U fiirso laba qof oo \(A\) iyo \(B\) aan isku dhalasho lahayn, halkaasoo \(A\) ka yar yahay \(B\) . Muuji: Waxaa jira dhab ahaan laba kooxood oo da'da \(a,b \in \mathbb{N}\) kuwaas oo kuwan soo socdaa ay khuseeyaan: \(2\cdot a = b\) . Waxaan marka hore u dhignay \(d \in \mathbb{R}^+\) farqiga da'da u dhexeeya \(A\) iyo \(B\) dhalashadii \(A\) \( d = d_0 + d_1 \) , \( d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[\) ; Waxaan hadda tixgelineynaa waqti aan macquul ahayn \(x \in \mathbb{R}^+\) ka dib dhalashada \(A\) oo leh \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
Waqtigan xaadirka ah, qeexitaan ahaan, \(a = \lfloor x \rfloor \) iyo \(b = \lfloor x+d \rfloor\) . Waxaan hadda go'aamineynaa dhammaan \(x\) wixii xajinaya:
$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$
Kiiska 1aad: \(0 \leq x_1 + d_1 < 1\):
Kadib $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$
Tani waxay ka dhigan tahay in $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ iyo $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ da'da koowaad $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ .
Kiiska 2aad: \( 1 \leq x_1 + d_1 < 2 \):
Kadib $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$
Tani waxay ka dhigan tahay in $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ iyo $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ kooxda labaad ee da 'da.
Ereyada la taaban karo, tani waxay la macno tahay, tusaale ahaan: Haddii hooyadaa ay ku dhashay da'da \(20\) , waxay dhab ahaantii laba jibaar ka weyn tahay da'daada \(40\) iyo \(42\) . Kiiska in hooyadaa ay \(n\) da' weyn tahay sidoo kale waa mid xiiso leh: Halkan waxaad dejisay \(n \in \mathbb{N}\) si aan macquul ahayn oo aad u hesho \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . Tani waxay shaqeysaa haddii iyo kaliya haddii farqiga da'da isugeynta \( \lfloor d \rfloor = d_0 \) uu yahay tiro badan oo ah \(n-1\) . Tusaale ahaan, kiiska kore, hooyadaa waa \(24\) sano \(6\) jeer da'daada.