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- 3. 𝟏 = 𝟏 𝟒 = 𝟐 𝟗 = 𝟑 𝟏𝟔 = 𝟒 𝟐𝟓 = 𝟓 𝟑𝟔 = 𝟔 𝟒𝟗 = 𝟕 𝟔𝟒 = 𝟖 𝟖𝟏 = 𝟗 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 𝟏𝟒𝟒 = 𝟏𝟐 𝟏𝟔𝟗 = 𝟏𝟑 𝟏𝟗𝟔 = 𝟏𝟒 𝟐𝟐𝟓 = 𝟏𝟓 𝟐𝟓𝟔 = 𝟏𝟔 𝟐𝟖𝟗 = 𝟏𝟕 𝟑𝟐𝟒 = 𝟏𝟖 𝟑𝟔𝟏 = 𝟏𝟗 𝟒𝟎𝟎 = 𝟐𝟎 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝑤ℎ𝑎𝑡 𝑡𝑤𝑜 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑑𝑜𝑒𝑠 𝟑𝟎 𝑙𝑖𝑒? 𝟑𝟎 < < 𝟐𝟓 𝟑𝟔 𝟑𝟎 < < 𝟓 𝟔 𝟑𝟎 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝟓 𝑎𝑛𝑑 𝟔.
- 4. 𝟏 = 𝟏 𝟒 = 𝟐 𝟗 = 𝟑 𝟏𝟔 = 𝟒 𝟐𝟓 = 𝟓 𝟑𝟔 = 𝟔 𝟒𝟗 = 𝟕 𝟔𝟒 = 𝟖 𝟖𝟏 = 𝟗 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 𝟏𝟒𝟒 = 𝟏𝟐 𝟏𝟔𝟗 = 𝟏𝟑 𝟏𝟗𝟔 = 𝟏𝟒 𝟐𝟐𝟓 = 𝟏𝟓 𝟐𝟓𝟔 = 𝟏𝟔 𝟐𝟖𝟗 = 𝟏𝟕 𝟑𝟐𝟒 = 𝟏𝟖 𝟑𝟔𝟏 = 𝟏𝟗 𝟒𝟎𝟎 = 𝟐𝟎 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝑤ℎ𝑎𝑡 𝑡𝑤𝑜 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑑𝑜𝑒𝑠 𝟏𝟔𝟑 𝑙𝑖𝑒? 𝟏𝟔𝟑 < < 𝟏𝟒𝟒 𝟏𝟔𝟗 𝟏𝟔𝟑 < < 𝟏𝟐 𝟏𝟑 𝟏𝟔𝟑 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝟏𝟐 𝑎𝑛𝑑 𝟏𝟑.
- 5. 𝟏 = 𝟏 𝟒 = 𝟐 𝟗 = 𝟑 𝟏𝟔 = 𝟒 𝟐𝟓 = 𝟓 𝟑𝟔 = 𝟔 𝟒𝟗 = 𝟕 𝟔𝟒 = 𝟖 𝟖𝟏 = 𝟗 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 𝟏𝟒𝟒 = 𝟏𝟐 𝟏𝟔𝟗 = 𝟏𝟑 𝟏𝟗𝟔 = 𝟏𝟒 𝟐𝟐𝟓 = 𝟏𝟓 𝟐𝟓𝟔 = 𝟏𝟔 𝟐𝟖𝟗 = 𝟏𝟕 𝟑𝟐𝟒 = 𝟏𝟖 𝟑𝟔𝟏 = 𝟏𝟗 𝟒𝟎𝟎 = 𝟐𝟎 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝑤ℎ𝑎𝑡 𝑡𝑤𝑜 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑑𝑜𝑒𝑠 − 𝟓𝟓 𝑙𝑖𝑒? − 𝟓𝟓 < < − 𝟔𝟒 − 𝟒𝟗 𝟓𝟓 < < −𝟖 −𝟕 − 𝟓𝟓 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 −𝟖 𝑎𝑛𝑑 − 𝟕.
- 6. 𝟏 = 𝟏 𝟒 = 𝟐 𝟗 = 𝟑 𝟏𝟔 = 𝟒 𝟐𝟓 = 𝟓 𝟑𝟔 = 𝟔 𝟒𝟗 = 𝟕 𝟔𝟒 = 𝟖 𝟖𝟏 = 𝟗 𝟏𝟎𝟎 = 𝟏𝟎 𝟏𝟐𝟏 = 𝟏𝟏 𝟏𝟒𝟒 = 𝟏𝟐 𝟏𝟔𝟗 = 𝟏𝟑 𝟏𝟗𝟔 = 𝟏𝟒 𝟐𝟐𝟓 = 𝟏𝟓 𝟐𝟓𝟔 = 𝟏𝟔 𝟐𝟖𝟗 = 𝟏𝟕 𝟑𝟐𝟒 = 𝟏𝟖 𝟑𝟔𝟏 = 𝟏𝟗 𝟒𝟎𝟎 = 𝟐𝟎 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝑤ℎ𝑎𝑡 𝑡𝑤𝑜 𝑐𝑜𝑛𝑠𝑒𝑐𝑢𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑑𝑜𝑒𝑠 − 𝟑𝟎𝟎 𝑙𝑖𝑒? − 𝟑𝟎𝟎 < < − 𝟑𝟐𝟒 − 𝟐𝟖𝟗 − 𝟑𝟎𝟎 < < −𝟏𝟖 −𝟏𝟕 − 𝟑𝟎𝟎 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 −𝟏𝟖 𝑎𝑛𝑑 − 𝟏𝟕.
- 7. 𝟏. ) 𝟑𝟓 𝟐. ) 𝟏𝟐𝟎 𝟑. ) 𝟔𝟓 𝟒. ) 𝟖𝟒 𝟓. ) 𝟏𝟓𝟓 𝟓 𝒂𝒏𝒅 𝟔 𝟏𝟎 𝒂𝒏𝒅 𝟏𝟏 𝟖 𝒂𝒏𝒅 𝟗 𝟗 𝒂𝒏𝒅 𝟏𝟎 𝟏𝟐 𝒂𝒏𝒅 𝟏𝟑
- 8. 𝟐 < < 𝟏 𝟒 𝟐 < < 𝟏 𝟐
- 9. 𝟐 < < 𝟏 𝟒 𝟐 < < 𝟏 𝟐 𝑰𝒇 𝟐 𝒊𝒔 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 𝟏, 𝒘𝒆 𝒘𝒊𝒍𝒍 𝒂𝒅𝒅 𝟎. 𝟏𝟎 − 𝟎. 𝟒𝟎 𝑰𝒇 𝟐 𝒊𝒔 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 𝟒, 𝒘𝒆 𝒘𝒊𝒍𝒍 𝒂𝒅𝒅 𝟎. 𝟓𝟎 − 𝟎. 𝟗𝟎 𝒙 𝒙𝟐 𝑻𝒂𝒓𝒈𝒆𝒕 𝟐 𝟏. 𝟒𝟎 (𝟏. 𝟒𝟎)𝟐 𝒔𝒕𝒊𝒍𝒍 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟐 𝟏. 𝟒𝟏 (𝟏. 𝟒𝟏)𝟐 = 𝟏. 𝟗𝟔 = 𝟏. 𝟗𝟖𝟖𝟏 𝒔𝒕𝒊𝒍𝒍 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟐 𝟏. 𝟒𝟐 (𝟏. 𝟒𝟐)𝟐 = 𝟐. 𝟎𝟏𝟔𝟒 𝒂 𝒍𝒊𝒕𝒕𝒍𝒆 𝒎𝒐𝒓𝒆 𝒕𝒉𝒂𝒏 𝟐 𝑇ℎ𝑒 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝟐 𝑖𝑛 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 𝑖𝑠 𝟐 ≈ 𝟏. 𝟒𝟏
- 10. 𝐿𝑒𝑡′ 𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑡ℎ𝑒 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝟐 𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒 𝟐 𝟐 < < 𝟏 𝟐
- 11. 𝟏𝟓 < < 𝟗 𝟏𝟔 𝟏𝟓 < < 𝟑 𝟒
- 12. 𝟏𝟓 < < 𝟗 𝟏𝟔 𝟏𝟓 < < 𝟑 𝟒 𝑰𝒇 𝟏𝟓 𝒊𝒔 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 𝟗, 𝒘𝒆 𝒘𝒊𝒍𝒍 𝒂𝒅𝒅 𝟎. 𝟏𝟎 − 𝟎. 𝟒𝟎 𝑰𝒇 𝟏𝟓 𝒊𝒔 𝒄𝒍𝒐𝒔𝒆𝒓 𝒕𝒐 𝟏𝟔, 𝒘𝒆 𝒘𝒊𝒍𝒍 𝒂𝒅𝒅 𝟎. 𝟓𝟎 − 𝟎. 𝟗𝟎 𝒙 𝒙𝟐 𝑻𝒂𝒓𝒈𝒆𝒕 𝟏𝟓 𝟑. 𝟖𝟎 (𝟑. 𝟖𝟎)𝟐 𝒔𝒕𝒊𝒍𝒍 𝒇𝒂𝒓 𝒇𝒓𝒐𝒎 𝟏𝟓 𝟑. 𝟖𝟔 (𝟑. 𝟖𝟔)𝟐 = 𝟏𝟒. 𝟒𝟒 = 𝟏𝟒. 𝟗𝟎 𝒔𝒕𝒊𝒍𝒍 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟏𝟓 𝟑. 𝟖𝟕 (𝟑. 𝟖𝟕)𝟐 = 𝟏𝟒. 𝟗𝟖 𝒔𝒕𝒊𝒍𝒍 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟏𝟓 𝑇ℎ𝑒 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝟏𝟓 𝑖𝑛 𝑛𝑒𝑎𝑟𝑒𝑠𝑡 ℎ𝑢𝑛𝑑𝑟𝑒𝑑𝑡ℎ𝑠 𝑖𝑠 𝟏𝟓 ≈ 𝟑. 𝟖𝟕 𝟑. 𝟖𝟖 (𝟑. 𝟖𝟖)𝟐 = 𝟏𝟓. 𝟎𝟓 𝒂 𝒍𝒊𝒕𝒕𝒍𝒆 𝒎𝒐𝒓𝒆 𝒕𝒉𝒂𝒏 𝟏𝟓
- 13. 𝐿𝑒𝑡′ 𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒 𝑡ℎ𝑒 𝑙𝑜𝑐𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝟏𝟓 𝑜𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒 𝟏𝟓 𝟏𝟓 < < 𝟑 𝟒