1 = 2, Where is the error ? I know...

Another trick to get 1 equals 2.

Category:

Entertainment

Tags:

• math
• trick
• arithmetic
• one
• two
• equals
• complex

[sun82friend]

We have two mistakes here!

1) sqrt(ab) = sqrt(a) . sqrt(b) iff a,b are real numbers

EX: 1 = sqrt(1) = sqrt( -1 * -1) Now if the above relation is true, then:

1=sqrt(-1) * sqrt(-1)

1= i * i

1=-1 !!!!! this is a counter example, the formal proof is not short!!

[LeconsdAnalyse]

The root function used in real analysis has domain, [0,infinity). So then, in taking the root of `-1` means that he is using the root function of COMPLEX analysis, which is multi-valued (ie, you have to choose a `branch`). In the clip, the expression `rt(-1/1)` is meaningless!! This is, in fact, the error.

[lukashoermann]

sqrt(-1)=i, -i

(-i^2=i^2=-1)

Those tricks always include some sloppy element.

[316derek316]

i^2 = -1?

[ivasenko48]

@luisoncpp In the previous comment, I meant "plane", not "plain" :)

Indeed C are of the form (x, y) [where x and y are real]. If you notice, real numbers all fit this definition (but y is always equal to 0 which is still a real number!).

We thus say that R are formally also C, as all numbers in the field R are enclosed within the field C. This is a well known corollary of the Fundamental Theorem of Algebra.

[ivasenko48]

@luisoncpp I am already familiar with the "proper" definition. If you read it closely (even the one on wikipedia), you will find that all real numbers are algebraically enclosed within the complex field. One way of explaining this is that all polynomials with real coefficients will produce roots contained within the complex field.

Another (perhaps easier) way of picturing this is that all numbers that can be plotted on the complex plain are complex. Real numbers fall on the real axis.

[luisoncpp]

@ivasenko48 In the practice it's ok to think real numbers(R) are complex numbers(C), because the complex numbers with imaginary part zero are isomorphic to R. But formally speaking R are not complex numbers.

Why? because C are ordened pair of real numbers, formally x+iy it's just an usuable way to write (x, y); it's not well known because is only used for construction not in practice.

If dont belive me check wikipedia in the article of complex numbers, the section of "Formal Construction"

[ivasenko48]

@luisoncpp Correct!

However, your cultural note is not.

1/z = conjugate (z)/|z|^2

[ivasenko48]

@luisoncpp Strictly speaking, a real number IS a complex number.

A complex number is defined as a number of the form x+iy, where x,y are real numbers. With real numbers, the imaginary part is 0 (0 is a REAL number by definition), therefore real numbers are also complex numbers.

[ivasenko48]

@SpookyBiology Actually, you are wrong. 'i' is a number (as well as being a symbol of course), just as "2" or "π" are technically symbols as well as numbers.

The term "number" is not defined as being inherently real.

Numbers by mathematical definition are NOT simply 0 1 2 3 4 5 6 7 8 9 and their typographical combinations.

[ivasenko48]

@Dvdbabcock Actually, you are wrong. You have it the wrong way around.

By definition, we take the notation "sqrt(x)" to be the positive square root. This leaves us with only +sqrt(x) when x is positive and real. This is called the principle square root.

However, the sqrt(-1) yields 2 answers: i and -i. To attempt to avoid ambiguity in the same way, we are to take the more positive root, BUT i and -i are equally as "positive" or "negative" as each other. So we have to take both answers.

[TehFinalSlice]

3:28

LOL, i(1/2i)=1/(-2) not 1/2