AT GCSE:
- DO use = to connect equivalent expressions, but
- DON’T use = to connect equivalent equations.
A reminder of the difference: an expression is an arrangement of letters, numbers and other mathematical symbols: but crucially an expression must not contain an equals sign! It might contain surds, $\pi$, indices, fractions, almost anything at all except for an equals sign (or the other comparison symbols strict inequalities $<,>$ and the weak inequalities $ \leq$ and $ \geq$ plus a few more at A-level).
EXAMPLE OF SIMPLIFYING AN EXPRESSION: All the following expressions are equal, and the = at the start of each line tell us this. Without the = this would just be a list of expressions with nothing to tell us they are equivalent: unhelpful!
$
\begin{aligned}
(x+3)(2x-4)+\sqrt{12}
&=x^2-4x+6x-12+\sqrt{4\times3}\\
&=x^2+2x-12+(\sqrt{4}\times\sqrt{3})\\
&=x^2+2x-12+(2\times\sqrt{3})\\
&=x^2+2x-12+(2\sqrt{3})
\end{aligned}
$
An equation is different because it consists of an equals sign between two expressions (the = is precisely why it is called an equation).
EXAMPLE OF SOLVING AN EQUATION:
$
\begin{aligned}
7(x+1)-9 &= 2(3x+1)\\
7x+7-9 &= 6x+2\\
7x-2&=6x+2\\
7x&=6x+4\\
x&=4
\end{aligned}
$
Note: each of these lines of working already contains an equals sign! (and the equals signs all line up beautifully: more on this later). To start each line with an additional equals sign would now be incorrect, because it would fall foul of our next two rules:
- DON’T have more then one = in the same line!
- DON’T use = to connect things are are unequal!
Here is a terrible example of bad maths which violates both of these rules:
$\left.
\begin{aligned}
3x&=6\\
= \; x&=2 \end{aligned}
\right\}
\quad \text{DON’T DO THIS!}$
Unfortunately we now have a chain of equals signs connecting 6 with 2. We are essentially saying that 6=2 (which I’m pretty sure is wrong!). Rule 3 (only one = per line) is there to help prevent us making this sort of blunder! NOTE: if you feel the need to link each line to the next when solving equations then it’s fine to use a therefore sign $\therefore$ (or even an implication sign $\implies$), like this:
$
\begin{array}{@{}l@{\quad}r@{\;}l@{}}
\phantom{\therefore} & 5(x-2) & =6\\
\therefore & 5x-10 & =6\\
\therefore & 5x & =-4\\
\therefore & x & =-0.8
\end{array}$
Which brings us nicely to our final two rules:
- DO have your equals signs lining up nicely!
- AVOID the “dangling equals sign”
Failing to line up your = looks ugly and horrible, like this:
$
\left.
\begin{array}{l}
12x+11=35\\
12x=24\\
x=2
\end{array}
\right\}
\quad \text{DON’T DO THIS!}
$
And a dangling equals sign (my term: not in general usage) is one which has nothing before it. This is wrong:
$=12$
Because what equals twelve?
But this is fine:
$\begin{aligned}
x&=5+7\\
&=12
\end{aligned}$
because this time the equals sign in front of the 12 is following on from the previous line (and 5+7 does equal 12).
AT A-LEVEL:
It’s not that I was lying to you on purpose, but let’s now amend rule number 3:
3A. IT’S FINE to have more then one = in the same line if you’re careful!
This change is because we (hopefully!) know the difference between an expression and an equation by now. So it’s ok to have several = on one line if say three things are equal $a=b=c$ or if each line of working is very small:
$32^{-2/5} = \frac{1}{32^{2/5}}=\frac{1}{(\sqrt[5]{32})^2}=\frac{1}{4}$
If we put each step on a fresh line this would just take up too much paper! So it’s ok to bend our earlier rule.
At A-level we also meet some new comparison symbols to go alongside the equals sign and the inequalities:
$\ne$ meaning “is not equal to”,
$\approx$ meaning “is approximately equal to”, and
$\equiv$ meaning “is identically equal to” or “always equals (for all values of x)”
For instance, we can use $\equiv$ to emphasize that the following is an identity (true for all values of x) as opposed to an equation (which we must solve to find the value(s) of x for which it is true):
$(x+3)^2\equiv x^2+6x+9$ ← THIS IS AN IDENTITY
Do use and enjoy the variety of comparison symbols: great maths needs great notation!
DID YOU KNOW?:
The equals sign = was invented by Welsh mathematician Robert Recorde in 1557 to mean “equals”: after all, what could be more equal than two short straight lines? Before 1557, mathematicians would write “is equal to” or the latin aequalis, which is definitely more work. Well done that Welshman! Read here for a great English mathematician, or here to find out my top things for a mathematician to do in Wales.