Whakaoti i te pi:11/2[31/2div21/3{11/4div(2+3frac{2}{3})}]

$$\pi : 1 \frac { 1 } { 2 } [ 3 \frac { 1 } { 2 } \div 2 \frac { 1 } { 3 } \{ 1 \frac { 1 } { 4 } \div ( 2 + 3 \frac { 2 } { 3 } ) \} ]$$

Aromātai

$$\frac{15\pi }{68}\approx 0.692998379$$

Ngā Upane Otinga

Whakawehe \pi ki te \frac{1\times 2+1}{2} mā te whakarea \pi ki te tau huripoki o \frac{1\times 2+1}{2}.

$$\frac{\pi \times 2}{1\times 2+1}\times \frac{\frac{3\times 2+1}{2}}{\frac{2\times 3+1}{3}}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakareatia te 1 ki te 2, ka 2.

$$\frac{\pi \times 2}{2+1}\times \frac{\frac{3\times 2+1}{2}}{\frac{2\times 3+1}{3}}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Tāpirihia te 2 ki te 1, ka 3.

$$\frac{\pi \times 2}{3}\times \frac{\frac{3\times 2+1}{2}}{\frac{2\times 3+1}{3}}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakawehe \frac{3\times 2+1}{2} ki te \frac{2\times 3+1}{3} mā te whakarea \frac{3\times 2+1}{2} ki te tau huripoki o \frac{2\times 3+1}{3}.

$$\frac{\pi \times 2}{3}\times \frac{\left(3\times 2+1\right)\times 3}{2\left(2\times 3+1\right)}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Me whakakore tahi te 1+2\times 3 i te taurunga me te tauraro.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakareatia te 1 ki te 4, ka 4.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Tāpirihia te 4 ki te 1, ka 5.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakareatia te 3 ki te 3, ka 9.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{2+\frac{9+2}{3}}$$

Tāpirihia te 9 ki te 2, ka 11.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{2+\frac{11}{3}}$$

Me tahuri te 2 ki te hautau \frac{6}{3}.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{\frac{6}{3}+\frac{11}{3}}$$

Tā te mea he rite te tauraro o \frac{6}{3} me \frac{11}{3}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{\frac{6+11}{3}}$$

Tāpirihia te 6 ki te 11, ka 17.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{\frac{17}{3}}$$

Whakawehe \frac{5}{4} ki te \frac{17}{3} mā te whakarea \frac{5}{4} ki te tau huripoki o \frac{17}{3}.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{5}{4}\times \frac{3}{17}$$

Me whakarea te \frac{5}{4} ki te \frac{3}{17} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{5\times 3}{4\times 17}$$

Mahia ngā whakarea i roto i te hautanga \frac{5\times 3}{4\times 17}.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{15}{68}$$

Me whakarea te \frac{3}{2} ki te \frac{15}{68} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\pi \times 2}{3}\times \frac{3\times 15}{2\times 68}$$

Mahia ngā whakarea i roto i te hautanga \frac{3\times 15}{2\times 68}.

$$\frac{\pi \times 2}{3}\times \frac{45}{136}$$

Me whakarea te \frac{\pi \times 2}{3} ki te \frac{45}{136} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\pi \times 2\times 45}{3\times 136}$$

Me whakakore tahi te 2\times 3 i te taurunga me te tauraro.

$$\frac{15\pi }{68}$$

Whakaroha

$$\frac{15 \pi}{68} = 0.6929983794683366$$

Ngā Upane Otinga

Whakawehe \pi ki te \frac{1\times 2+1}{2} mā te whakarea \pi ki te tau huripoki o \frac{1\times 2+1}{2}.

$$\frac{\pi \times 2}{1\times 2+1}\times \frac{\frac{3\times 2+1}{2}}{\frac{2\times 3+1}{3}}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakareatia te 1 ki te 2, ka 2.

$$\frac{\pi \times 2}{2+1}\times \frac{\frac{3\times 2+1}{2}}{\frac{2\times 3+1}{3}}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Tāpirihia te 2 ki te 1, ka 3.

$$\frac{\pi \times 2}{3}\times \frac{\frac{3\times 2+1}{2}}{\frac{2\times 3+1}{3}}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakawehe \frac{3\times 2+1}{2} ki te \frac{2\times 3+1}{3} mā te whakarea \frac{3\times 2+1}{2} ki te tau huripoki o \frac{2\times 3+1}{3}.

$$\frac{\pi \times 2}{3}\times \frac{\left(3\times 2+1\right)\times 3}{2\left(2\times 3+1\right)}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Me whakakore tahi te 1+2\times 3 i te taurunga me te tauraro.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{1\times 4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakareatia te 1 ki te 4, ka 4.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{4+1}{4}}{2+\frac{3\times 3+2}{3}}$$

Tāpirihia te 4 ki te 1, ka 5.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{2+\frac{3\times 3+2}{3}}$$

Whakareatia te 3 ki te 3, ka 9.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{2+\frac{9+2}{3}}$$

Tāpirihia te 9 ki te 2, ka 11.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{2+\frac{11}{3}}$$

Me tahuri te 2 ki te hautau \frac{6}{3}.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{\frac{6}{3}+\frac{11}{3}}$$

Tā te mea he rite te tauraro o \frac{6}{3} me \frac{11}{3}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{\frac{6+11}{3}}$$

Tāpirihia te 6 ki te 11, ka 17.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{\frac{5}{4}}{\frac{17}{3}}$$

Whakawehe \frac{5}{4} ki te \frac{17}{3} mā te whakarea \frac{5}{4} ki te tau huripoki o \frac{17}{3}.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{5}{4}\times \frac{3}{17}$$

Me whakarea te \frac{5}{4} ki te \frac{3}{17} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{5\times 3}{4\times 17}$$

Mahia ngā whakarea i roto i te hautanga \frac{5\times 3}{4\times 17}.

$$\frac{\pi \times 2}{3}\times \frac{3}{2}\times \frac{15}{68}$$

Me whakarea te \frac{3}{2} ki te \frac{15}{68} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\pi \times 2}{3}\times \frac{3\times 15}{2\times 68}$$

Mahia ngā whakarea i roto i te hautanga \frac{3\times 15}{2\times 68}.

$$\frac{\pi \times 2}{3}\times \frac{45}{136}$$

Me whakarea te \frac{\pi \times 2}{3} ki te \frac{45}{136} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

$$\frac{\pi \times 2\times 45}{3\times 136}$$

Me whakakore tahi te 2\times 3 i te taurunga me te tauraro.

$$\frac{15\pi }{68}$$