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Math-7-Lesson-9-Irrational-Numbers.pdf

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This document contains examples of finding the location of numbers on number lines between consecutive integers. It asks the reader to determine between which two consecutive integers various numbers fall, such as 30, 163, -55, and -300. It then shows the work and answers. It also gives an example of approximating the location of the number 2 on a number line between 1 and 2 using successive guesses and squaring the guesses.

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𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  πŸ‘πŸŽ 𝑙𝑖𝑒? πŸ‘πŸŽ < < πŸπŸ“ πŸ‘πŸ” πŸ‘πŸŽ < < πŸ“ πŸ” πŸ‘πŸŽ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 πŸ“ π‘Žπ‘›π‘‘ πŸ”.
𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  πŸπŸ”πŸ‘ 𝑙𝑖𝑒? πŸπŸ”πŸ‘ < < πŸπŸ’πŸ’ πŸπŸ”πŸ— πŸπŸ”πŸ‘ < < 𝟏𝟐 πŸπŸ‘ πŸπŸ”πŸ‘ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 𝟏𝟐 π‘Žπ‘›π‘‘ πŸπŸ‘.
𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  βˆ’ πŸ“πŸ“ 𝑙𝑖𝑒? βˆ’ πŸ“πŸ“ < < βˆ’ πŸ”πŸ’ βˆ’ πŸ’πŸ— πŸ“πŸ“ < < βˆ’πŸ– βˆ’πŸ• βˆ’ πŸ“πŸ“ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 βˆ’πŸ– π‘Žπ‘›π‘‘ βˆ’ πŸ•.
𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  βˆ’ πŸ‘πŸŽπŸŽ 𝑙𝑖𝑒? βˆ’ πŸ‘πŸŽπŸŽ < < βˆ’ πŸ‘πŸπŸ’ βˆ’ πŸπŸ–πŸ— βˆ’ πŸ‘πŸŽπŸŽ < < βˆ’πŸπŸ– βˆ’πŸπŸ• βˆ’ πŸ‘πŸŽπŸŽ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 βˆ’πŸπŸ– π‘Žπ‘›π‘‘ βˆ’ πŸπŸ•.
𝟏. ) πŸ‘πŸ“ 𝟐. ) 𝟏𝟐𝟎 πŸ‘. ) πŸ”πŸ“ πŸ’. ) πŸ–πŸ’ πŸ“. ) πŸπŸ“πŸ“ πŸ“ 𝒂𝒏𝒅 πŸ” 𝟏𝟎 𝒂𝒏𝒅 𝟏𝟏 πŸ– 𝒂𝒏𝒅 πŸ— πŸ— 𝒂𝒏𝒅 𝟏𝟎 𝟏𝟐 𝒂𝒏𝒅 πŸπŸ‘
𝟐 < < 𝟏 πŸ’ 𝟐 < < 𝟏 𝟐
𝟐 < < 𝟏 πŸ’ 𝟐 < < 𝟏 𝟐 𝑰𝒇 𝟐 π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 𝟏, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. 𝟏𝟎 βˆ’ 𝟎. πŸ’πŸŽ 𝑰𝒇 𝟐 π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 πŸ’, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. πŸ“πŸŽ βˆ’ 𝟎. πŸ—πŸŽ 𝒙 π’™πŸ π‘»π’‚π’“π’ˆπ’†π’• 𝟐 𝟏. πŸ’πŸŽ (𝟏. πŸ’πŸŽ)𝟐 π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟐 𝟏. πŸ’πŸ (𝟏. πŸ’πŸ)𝟐 = 𝟏. πŸ—πŸ” = 𝟏. πŸ—πŸ–πŸ–πŸ π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟐 𝟏. πŸ’πŸ (𝟏. πŸ’πŸ)𝟐 = 𝟐. πŸŽπŸπŸ”πŸ’ 𝒂 π’π’Šπ’•π’•π’π’† π’Žπ’π’“π’† 𝒕𝒉𝒂𝒏 𝟐 π‘‡β„Žπ‘’ π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝟐 𝑖𝑛 π‘›π‘’π‘Žπ‘Ÿπ‘’π‘ π‘‘ β„Žπ‘’π‘›π‘‘π‘Ÿπ‘’π‘‘π‘‘β„Žπ‘  𝑖𝑠 𝟐 β‰ˆ 𝟏. πŸ’πŸ
𝐿𝑒𝑑′ 𝑠 π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘™π‘œπ‘π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ 𝟐 π‘œπ‘› π‘‘β„Žπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ 𝑙𝑖𝑛𝑒 𝟐 𝟐 < < 𝟏 𝟐
πŸπŸ“ < < πŸ— πŸπŸ” πŸπŸ“ < < πŸ‘ πŸ’
πŸπŸ“ < < πŸ— πŸπŸ” πŸπŸ“ < < πŸ‘ πŸ’ 𝑰𝒇 πŸπŸ“ π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 πŸ—, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. 𝟏𝟎 βˆ’ 𝟎. πŸ’πŸŽ 𝑰𝒇 πŸπŸ“ π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 πŸπŸ”, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. πŸ“πŸŽ βˆ’ 𝟎. πŸ—πŸŽ 𝒙 π’™πŸ π‘»π’‚π’“π’ˆπ’†π’• πŸπŸ“ πŸ‘. πŸ–πŸŽ (πŸ‘. πŸ–πŸŽ)𝟐 π’”π’•π’Šπ’π’ 𝒇𝒂𝒓 π’‡π’“π’π’Ž πŸπŸ“ πŸ‘. πŸ–πŸ” (πŸ‘. πŸ–πŸ”)𝟐 = πŸπŸ’. πŸ’πŸ’ = πŸπŸ’. πŸ—πŸŽ π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 πŸπŸ“ πŸ‘. πŸ–πŸ• (πŸ‘. πŸ–πŸ•)𝟐 = πŸπŸ’. πŸ—πŸ– π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 πŸπŸ“ π‘‡β„Žπ‘’ π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ πŸπŸ“ 𝑖𝑛 π‘›π‘’π‘Žπ‘Ÿπ‘’π‘ π‘‘ β„Žπ‘’π‘›π‘‘π‘Ÿπ‘’π‘‘π‘‘β„Žπ‘  𝑖𝑠 πŸπŸ“ β‰ˆ πŸ‘. πŸ–πŸ• πŸ‘. πŸ–πŸ– (πŸ‘. πŸ–πŸ–)𝟐 = πŸπŸ“. πŸŽπŸ“ 𝒂 π’π’Šπ’•π’•π’π’† π’Žπ’π’“π’† 𝒕𝒉𝒂𝒏 πŸπŸ“
𝐿𝑒𝑑′ 𝑠 π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘™π‘œπ‘π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ πŸπŸ“ π‘œπ‘› π‘‘β„Žπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ 𝑙𝑖𝑛𝑒 πŸπŸ“ πŸπŸ“ < < πŸ‘ πŸ’

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Math-7-Lesson-9-Irrational-Numbers.pdf

  • 1.
  • 2.
  • 3. 𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  πŸ‘πŸŽ 𝑙𝑖𝑒? πŸ‘πŸŽ < < πŸπŸ“ πŸ‘πŸ” πŸ‘πŸŽ < < πŸ“ πŸ” πŸ‘πŸŽ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 πŸ“ π‘Žπ‘›π‘‘ πŸ”.
  • 4. 𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  πŸπŸ”πŸ‘ 𝑙𝑖𝑒? πŸπŸ”πŸ‘ < < πŸπŸ’πŸ’ πŸπŸ”πŸ— πŸπŸ”πŸ‘ < < 𝟏𝟐 πŸπŸ‘ πŸπŸ”πŸ‘ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 𝟏𝟐 π‘Žπ‘›π‘‘ πŸπŸ‘.
  • 5. 𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  βˆ’ πŸ“πŸ“ 𝑙𝑖𝑒? βˆ’ πŸ“πŸ“ < < βˆ’ πŸ”πŸ’ βˆ’ πŸ’πŸ— πŸ“πŸ“ < < βˆ’πŸ– βˆ’πŸ• βˆ’ πŸ“πŸ“ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 βˆ’πŸ– π‘Žπ‘›π‘‘ βˆ’ πŸ•.
  • 6. 𝟏 = 𝟏 πŸ’ = 𝟐 πŸ— = πŸ‘ πŸπŸ” = πŸ’ πŸπŸ“ = πŸ“ πŸ‘πŸ” = πŸ” πŸ’πŸ— = πŸ• πŸ”πŸ’ = πŸ– πŸ–πŸ = πŸ— 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 πŸπŸ’πŸ’ = 𝟏𝟐 πŸπŸ”πŸ— = πŸπŸ‘ πŸπŸ—πŸ” = πŸπŸ’ πŸπŸπŸ“ = πŸπŸ“ πŸπŸ“πŸ” = πŸπŸ” πŸπŸ–πŸ— = πŸπŸ• πŸ‘πŸπŸ’ = πŸπŸ– πŸ‘πŸ”πŸ = πŸπŸ— πŸ’πŸŽπŸŽ = 𝟐𝟎 𝐡𝑒𝑑𝑀𝑒𝑒𝑛 π‘€β„Žπ‘Žπ‘‘ π‘‘π‘€π‘œ π‘π‘œπ‘›π‘ π‘’π‘π‘’π‘‘π‘–π‘£π‘’ π‘–π‘›π‘‘π‘’π‘”π‘’π‘Ÿπ‘  π‘‘π‘œπ‘’π‘  βˆ’ πŸ‘πŸŽπŸŽ 𝑙𝑖𝑒? βˆ’ πŸ‘πŸŽπŸŽ < < βˆ’ πŸ‘πŸπŸ’ βˆ’ πŸπŸ–πŸ— βˆ’ πŸ‘πŸŽπŸŽ < < βˆ’πŸπŸ– βˆ’πŸπŸ• βˆ’ πŸ‘πŸŽπŸŽ 𝑙𝑖𝑒𝑠 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 βˆ’πŸπŸ– π‘Žπ‘›π‘‘ βˆ’ πŸπŸ•.
  • 7. 𝟏. ) πŸ‘πŸ“ 𝟐. ) 𝟏𝟐𝟎 πŸ‘. ) πŸ”πŸ“ πŸ’. ) πŸ–πŸ’ πŸ“. ) πŸπŸ“πŸ“ πŸ“ 𝒂𝒏𝒅 πŸ” 𝟏𝟎 𝒂𝒏𝒅 𝟏𝟏 πŸ– 𝒂𝒏𝒅 πŸ— πŸ— 𝒂𝒏𝒅 𝟏𝟎 𝟏𝟐 𝒂𝒏𝒅 πŸπŸ‘
  • 9. 𝟐 < < 𝟏 πŸ’ 𝟐 < < 𝟏 𝟐 𝑰𝒇 𝟐 π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 𝟏, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. 𝟏𝟎 βˆ’ 𝟎. πŸ’πŸŽ 𝑰𝒇 𝟐 π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 πŸ’, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. πŸ“πŸŽ βˆ’ 𝟎. πŸ—πŸŽ 𝒙 π’™πŸ π‘»π’‚π’“π’ˆπ’†π’• 𝟐 𝟏. πŸ’πŸŽ (𝟏. πŸ’πŸŽ)𝟐 π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟐 𝟏. πŸ’πŸ (𝟏. πŸ’πŸ)𝟐 = 𝟏. πŸ—πŸ” = 𝟏. πŸ—πŸ–πŸ–πŸ π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟐 𝟏. πŸ’πŸ (𝟏. πŸ’πŸ)𝟐 = 𝟐. πŸŽπŸπŸ”πŸ’ 𝒂 π’π’Šπ’•π’•π’π’† π’Žπ’π’“π’† 𝒕𝒉𝒂𝒏 𝟐 π‘‡β„Žπ‘’ π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ 𝟐 𝑖𝑛 π‘›π‘’π‘Žπ‘Ÿπ‘’π‘ π‘‘ β„Žπ‘’π‘›π‘‘π‘Ÿπ‘’π‘‘π‘‘β„Žπ‘  𝑖𝑠 𝟐 β‰ˆ 𝟏. πŸ’πŸ
  • 10. 𝐿𝑒𝑑′ 𝑠 π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘™π‘œπ‘π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ 𝟐 π‘œπ‘› π‘‘β„Žπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ 𝑙𝑖𝑛𝑒 𝟐 𝟐 < < 𝟏 𝟐
  • 12. πŸπŸ“ < < πŸ— πŸπŸ” πŸπŸ“ < < πŸ‘ πŸ’ 𝑰𝒇 πŸπŸ“ π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 πŸ—, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. 𝟏𝟎 βˆ’ 𝟎. πŸ’πŸŽ 𝑰𝒇 πŸπŸ“ π’Šπ’” 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 πŸπŸ”, π’˜π’† π’˜π’Šπ’π’ 𝒂𝒅𝒅 𝟎. πŸ“πŸŽ βˆ’ 𝟎. πŸ—πŸŽ 𝒙 π’™πŸ π‘»π’‚π’“π’ˆπ’†π’• πŸπŸ“ πŸ‘. πŸ–πŸŽ (πŸ‘. πŸ–πŸŽ)𝟐 π’”π’•π’Šπ’π’ 𝒇𝒂𝒓 π’‡π’“π’π’Ž πŸπŸ“ πŸ‘. πŸ–πŸ” (πŸ‘. πŸ–πŸ”)𝟐 = πŸπŸ’. πŸ’πŸ’ = πŸπŸ’. πŸ—πŸŽ π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 πŸπŸ“ πŸ‘. πŸ–πŸ• (πŸ‘. πŸ–πŸ•)𝟐 = πŸπŸ’. πŸ—πŸ– π’”π’•π’Šπ’π’ 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 πŸπŸ“ π‘‡β„Žπ‘’ π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ πŸπŸ“ 𝑖𝑛 π‘›π‘’π‘Žπ‘Ÿπ‘’π‘ π‘‘ β„Žπ‘’π‘›π‘‘π‘Ÿπ‘’π‘‘π‘‘β„Žπ‘  𝑖𝑠 πŸπŸ“ β‰ˆ πŸ‘. πŸ–πŸ• πŸ‘. πŸ–πŸ– (πŸ‘. πŸ–πŸ–)𝟐 = πŸπŸ“. πŸŽπŸ“ 𝒂 π’π’Šπ’•π’•π’π’† π’Žπ’π’“π’† 𝒕𝒉𝒂𝒏 πŸπŸ“
  • 13. 𝐿𝑒𝑑′ 𝑠 π‘Žπ‘π‘π‘Ÿπ‘œπ‘₯π‘–π‘šπ‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘™π‘œπ‘π‘Žπ‘‘π‘–π‘œπ‘› π‘œπ‘“ πŸπŸ“ π‘œπ‘› π‘‘β„Žπ‘’ π‘›π‘’π‘šπ‘π‘’π‘Ÿ 𝑙𝑖𝑛𝑒 πŸπŸ“ πŸπŸ“ < < πŸ‘ πŸ’