# The order of equalities

First note that I am not a mathematician. I do use it for my studies, but I am not reading anything remotely complicated in regards to maths. That said, here is my question:

Today I found myself wanting to write a probability, first in terms of a fraction and then in terms of a decimal value. I wrote down this:

$$P \geq \left( \frac{1}{2} \right)^3 = 0.125$$

That is to say, the probability is at least a half to the power of three which equals $0.125$, but I am afraid that I am in fact writing the probability is at least a half to the power of three, which means that the probability equals $0.125$ which would be incorrect.

Does the equals sign bind stronger than greater-than-equals, meaning the first interpretation is correct, or do the have the same strength so that the second interpretation is the correct one?

If the latter is true, is there a different way to express the original meaning without using biimplication? I know I can write something like

$$P \geq \left( \frac{1}{2} \right)^3 \Leftrightarrow P \geq 0.125$$

but I like the other format better. If it had to do with multiplication instead of equalities I could use paranthesis to express the order of operations, but I do not think I have ever seen this used for clarifying the "order of equalitites"?

Thank you.

• The first interpretation is the one that would be commonly understood. Sentences of the form $a\ge b=c$ occur often. The sentence would not be interpreted as claiming that $a=c$. – André Nicolas Nov 7 '13 at 8:27
• In a chain of $=$ and $\leq$, you simply read them from left to right, so that each sign "acts" on the two numbers that lie on the left and on the right. But there are forbidden things like $a \leq b \geq c$, that we do not usually accept (although there is no real confusion at all, since it means $a \leq b$ and $b \geq c$). – Siminore Nov 7 '13 at 9:01

## 1 Answer

You may be overthinking this. If your probability $P \geq \left( \frac{1}{2} \right)^3$, then $P \geq 0.125$, and there is nothing mathematically wrong with saying that $P \geq \left( \frac{1}{2} \right)^3 = 0.125$. You can leave that first statement as is with no risk of having someone pick on it. Anyone understanding what you are doing will understand your inference. People write similar phrases in published papers all the time.

• I will write it as I initially intended, but it would be good to know if this a rule set in stone, or if the sentence is in fact ambiguous as it stands (even if people tend to interpret it the correct way)? – Thorkil Holm-Jacobsen Nov 7 '13 at 8:39
• Yes it is a rule set in stone, and no it is not ambiguous. In proofs we do this all the time. The scope of inequalities/equalities go no further than the surrounding terms. – J. W. Perry Nov 7 '13 at 9:14
• Thank you. Do you have a reference for this? – Thorkil Holm-Jacobsen Nov 7 '13 at 9:27
• Interesting comment regarding a reference. I think you may be the first person to have ever asked this question (the core of your question) in a formal manner on a public forum. The question of scope on an equality/inequality symbol is not arguable, just universally understood, but it seems truly original. Maybe history has an answer. Hail MSE. – J. W. Perry Nov 7 '13 at 10:06