Why Is Equal To Sign Written With Three Lines Sometimes?

[Sachin114]

why is equal to sign written with three lines sometimes?

[gangesboy]

If you google the question you will find several sites offering explanations. However, the three line version is use to express something that is an equivalent. For example: 2/3 is equivalent to 4/6 (the same value).

[gangesboy]

"used"

[Buenchico]

Explained quite well here:
http://pediaa.com/difference-between-equal-and-equivalent/

I have to admit though, that despite holding a university degree in mathematics, I might have got it wrong from time to time!

[Old_Geezer]

Interesting. I may have been using the tilde incorrectly as I've occasionally used it to mean approximately equal/in the same ballpark. That must've confused some :-O

[jno]

that also seems to be correct, OG

https://en.wikipedia.org/wiki/Tilde#Mathematics

have you been pronouncng it as twiddle?

[douglas9401]

Because some are more equal than others?

[Buenchico]

It's a double tilde for 'is approximately equal to', O_G.

e.g. my bank balance ≈ 0

[fiction-factory]

I always think of ≡ as an identity sign meaning "always identical to".
For example the following are always true:
(a + b)² ≡ a²+ b² +2ab
a + b ≡ b+a

3y+4=19 is an equation however and not an identity since it is only true when y = 5

However, like Buenchico, I don't always find it easy to know when to use it.
I'm not sure, for example, why we write 6+4=10 rather than 6+4 ≡ 10

[Old_Geezer]

Thanks for that jno.
Try as I might I'm not seeing that on my keyboard Chris ;-) But yes, sounds about right.

[Old_Geezer]

Because the result is exactly the same ff ?

[THECORBYLOON]

I have taken it to mean that in the example

a + b + c ≡ x² + y² + z²

whatever the value of the terms on the left-hand side, the values on the right-hand side are exactly equal in the same order. So a = x², b = y² , c = z² .

[THECORBYLOON]

FF, if you write 6 + 4 ≡ a + b, a MUST be 6 and b MUST be 4.

If you write 6 + 4 = a + b, a and b can be any combination that totals ten.

[fiction-factory]

As soon as I typed my 6+4 =10 example I started to wonder whether it could be an identity since 3+7=10, 5+5=10 and -25+35= 10 .
I am not sure about the terms having to be in the same order in an identity though, THECORBEYLOON

[THECORBYLOON]

I have always taken "=" to mean "the same as" and "≡" as meaning "exactly the same as" so whatever the terms on the left-hand side, each one is exactly the same as that on the right but in that same order.

[fiction-factory]

I agree that "=" means "the same as" and "≡" means "exactly the same as" but am still not sure about the order
Are you saying a+b +c ≢ c + a + b ?

[jim360]

Corbyloon has to be wrong about the ordering issue, because the definition of ≡, as with =, is that it's a binary operator, linking what is on the left with what is on the right. Hence (a+b) ≡ (x+y) implies a relation between the quantities (a+b) and (x+y), and not between their constituent parts.

As far as I can tell, the problem is that ≡ has multiple uses, and is used inconsistently anyway. Such is always the way with mathematical notation, really.

Roughly speaking, you could or should favour using ≡ over = in the following cases:

1) Equalities of shape, where ≡ is taken to mean congruence;
2) Modular relations, eg a ≡ b (mod c);
3) Identity statements in algebra that don't depend on the values the variables take, eg (cos x)^2 + (sin x)^2 ≡ 1, since this is a statement that doesn't actually fix a value of x.
4) As a definition of some new variable, eg let u ≡ x^2 - 3x.

The former two are usually consistent notation choices, but in (3) and (4) it's often not bothered with to distinguish between = and ≡.

[vascop]

one thing that seems to have been missed is that the equivalence sign is stronger than the normal = sign. If a equiv b then it follows that a=b but not vice versa.
i disagree with jim about cos^2+sin^2 because there's nothing wrong with saying when theta=30 degrees cos^2(theta)+sin^2(theta)=1. this is clearly true and the person saying it may well be unaware that it is true for all values of theta.

[Hopkirk]

My brain hurts

[jim360]

But then, vascop, that's a different case, because you have fixed a value of x from the start (and never mind the point that maths notation shouldn't be based on people who don't know about it). However, since you didn't need to do that in order to make (cos x)^2 + (sin x)^2 = 1, as a starting point it's an identity rather than an equation.

Having said that, I tend to write it with two lines not three. In most practical cases, it isn't going to matter.

One recent example where it did was the following problem:

Let - 8x + p ≡ 2xq + q^2

In standard mathematical rules, x is the "free variable" in this problem, and p and q are some cofficients with fixed, but unknown, values. Hence, the identity ≡ between the two sides, rather than the equality =, allows us to assume that the statement above is true for any value of x. Hence, what looks like one equation in three unknowns becomes (by, say, choosing any two values for x you like), two equations in two unknowns. As an identity, the statement above implies that q=-4 and p=16; an an equality, you could find a form for p in terms of q and x, but nothing else.

[Peter Pedant]

thanks Jim

my math master in the sixties was v keen on Jim's type 3

that the triple was used for ( true for all x )

so "ecks equals three " got an equals as it only was true for three

two ecks plus three was identical to ecks plus ecks plus three
that one got a triple as it was true for all x ( was an identity )

modular arith Jim - I know it is elementary number theory but it still makes my brain hurt ....