# Equality vs Approximation which one has the precedence?

Which one of these statements is true?

4/3 ≈ 1.3 = 8/6


(i.e. 4/3 ≈ 1.3 and 4/3 = 8/6)

4/3 ≈ 1.3 ≈ 8/6


Edit:

Is this statement true or false?

4/3 ≈ 1.3 ≈ 8/6 = 12/9 = 16/12 ≈ 1 ≈ 1.3 ≈ 4/3


Edit:

This sign denotes approximation. And this sign = obviously denotes equality.

• @TheCount You actually answered my question. But I need some kind of proof for the answers. – user37421 Jan 1 '17 at 12:01
• I'm not sure what that means. Proof of what? – The Count Jan 1 '17 at 16:09
• @TheCount I mean resources for example. – user37421 Jan 1 '17 at 18:58
• Well, how do you define the symbol $\approx$? Without a clear definition, there is no way to say if $4/3\approx 1.3$ is true or not. And unfortunately, I don't think there is a precise definition for it. – Jack Jan 3 '17 at 15:26

4/3 ≈ 1.3 ≈ 8/6 = 12/9 = 16/12 ≈ 1 ≈ 1.3 ≈ 4/3 is shorthand for (4/3 ≈ 1.3 and 1.3 ≈ 8/6 and 8/6 = 12/9 and 12/9 = 16/12 and 16/12 ≈ 1 and 1 ≈ 1.3 and 1.3 ≈ 4/3). Your compound statement is true iff each part is true.

• What is a compound statement? – user37421 Dec 31 '16 at 3:12
• It is a statement with multiple claims in it. – The Count Dec 31 '16 at 3:35

The second is true. In the second, you wrote that both $4/3$ and $8/6$ are 'close enough for our purposes' to $1.3$. In the first, you have that $1.3=8/6$, which it objectively does not.

I just reread your title. Each and every relative statement, involving $\cong, =, \equiv, ~, \leq$, etc. should be true along the way. Have I resolved your concern?

Edit: Responding to your added portion of your question, I would say that your statement at the end is true, provided you don't mind saying that $1$ is 'close enough' to $1.3$, for example. That very well may be true in some applications, but most people would do a double-take. Although, it is a matter of preference and opinion at that point. Also of note: your final conclusion seems to be that $4/3$ is 'close enough' to itself, which it darn well better be!

• So, they have the precedence from the left to the right. – user37421 Dec 31 '16 at 2:49
• 4/3 ≈ 1.3 = 8/6 is shorthand for the statement (4/3 ≈ 1.3 and 1.3 = 8/6). Since the second part is false, the statement is false. – Browning Dec 31 '16 at 2:55
• @user37421 no, they don't. consider the statement "numbers are even". even though it is partially true, it is not entirely true, and is therefore false. the statement "the sky is blue, the grass is green, and the sun is purple" would also be considered false. – The Count Dec 31 '16 at 3:37