?\) (Evan)
• \(\infty \times 0 = ?\) (Aareyan)
• Is \( 0 ^{\infty} \) an indeterminate form?

• Is \( 1 \text{ m}^2 = 100 \text{ cm}^2?\)

• If \(2\) triangles have the same angles (AAA), does it mean they are congruent?
  Answer: No. Both triangles can be of different scales. AAA, however, means that both triangles are similar.

• If \(2\) triangles pass the similarity test ASS, does it mean they are congruent? Or does it mean they are similar?
  Answer: No to both. Consider the following diagram:

• Since \(\angle ABC\) is acute, both triangles \(ABC\) and \(ABD\) pass the similarity test ASS. Hence, the triangles do not necessary or congruent.

• What happens when we reflect along a line that is not vertical or horizontal?

• Can a triangle have an angle of 0 degrees?

• Can a triangle have two right angles?

• Do angles in a spherical triangle sum to 180 degrees?

• Can a right triangle with hypotenuse 10 have a height of 6?

• Is the area of a trapezium equal to base times height?

• Is the area of a rhombus equal to half the product of the diagonals?

• For all triangles, do we have \( a + b \geq c?\)

• For all triangles, do we have \( a^2 + b^2 \geq c^2?\) - Relate to cosine rule

• In a triangle, is it true that the angle opposite the longer leg must be larger than the angle opposite the smaller leg?

• Surface area of a slanted cylinder in progress

• Volume of a slanted cylinder

• Is every cyclic polygon a regular polygon?

• Is every equiangular polygon a regular polygon?

• Is every equilateral polygon a regular polygon?

1. After I flip a coin, it is either heads or tails. Thus the probability of heads is either 0 or 1.

2. The Monty Hall Problem.

3. The Gambler's Fallacy: Believing that randomness keeps some variables due.
   When you toss a coin six times and get consecutive heads, tails aren't due. The next toss is as likely to provide heads as tails.

4. Suppose that we have events A and B in chronological order. The outcome of event B does not affect the outcome of event A. Does knowledge of the outcome of event B affect the probability of event A's outcome?

Dynamics

• Everything that moves must eventually come to a stop. In order for continuous motion, there must always be a force acting on it.
• Do heavier objects fall faster than lighter objects?
• If an object experiences no force, then its acceleration is 0.
  Answer: The statement is true, According to Newton's law of motion Net force equals the product of mass and acceleration. Thus, if force is zero then acceleration is zero. The catch here can be, even if the forces are applied, still the acceleration can be zero.
• If an object is moving, then a net force must be acting on it. In progress
• Since every action has an equal and opposite reaction, nothing will be moved. In progress
• If a body experiences a force in a direction, then it must be moving in that direction.
• A car's gas mileage is independent of the number of passengers. In progress
• Does force of friction between two objects depend on the area of contact?

Kinematics

• Acceleration is the rate of change of speed.
• An object performing uniform circular motion does not accelerate.
• is the acceleration of an object at rest zero?
• If two particles are thrown horizontally with different speeds, the one with a higher speed reaches the ground first.
• When a ball is thrown upwards from the surface, at the highest point of its motion, it is at rest.
• Average speed = (initial speed + final speed)/2

Conservation Laws

• Law of conservation of kinetic energy

Units and Measurements

Rotational Motion

• A rolling ball comes to a stop even when no external force acts on it. This is a violation of Newton's first law of motion.

Oscillation and Waves

Gravity and Space

• There is no gravity in space.
• A larger planet will have a larger gravitational pull.
• Outer space is a complete vacuum.
• Planetary motion is completely circular.
• The earth is flat.

Thermodynamics

• Is heat the same as temperature?
• Does temperature depend upon the frame of reference?
• A black body is black in color.

Quantum Mechanics

• We can travel faster than the speed of light.
• Relative speeds no longer approximate a vector sum when it approaches the speed of light.
• Neutrinos travel faster than the speed of light.
• Are photons mass-less?

Electrodynamics

• If an object has positive charge, has it gained protons? in progress

Circuits

Magnetism

• Is a perfect transparent material black?
• Light travels in straight lines.
• Optically denser medium means it is denser by mass too.

1. Does \(A \Rightarrow B\) imply \(\neg A \Rightarrow \neg B?\)

No, it does not.

Let statement \(A\) be the statement "I am a boy" and statement \(B\) be the statement "I am a human." \(A\) implies \(B,\) but not \(A\) does not imply not \(B\) (i.e. "I am not a boy" does not imply "I am not a human"). Of course, I may not be human, but I could be, and thus \(\text{not } A\) does not imply \(\text{not } B.\)

1. Do the following two different codes in C++ do the same thing?

1
2

int i=5;
cout<<++i;

1
2

int i=5;
cout<<i++;

The answer is \( \color{red}{\textbf{No!}}\)

\(++\) is a unary operator which increments the operand \(i.\) Both \(++i\) and \(i++\) are valid statements but the difference is that \(++i\) increments \(i\) before the function is performed such as cout where as \(i++\) increments \(i\) after performing the function (like cout). More clearly, the first code statement is equivalent to

1 2 3

int i=5; i=i+1; cout<<i;

whereas the second one is equivalent to

1 2 3

int i=5; cout<<i; i=i+1;

Bottomline: Do not increment and use the same variable in the same statement. This is considered a bad coding standard (except in competitive programming).