Direct and LHS to RHS proofs

In direct proofs, we start with a fact (whether it be an equation, an inequality, or a statement or sentence), and use algebra and other techniques to derive additional facts.

For an equation, one way is to start with one true equation, or many of them, then simultaneously arrive at both the left and right sides of the desired equation.

Another way is to start from a single side, typically the more complicated side, then manipulate this expression to equal to the other side. This variation is more commonly known as “LHS to RHS proof”, ie left-hand side to right-hand side. “RHS to LHS proof” is also accepted.

IB may refer to either method with the command term “show that”.

Example: Show that 6+20=1+5\sqrt{6 + \sqrt{20}} = 1 + \sqrt{5}.

Using RHS to LHS method we have

1+5=(1+5)2=1+25+5=6+5⋅22=6+20■\begin{align*} 1 + \sqrt{5} &= \sqrt{\left(1 + \sqrt{5}\right)^2} \\ &= \sqrt{1 + 2\sqrt{5} + 5} \\ &= \sqrt{6 + \sqrt{5 \cdot 2^2}} \\ &= \sqrt{6 + \sqrt{20}} \qed \end{align*}
General way to "de-nest" a nested radical 

Example: Show that for prime numbers greater than 33, it is either one more than a multiple of 66, or one less than a multiple of 66.

All positive integers can be written as 6n+k{6n + k}, n∈N{n\in\mathbb N}, k∈{0,1,2,3,4,5}{k\in\{0, 1, 2, 3, 4, 5\}}, ie kk is the remainder when the integer is divided by 66.

When the integer is 6n6n, it is divisible by 66.

When the integer is 6n+26n + 2 or 6n+46n + 4, the integer will be divisible by 22.

When the integer is 6n+36n + 3, the integer will be divisible by 33.

This means 6n+16n + 1 and 6n+56n + 5 are not divisible by 22 or 33, hence may be prime. Furthermore, 6n+5=6(n+1)−1{6n + 5 = 6(n + 1) - 1}. Hence, all primes greater than 22 and 33 must be either one more or one less than a multiple of 6■6\qed

This does not show that all such numbers are prime, but merely have a chance to be prime.

Tips

  1. Your proof must only use the givens or derive one side of the equation from the other, meaning that you cannot assume the target statement is true. “Show that” problems are derivations, not verifications.
  2. As IB accepts both LHS to RHS, and RHS to LHS, you should start with the more complicated side then simplify to the more simplified side.
  3. You can work backwards, as long as you cross out such portions, and rewrite them in the proper order in your solution. Remember, the reading order is not necessarily the writing order. Trust me I got a 5 in English A.
  4. In HL, you may also have to prove an inequality. This typically involves first finding that one side is greater than some intermediate result, which is then shown to be greater than the other side.
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