اراﺋﻪ و اﺛﺒﺎت روﺷﯽ ﺑﺮاي ﻧﻤﺎﯾﺶ ﻣﻌﮑﻮس اﻋﺪاد ﺣﻘﯿﻘﯽ روي ﻣﺤﻮر اﻋﺪاد ﺑﺎ اﺑﺰارﻫﺎي اﻗﻠﯿﺪﺳﯽ

Providing and Proving a Method for the Representation of the Inverse of Real Numbers on the Number Line Using Euclidean Tools

در ﻫﻨﺪﺳﻪ اﻗﻠﯿﺪﺳﯽ، ﺑﺮرﺳﯽ اﻣﮑﺎنﭘﺬﯾﺮ ﺑﻮدن ﺗﺮﺳﯿﻤﺎت ﻫﻨﺪﺳﯽ ﺗﻨﻬﺎ ﺑﺎ اﺳﺘﻔﺎده از ﺳﺘﺎره )ﺧﻂﮐﺶ ﻏﯿﺮ ﻣﺪرج( و ﭘﺮﮔﺎر از دﯾﺮﺑﺎز ﻣﻮرد ﺑﺤﺚ و ﮐﻨﮑﺎش ﺑﻮده اﺳﺖ. ﺛﺎﺑﺖ ﺷﺪه اﺳﺖ ﮐﻪ ﺣﻞ ﺑﺮﺧﯽ ﻣﺴﺎﺋﻞ ﮐﻼﺳﯿﮏ اﯾﻦ ﺷﺎﺧﻪ از داﻧﺶ رﯾﺎﺿﯿﺎت ﺗﻨﻬﺎ ﺑﺎ ﮐﺎرﺑﺮد دو اﺑﺰار ﯾﺎدﺷﺪه در ﺣﺎﻟﺖ ﮐﻠﯽ ﻧﺎﻣﻤﮑﻦ اﺳﺖ؛ ﮔﺮﭼﻪ ﻣﻤﮑﻦ اﺳﺖ ﻣﺴﺎﺋﻞ ﻣﺰﺑﻮر در ﻣﻮارد ﺧﺎﺻﯽ ﺣﻞﭘﺬﯾﺮ ﺑﺎﺷﻨﺪ. ﺗﺜﻠﯿﺚ زاوﯾﻪ )ﺗﻘﺴﯿﻢ زاوﯾﻪي دلﺧﻮاه ﺑﻪ ﺳﻪ زاوﯾﻪي ﺑﺮاﺑﺮ(، ﺗﻀﻌﯿﻒ ﻣﮑﻌﺐ )ﺗﺮﺳﯿﻢ ﻣﮑﻌﺒﯽ ﺑﺎ ﺣﺠﻢ دو ﺑﺮاﺑﺮ ﻣﮑﻌﺐ داده ﺷﺪه( و ﺗﺮﺑﯿﻊ داﯾﺮه )رﺳﻢ ﻣﺮﺑﻌﯽ ﺑﺎ ﻣﺴﺎﺣﺖ ﺑﺮاﺑﺮ ﺑﺎ داﯾﺮهي ﻣﻔﺮوض( از اﯾﻦ دﺳﺖ ﻣﺴﺎﺋﻞ ﻫﺴﺘﻨﺪ ﮐﻪ ﺑﺮاي ﺳﺪهﻫﺎي ﭘﯿﺎﭘﯽ رﯾﺎﺿﯽداﻧﺎن ﺗﺎزهﮐﺎر و ﺣﺮﻓﻪاي را ﺑﻪ ﭼﺎﻟﺶ ﻣﯽﮐﺸﯿﺪﻧﺪ. ﺑﺎ اﯾﻦ ﺣﺎل، ﺑﺴﯿﺎري ﺗﺮﺳﯿﻤﺎت ﻫﻨﺪﺳﯽ ﻧﯿﺰ ﺑﺎ ﺧﻂﮐﺶ و ﭘﺮﮔﺎر ﻗﺎﺑﻞ اﻧﺠﺎماﻧﺪ؛ ﻧﻤﺎﯾﺶ ﻣﻮﻗﻌﯿﺖ ﺑﺮﺧﯽ اﻋﺪاد ﮔﻨﮓ ﺑﺮ روي ﻣﺤﻮر اﻋﺪاد ﭼﻨﯿﻦ ﻫﺴﺘﻨﺪ. ﺑﺎ روشﻫﺎي ﺳﺎدهاي ﻣﯽﺗﻮان√2, √3 و ﺑﻪ ﻃﻮر ﻋﻤﻮﻣﯽ رﯾﺸﻪي ﻫﺮ ﻋﺪد ﻃﺒﯿﻌﯽ را روي ﻣﺤﻮر اﻋﺪاد ﻧﺸﺎن داد. در اﯾﻦ ﻣﻘﺎﻟﻪ روﺷﯽ ﺑﺮاي ﻧﻤﺎﯾﺶ ﻣﻌﮑﻮس ﻫﺮ ﻋﺪد ﺣﻘﯿﻘﯽ، اﻋﻢ از ﮔﻮﯾﺎ و ﮔﻨﮓ، ﮐﻪ ﻣﻮﻗﻌﯿﺖ آن ﺑﺮ روي ﻣﺤﻮر اﻋﺪاد داده ﺷﺪه ﺑﺎﺷﺪ، اراﺋﻪ و اﺛﺒﺎت ﻣﯽﺷﻮد و ﻧﺘﺎﯾﺞ ﺟﺒﺮي ﺣﺎﺻﻞ از آن ﻧﯿﺰ ﺗﺒﯿﯿﻦ ﻣﯽﮔﺮدد.

In Euclidean geometry, the study of the feasibility of geometric drawings using only straightedge and compass has long been debated. It has been proved that it is impossible to solve some classical problems of this branch of mathematics by using only the two mentioned tools in general; however, these issues may be soluble in certain cases. Angle trisection (dividing an arbitrary angle into three equal angles), doubling the cube (drawing a cube with twice the volume of a given cube) and squaring the circle (drawing a square with an area equal to a given circle) are among these problems which have been challenging novices and professional mathematicians for centuries. However, many geometric drawings can also be done with a straightedge and compass; like showing the position of some irrational numbers on the number line. √2, √3 and, in general, the square root of any natural number can be shown on the number line by simple methods. In this paper, a method for finding the position of the inverse of any real number, including rational and irrational, whose position is given on the number line, is presented and proved, besides, its algebraic results are explained.