The image is a close-up, top-down shot of a page from a textbook, likely related to mathematics, written in Bengali. The page displays mathematical proofs and definitions. On the left side, under the heading "সমাধান" (Solution), a proof demonstrates that if |a-b| < ε, then a-b < ε and a-b > -ε, which further implies a < b+ε and a > b-ε. This is followed by an implication that b-ε < a < b+ε. On the right side, under a similar heading, another proof shows that if b-ε < a < b+ε, then a-b < ε and a-b > -ε. This leads to ±(a-b) < ε, and finally |a-b| < ε. Below this, a new problem is presented: "সমস্যা-১.১০। যদি r একটি অশূন্য মূলদ সংখ্যা হয় এবং x একটি অমূলদ সংখ্যা হয় তবে দেখাও যে r + x এবং rx অমূলদ সংখ্যা।" (Problem-1.10. If r is a non-zero rational number and x is an irrational number, then show that r+x and rx are irrational numbers.) This problem is followed by "(যদি-২০০৮, ২০১২, ২০১৩, ২০১৯)". Under the solution for this problem, it's stated, "ধরা যাক, r + x অমূলদ সংখ্যা নয় অর্থাৎ মূলদ সংখ্যা"। (Let r+x not be an irrational number, meaning it is a rational number.) This is followed by "এবং r + x = z", and then "⇒ z মূলদ সংখ্যা.........(i)". Further down, under "এখানে" (Here), it states "x = z - r যেহেতু z ও r মূলদ সংখ্যা;" (x = z - r since z and r are rational numbers;). The right side of the solution states, "ধরা যাক rx মূলদ সংখ্যা" (Let rx be a rational number). This is followed by "⇒ rx = y হলে y মূলদ সংখ্যা।" (⇒ if rx = y then y is a rational number.). Then, "⇒ x = y/r যেহেতু y ও r মূলদ সংখ্যা" (⇒ x = y/r since y and r are rational numbers). Finally, under the "Therefore" symbol (∴), it says "y/r মূলদ সংখ্যা," (y/r is a rational number,), followed by "⇒ x মূলদ সংখ্যা কিন্তু এই শর্ত বিরোधी" (⇒ x is a rational number, but this condition is contradictory). in Unknown, Unknown

FM-rewax2

Sep 20, 2026, 5:27 AM

Unknown, Unknown

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The image is a close-up, top-down shot of a page from a textbook, likely related to mathematics, written in Bengali. The page displays mathematical proofs and definitions. On the left side, under the heading "সমাধান" (Solution), a proof demonstrates that if |a-b| < ε, then a-b < ε and a-b > -ε, which further implies a < b+ε and a > b-ε. This is followed by an implication that b-ε < a < b+ε. On the right side, under a similar heading, another proof shows that if b-ε < a < b+ε, then a-b < ε and a-b > -ε. This leads to ±(a-b) < ε, and finally |a-b| < ε. Below this, a new problem is presented: "সমস্যা-১.১০। যদি r একটি অশূন্য মূলদ সংখ্যা হয় এবং x একটি অমূলদ সংখ্যা হয় তবে দেখাও যে r + x এবং rx অমূলদ সংখ্যা।" (Problem-1.10. If r is a non-zero rational number and x is an irrational number, then show that r+x and rx are irrational numbers.) This problem is followed by "(যদি-২০০৮, ২০১২, ২০১৩, ২০১৯)". Under the solution for this problem, it's stated, "ধরা যাক, r + x অমূলদ সংখ্যা নয় অর্থাৎ মূলদ সংখ্যা"। (Let r+x not be an irrational number, meaning it is a rational number.) This is followed by "এবং r + x = z", and then "⇒ z মূলদ সংখ্যা.........(i)". Further down, under "এখানে" (Here), it states "x = z - r যেহেতু z ও r মূলদ সংখ্যা;" (x = z - r since z and r are rational numbers;). The right side of the solution states, "ধরা যাক rx মূলদ সংখ্যা" (Let rx be a rational number). This is followed by "⇒ rx = y হলে y মূলদ সংখ্যা।" (⇒ if rx = y then y is a rational number.). Then, "⇒ x = y/r যেহেতু y ও r মূলদ সংখ্যা" (⇒ x = y/r since y and r are rational numbers). Finally, under the "Therefore" symbol (∴), it says "y/r মূলদ সংখ্যা," (y/r is a rational number,), followed by "⇒ x মূলদ সংখ্যা কিন্তু এই শর্ত বিরোधी" (⇒ x is a rational number, but this condition is contradictory).

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