Meros a kefalaio 7

ª∂ƒ√™ ∞

£ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ·ÚÈıÌÔ› 7.1

7.2

7.3

7.4

7.5

7.6

7.7

£ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ·ÚÈıÌÔ› (ƒËÙÔ› ·ÚÈıÌÔ›) - ∏ ¢ı›· ÙˆÓ ÚËÙÒÓ - ΔÂÙÌË̤ÓË ÛËÌ›Ԣ ñ ∫·Ù·ÓÔÒ ÙËÓ ·Ó¿ÁÎË ÂÈÛ·ÁˆÁ‹˜ ÙˆÓ ·ÚÓËÙÈÎÒÓ ·ÚÈıÌÒÓ ñ ∂ÎÊÚ¿˙ˆ ÌÂÁ¤ıË ‹ ÌÂÙ·‚ÏËÙ¤˜ ÌÂÁÂıÒÓ, Ì ıÂÙÈÎÔ‡˜ ‹ ·ÚÓËÙÈÎÔ‡˜ ·ÚÈıÌÔ‡˜ ñ ¶·ÚÈÛÙ¿Óˆ ¤Ó· ÚËÙfi Ì ÛËÌÂ›Ô ÂÓfi˜ ¿ÍÔÓ·

∞fiÏ˘ÙË ÙÈÌ‹ ÚËÙÔ‡ - ∞ÓÙ›ıÂÙÔÈ ÚËÙÔ› - ™‡ÁÎÚÈÛË ÚËÙÒÓ

ñ °ÓˆÚ›˙ˆ ÙËÓ ¤ÓÓÔÈ· Ù˘ ·fiÏ˘Ù˘ ÙÈÌ‹˜ ÂÓfi˜ ÚËÙÔ‡ Î·È Ì ÙË ‚Ô‹ıÂÈ· ·˘Ù‹˜ Î·È ÙÔ˘ ÚÔÛ‹ÌÔ˘ ÙÔ˘ ÚËÙÔ‡, ·ÓÙÈÛÙÔȯ›˙ˆ ÙÔÓ ÚËÙfi Ì ¤Ó· ÛËÌÂ›Ô ÙÔ˘ ¿ÍÔÓ· ñ μÚ›ÛΈ Ì ·ÎÚ›‚ÂÈ· ‹ Ì ÚÔÛ¤ÁÁÈÛË ÙÔÓ ÚËÙfi Ô˘ ·ÓÙÈÛÙÔȯ› Û ¤Ó· ÛËÌÂ›Ô ÙÔ˘ ¿ÍÔÓ· ñ °ÓˆÚ›˙ˆ ÔÈÔÈ ÚËÙÔ› Â›Ó·È ·ÓÙ›ıÂÙÔÈ Î·È ÔÈ· Â›Ó·È Ë Û¯ÂÙÈ΋ ÙÔ˘˜ ı¤ÛË ÛÙÔÓ ¿ÍÔÓ· ñ ™˘ÁÎÚ›Óˆ ‰‡Ô ÚËÙÔ‡˜ Î·È ÁÓˆÚ›˙ˆ ÙËÓ ı¤ÛË ÙÔ˘˜ ¿Óˆ ÛÙÔÓ ¿ÍÔÓ· ñ ¢È·Ù¿ÛÛˆ ‰‡Ô ‹ ÂÚÈÛÛfiÙÂÚÔ˘˜ ÚËÙÔ‡˜

¶ÚfiÛıÂÛË ÚËÙÒÓ ·ÚÈıÌÒÓ

ñ μÚ›ÛΈ ÙÔ ¿ıÚÔÈÛÌ· ‰‡Ô ÚËÙÒÓ ·ÚÈıÌÒÓ ñ μÚ›ÛΈ ÙÔ ¿ıÚÔÈÛÌ· ÔÏÏÒÓ ÚËÙÒÓ ·ÚÈıÌÒÓ ñ °ÓˆÚ›˙ˆ ÙȘ ȉÈfiÙËÙ˜ Ù˘ ÚfiÛıÂÛ˘ Î·È ÙË ÛËÌ·Û›· ÙÔ˘˜ ÛÙÔÓ ˘ÔÏÔÁÈÛÌfi ·ıÚÔÈÛÌ¿ÙˆÓ ÔÏÏÒÓ ÚÔÛıÂÙ¤ˆÓ

∞Ê·›ÚÂÛË ÚËÙÒÓ ·ÚÈıÌÒÓ

ñ °ÓˆÚ›˙ˆ fiÙÈ Ë ‰È·ÊÔÚ¿ ·–‚, ÔÚ›˙ÂÙ·È ˆ˜ Ë ÌÔÓ·‰È΋ χÛË Ù˘ Â͛ۈÛ˘ ‚+x=·, ‰ËÏ·‰‹ fiÙÈ ÈÛ¯‡ÂÈ Ë ÈÛÔ‰˘Ó·Ì›·: ‚+x=· ⇔ x=·–‚ ñ μÚ›ÛΈ ÙË ‰È·ÊÔÚ¿ ‰‡Ô ÚËÙÒÓ ·ÚÈıÌÒÓ ñ ÀÔÏÔÁ›˙ˆ ·ÚÈıÌËÙÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ Ì ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ ñ ∫¿Óˆ ··ÏÔÈÊ‹ ·ÚÂÓı¤ÛˆÓ

¶ÔÏÏ·Ï·ÛÈ·ÛÌfi˜ ÚËÙÒÓ ·ÚÈıÌÒÓ

ñ ñ ñ ñ

μÚ›ÛΈ ÙÔ ÁÈÓfiÌÂÓÔ ‰‡Ô ÚËÙÒÓ ·ÚÈıÌÒÓ °ÓˆÚ›˙ˆ Î·È ÂÊ·ÚÌfi˙ˆ ÙȘ ȉÈfiÙËÙ˜ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ÚËÙÒÓ ·ÚÈıÌÒÓ ÀÔÏÔÁ›˙ˆ ·ÚÈıÌËÙÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ∂Ê·ÚÌfi˙ˆ ÙËÓ ÂÈÌÂÚÈÛÙÈ΋ ȉÈfiÙËÙ·

¢È·›ÚÂÛË ÚËÙÒÓ ·ÚÈıÌÒÓ

ñ °ÓˆÚ›˙ˆ fiÙÈ ÙÔ ËÏ›ÎÔ · : ‚, ÔÚ›˙ÂÙ·È ˆ˜ Ë ÌÔÓ·‰È΋ χÛË Ù˘ Â͛ۈÛ˘ ‚ x=·, ‰ËÏ·‰‹ fiÙÈ ÈÛ¯‡ÂÈ Ë ÈÛÔ‰˘Ó·Ì›· ‚ x=· ⇔ x=·:‚ ñ μÚ›ÛΈ ÙÔ ËÏ›ÎÔ ‰‡Ô ÚËÙÒÓ ·ÚÈıÌÒÓ ñ °ÓˆÚ›˙ˆ fiÙÈ ÙÔ ÁÈÓfiÌÂÓÔ Î·È ÙÔ ËÏ›ÎÔ ‰‡Ô ÚËÙÒÓ ·ÚÈıÌÒÓ Â›Ó·È ÔÌfiÛËÌÔÈ ·ÚÈıÌÔ› ñ ∫·Ù·ÓÔÒ ÙÔ ËÏ›ÎÔ ‰‡Ô ÚËÙÒÓ Î·È ˆ˜ ÏfiÁÔ

¢Âη‰È΋ ÌÔÚÊ‹ ÚËÙÒÓ ·ÚÈıÌÒÓ

ñ ¢È·ÎÚ›Óˆ ÙÔ˘˜ ÚËÙÔ‡˜ ˆ˜ ‰Âη‰ÈÎÔ‡˜ ‹ ÂÚÈÔ‰ÈÎÔ‡˜ ‰Âη‰ÈÎÔ‡˜ ñ ªÂÙ·ÙÚ¤ˆ ¤Ó· ÎÏ¿ÛÌ· Û ‰Âη‰ÈÎfi ‹ ÂÚÈÔ‰ÈÎfi ‰Âη‰ÈÎfi Î·È ·ÓÙÈÛÙÚfiʈ˜

7.8

¢˘Ó¿ÌÂȘ ÚËÙÒÓ ·ÚÈıÌÒÓ Ì ÂÎı¤ÙË Ê˘ÛÈÎfi ñ ñ

ñ

ñ

7.9

°ÓˆÚ›˙ˆ ÙËÓ ¤ÓÓÔÈ· Ù˘ ‰‡Ó·Ì˘ ·Ó, Ì · ÚËÙfi Î·È Ó Ê˘ÛÈÎfi Î·È ˘ÔÏÔÁ›˙ˆ Ù¤ÙÔȘ ‰˘Ó¿ÌÂȘ. °ÓˆÚ›˙ˆ ÙȘ ȉÈfiÙËÙ˜ ÙˆÓ ‰˘Ó¿ÌÂˆÓ Ì ÂÎı¤ÙË Ê˘ÛÈÎfi Î·È ÙȘ ÂÊ·ÚÌfi˙ˆ ÛÙÔÓ ˘ÔÏÔÁÈÛÌfi ·ÚÈıÌËÙÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ °ÓˆÚ›˙ˆ ÙËÓ ¤ÓÓÔÈ· Ù˘ ‰‡Ó·Ì˘ ·–Ó, Ì ÙÔÓ ÚËÙfi · 0 Î·È Ó, Ê˘ÛÈÎfi Î·È ˘ÔÏÔÁ›˙ˆ Ù¤ÙÔȘ ‰˘Ó¿ÌÂȘ ∂ÎÙÂÏÒ ÙȘ Ú¿ÍÂȘ Ì ÙËÓ ÚÔ‚ÏÂfiÌÂÓË ÚÔÙÂÚ·ÈfiÙËÙ· ÙˆÓ ڿ͈Ó

¢˘Ó¿ÌÂȘ ÚËÙÒÓ ·ÚÈıÌÒÓ Ì ÂÎı¤ÙË ·Î¤Ú·ÈÔ ñ

°ÓˆÚ›˙ˆ ÙȘ ȉÈfiÙËÙ˜ ÙˆÓ ‰˘Ó¿ÌÂˆÓ Ì ÂÎı¤ÙË ·Î¤Ú·ÈÔ Î·È ˘ÔÏÔÁ›˙ˆ ·ÚÈıÌËÙÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ Ì ‰˘Ó¿ÌÂȘ

ñ

°ÓˆÚ›˙ˆ fiÙÈ:

·

Ó

‚

–Ó

‚ = ·

Î·È Ì ÙË ‚Ô‹ıÂÈ·

Ù˘ ÈÛfiÙËÙ·˜ ·˘Ù‹˜, ˘ÔÏÔÁ›˙ˆ ‰˘Ó¿ÌÂȘ Ì ‚¿ÛË ÎÏ·ÛÌ·ÙÈÎfi ·ÚÈıÌfi Î·È ÂÎı¤ÙË ·ÚÓËÙÈÎfi ·Î¤Ú·ÈÔ

À¶∞Δπ∞ (370 - 415

ª .X.)

7.10 Δ˘ÔÔÈË̤ÓË ÌÔÚÊ‹ ÌÂÁ¿ÏˆÓ Î·È ÌÈÎÚÒÓ ·ÚÈıÌÒÓ. ñ

°Ú¿Êˆ Ôχ “ÌÂÁ¿ÏÔ˘˜” Î·È “ÌÈÎÚÔ‡˜” ·ÚÈıÌfi˘˜ ÛÂ Ù˘ÔÔÈË̤ÓË ÌÔÚÊ‹

7o ∫ ∂ º ∞ § ∞ π √

- 114 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞.7.1. £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› (ƒËÙÔ› ·ÚÈıÌÔ›) H ¢ı›· ÙˆÓ ÚËÙÒÓ - ΔÂÙÌË̤ÓË ÛËÌ›Ԣ OÈ ·ÚÈıÌÔ› Ô˘ ÁÓˆÚ›Û·ÌÂ, ̤¯ÚÈ ÙÒÚ·, ÔÈ Ê˘ÛÈÎÔ›, ÔÈ ‰Âη‰ÈÎÔ› Î·È ÔÈ ÎÏ·ÛÌ·ÙÈÎÔ›, ÌÔÚÔ‡Ó Ó· ÂÎÊÚ¿ÛÔ˘Ó Ù· Ê˘ÛÈο ÌÂÁ¤ıË, fi¯È fï˜ Î·È fiϘ ÙȘ ·ÓıÚÒÈÓ˜ ‰Ú·ÛÙËÚÈfiÙËÙ˜ Î·È Î·Ù·ÛÙ¿ÛÂȘ. ÕÏψÛÙÂ, fiÙ·Ó ÙÔÔıÂÙ‹Û·Ì ÙÔ˘˜ Ê˘ÛÈÎÔ‡˜ ·ÚÈıÌÔ‡˜ ¿Óˆ Û ÌÈ· ¢ı›·, ‹Ú·Ì ·˘ı·›ÚÂÙ· ¤Ó· ÛËÌÂ›Ô ˆ˜ ·Ú¯‹ Î·È Û˘Ó¯›Û·Ì “‰ÂÍÈ¿”, Û ›Û˜ ·ÔÛÙ¿ÛÂȘ, Ó· ÁÚ¿ÊÔ˘Ì ÙÔ˘˜ ÁÓˆÛÙÔ‡˜ Ì·˜ ·ÚÈıÌÔ‡˜. ∫¿ı “‰ÂÍÈ¿” fï˜ ÌÈ·˜ ·Ú¯‹˜, ÚÔ¸Ôı¤ÙÂÈ Î·È ÙÔ “·ÚÈÛÙÂÚ¿”. Ÿˆ˜ ÙÔ ‰ÂÍ› ¯¤ÚÈ ÚÔ¸Ôı¤ÙÂÈ ÙÔ ·ÚÈÛÙÂÚfi ÙÔ˘. ∫·È ·ÎfiÌ·, ‰ÂÓ ı· ˘‹Ú¯Â ÙÔ “¿Óˆ” ¯ˆÚ›˜ ÙÔ “οو”, ÙÔ “˙ÂÛÙfi” ¯ˆÚ›˜ ÙÔ “ÎÚ‡Ô” Î.Ï. ŒÙÛÈ, ·Ó ÙÔ ÚÒÙÔ ÙÔ ԇ̠“ıÂÙÈÎfi” ı· Ú¤ÂÈ Ó· ˘¿Ú¯ÂÈ ÙÔ ·ÓÙ›ıÂÙfi ÙÔ˘ Î·È Ó· ÙÔ ԇ̠“·ÚÓËÙÈÎfi”. ∂›Û˘, Û οı ÛÙÔÈ¯Â›Ô ÙÔ˘ ÂÓfi˜ ¯ÚÂÈ¿˙ÂÙ·È Ó· ·ÓÙÈÛÙÔȯ‹ÛÔ˘Ì ·ÎÚÈ‚Ò˜ ¤Ó· ÛÙÔÈ¯Â›Ô ÙÔ˘ ¿ÏÏÔ˘ Î·È Ó· ‚Úԇ̠ÙÚfiÔ Ó· ÙÔ Û˘Ì‚ÔÏ›ÛÔ˘ÌÂ. ¢Â ı· Â›Ó·È ‰‡ÛÎÔÏÔ, ·ÊÔ‡ Â›Ó·È Ú¿ÁÌ·Ù· Ô˘ Ù· ‚ÈÒÓÔ˘Ì ηıËÌÂÚÈÓ¿. Ÿˆ˜ .¯. Ë Úfi‚ÏÂ„Ë ÙÔ˘ ηÈÚÔ‡ ÛÙË ‰Ú·ÛÙËÚÈfiÙËÙ· Ô˘ ·ÎÔÏÔ˘ı›.

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 1 Ë H ÌÂÙˆÚÔÏÔÁÈ΋ ˘ËÚÂÛ›· ÚÔ¤‚Ï„ fiÙÈ ÔÈ ıÂÚÌÔÎڷۛ˜ ÛÙȘ ‰È¿ÊÔÚ˜ fiÏÂȘ ı· Â›Ó·È ·˘Ù¤˜ Ô˘ ·Ó·ÁÚ¿ÊÔÓÙ·È ÛÙÔÓ ·Ú·Î¿Ùˆ ›Ó·Î·: ∞ı‹Ó· £ÂÛÛ·ÏÔÓ›ÎË πˆ¿ÓÓÈÓ· ¶¿ÙÚ· ∞ÏÂÍ·Ó‰ÚÔ‡ÔÏË ºÏÒÚÈÓ· ΔÚ›ÔÏË Ã·ÓÈ¿

➣

7Æ 3Æ 5Æ 2Æ 8Æ 10Æ 6Æ 11Æ

¿Óˆ ·fi ÙÔ Ìˉ¤Ó οو ·fi ÙÔ Ìˉ¤Ó οو ·fi ÙÔ Ìˉ¤Ó ¿Óˆ ·fi ÙÔ Ìˉ¤Ó οو ·fi ÙÔ Ìˉ¤Ó οو ·fi ÙÔ Ìˉ¤Ó οو ·fi ÙÔ Ìˉ¤Ó ¿Óˆ ·fi ÙÔ Ìˉ¤Ó

¶ÚÔÛ¿ıËÛ ӷ ÛËÌÂÈÒÛÂȘ ÛÙÔ ¯¿ÚÙË ·ÚÈıÌÔ‡˜ Ô˘ Ó· ÂÎÊÚ¿˙Ô˘Ó ÙȘ Û˘ÁÎÂÎÚÈ̤Ó˜ ıÂÚÌÔÎڷۛ˜.

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 2 Ë

3 2

™ÙÔÓ ·ÓÂÏ΢ÛÙ‹Ú· ÂÓfi˜ Áηڿ˙ ˘¿Ú¯Ô˘Ó Ù· ÎÔ˘ÌÈ¿ Ô˘ ‚ϤÂȘ ‰›Ï·.

1

➣

0

ΔÈ ÂÎÊÚ¿˙Ô˘Ó ÔÈ ·ÚÈıÌÔ› Ô˘ Â›Ó·È ÁÚ·Ì̤ÓÔÈ ÛÙ· ÎÔ˘ÌÈ¿;

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 3 Ë ™ÙÔ ‰ÈÏ·Ófi Û¯‹Ì· ·Ú·ÙËÚԇ̠fiÙÈ ÙÔ ·ÂÚÔÏ¿ÓÔ ÂÙ¿ÂÈ ÛÙ· 200 m Î·È Ô Î·Ú¯·Ú›·˜ ‚Ú›ÛÎÂÙ·È Û ‚¿ıÔ˜ 200 m οو ·fi ÙËÓ ÂÈÊ¿ÓÂÈ· Ù˘ ı¿Ï·ÛÛ·˜; 0m

-1 -2

➣ ¶ÚÔÛ¿ıËÛ ӷ ÂÎÊÚ¿ÛÂȘ Ì ηٿÏÏËÏÔ˘˜ ·ÚÈıÌÔ‡˜ ÙȘ

ı¤ÛÂȘ ÙÔ˘ ·ÂÚÔÏ¿ÓÔ˘ Î·È ÙÔ˘ ηگ·Ú›· Û ۯ¤ÛË Ì ÙËÓ ÂÈÊ¿ÓÂÈ· Ù˘ ı¿Ï·ÛÛ·˜;

™ÎÂÊÙfiÌ·ÛÙ AÓ Ô ·ÚÈıÌfi˜ 0 ÂÎÊÚ¿˙ÂÈ ÙË ı¤ÛË Ù˘ ÂÈÊ¿ÓÂÈ·˜ Ù˘ ı¿Ï·ÛÛ·˜, ÙfiÙ ÙÔ ·ÂÚÔÏ¿ÓÔ ÂÙ¿ÂÈ ÛÙ· +200 m Î·È Ô Î·Ú¯·Ú›·˜ ‚Ú›ÛÎÂÙ·È ÛÙ· –200 m.

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 115 -

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì ªÂÙ¿ ·fi Ù· ·Ú·¿Óˆ Û˘ÌʈÓԇ̠fiÙÈ: 䊉

䉬

Δ· ۇ̂ÔÏ· «+» Î·È « – » ϤÁÔÓÙ·È ÚfiÛËÌ·. °Ú¿ÊÔÓÙ·È ÚÈÓ ·fi ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Î·È ÙÔ˘˜ ¯·Ú·ÎÙËÚ›˙Ô˘Ó, ·ÓÙ›ÛÙÔȯ·, ˆ˜ ıÂÙÈÎÔ‡˜ ‹ ·ÚÓËÙÈÎÔ‡˜.

+3, +

3 , +15,7 Î·È +0,352 4 ıÂÙÈÎÔ› ·ÚÈıÌÔ›

–2, –10, –28,95 Î·È –0,098 ·ÚÓËÙÈÎÔ› ·ÚÈıÌÔ›

√È ·ÚÈıÌÔ› Ô˘ Û˘Ó·ÓÙ‹Û·Ì ̤¯ÚÈ ÙÒÚ· ‹Ù·Ó ™Â ÂÚÈÙÒÛÂȘ Ô˘ ÌfiÓÔ ıÂÙÈÎÔ› Î·È ÂÔ̤ӈ˜ ‰ÂÓ ˘‹Ú¯Â ·Ó¿ÁÎË ·Ó·ÊÂÚfiÌ·ÛÙ ÌfiÓÔ Û ӷ ¯ÚËÛÈÌÔÔÈԇ̠ÚfiÛËÌÔ. ∏ ÂÈÛ·ÁˆÁ‹ ÙˆÓ ıÂÙÈÎÔ‡˜ ·ÚÈıÌÔ‡˜, ÌÔÚԇ̠·ÚÓËÙÈÎÒÓ ·ÚÈıÌÒÓ ‰ËÌÈÔ˘ÚÁ› ÙËÓ ·Ó¿ÁÎË Ù˘ Ó· ·Ú·Ï›„Ô˘Ì ÙÔ ÚfiÛËÌÔ + .¯. ·ÓÙ› Ó· ÁÚ¿„Ô˘Ì +7 ÙÔÔı¤ÙËÛ˘ ÚfiÛËÌÔ˘ ÌÚÔÛÙ¿ ·fi fiÏÔ˘˜ ·Ú·Ï›Ô˘Ì ÙÔ + Î·È ÙÔ˘˜ ·ÚÈıÌÔ‡˜. ŒÙÛÈ Á›ÓÂÙ·È Ê·ÓÂÚfi ÔÈÔÈ ÁÚ¿ÊÔ˘Ì 7 . ·ÚÈıÌÔ› Â›Ó·È ÔÈ ıÂÙÈÎÔ› Î·È ÔÈÔÈ ÔÈ ·ÚÓËÙÈÎÔ›.

䉬

ΔÔ Ìˉ¤Ó ‰ÂÓ Â›Ó·È Ô‡Ù ıÂÙÈÎfi˜ Ô‡Ù ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜.

䊉

√ÌfiÛËÌÔÈ Ï¤ÁÔÓÙ·È ÔÈ ·ÚÈıÌÔ› Ô˘ ¤¯Ô˘Ó ÙÔ ›‰ÈÔ ÚfiÛËÌÔ.

.¯. ÔÈ ·ÚÈıÌÔ› –7, –0,58 Î·È – ÔÈ ·ÚÈıÌÔ› +1,25, +

3 Â›Ó·È ÔÌfiÛËÌÔÈ Î·È 4

10 Î·È +5 Â›Ó·È ÔÌfiÛËÌÔÈ.

7

䊉

∂ÙÂÚfiÛËÌÔÈ Ï¤ÁÔÓÙ·È ÔÈ ·ÚÈıÌÔ› Ô˘ ¤¯Ô˘Ó ‰È·ÊÔÚÂÙÈÎfi ÚfiÛËÌÔ.

䊉

∞ΤڷÈÔÈ ·ÚÈıÌÔ› Â›Ó·È ÔÈ Ê˘ÛÈÎÔ› ·ÚÈıÌÔ› Ì·˙› Ì ÙÔ˘˜ ·ÓÙ›ÛÙÔÈ¯Ô˘˜ ·ÚÓËÙÈÎÔ‡˜ ·ÚÈıÌÔ‡˜.

䊉

ƒËÙÔ› ·ÚÈıÌÔ› Â›Ó·È fiÏÔÈ ÔÈ ÁÓˆÛÙÔ› Ì·˜ ¤ˆ˜ ÙÒÚ· ·ÚÈıÌÔ›: Ê˘ÛÈÎÔ›, ÎÏ¿ÛÌ·Ù· Î·È ‰Âη‰ÈÎÔ› Ì·˙› Ì ÙÔ˘˜ ·ÓÙ›ÛÙÔÈ¯Ô˘˜ ·ÚÓËÙÈÎÔ‡˜ ·ÚÈıÌÔ‡˜.

√È ·ÚÈıÌÔ› –7 Î·È +0,58 Â›Ó·È ÂÙÂÚfiÛËÌÔÈ ·ÏÏ¿ Î·È 10 ÔÈ ·ÚÈıÌÔ› –1,25 Î·È + Â›Ó·È ÂÙÂÚfiÛËÌÔÈ.

7 ..., –3, –2, –1, 0, 1, 2, 3, ...

√È ·ÚÓËÙÈÎÔ› ·ÚÈıÌÔ› ÂÌÊ·Ó›˙ÔÓÙ·È ÁÈ· ÚÒÙË ÊÔÚ¿ Û ¤Ó· ∫ÈÓ¤˙ÈÎÔ Ì·ıËÌ·ÙÈÎfi ‚È‚Ï›Ô, Ì ٛÙÏÔ “ª·ıËÌ·ÙÈο Û ÂÓÓ¤· μÈ‚Ï›·” (“ΔÛÈÔ˘-ÙÛ·ÓÁÎ-ÛÔ˘¿Ó ÛÔ˘”), Ô˘ ÙÔÔıÂÙÂ›Ù·È ¯ÚÔÓÈο ÛÙËÓ ÂÚ›Ô‰Ô Ù˘ ‰˘Ó·ÛÙ›·˜ ÙˆÓ Ã·Ó (206 .Ã. - 220 Ì.Ã.). ™ÙËÓ E˘Úˆ·˚΋ ·Ú¿‰ÔÛË Ô ÌÂÁ¿ÏÔ˜ ŒÏÏËÓ·˜ Ì·ıËÌ·ÙÈÎfi˜ ¢ÈfiÊ·ÓÙÔ˜ (200 - 284 Ì.Ã.), Ô˘ ¿ÎÌ·Û ÛÙËÓ ∞ÏÂÍ¿Ó‰ÚÂÈ· Á‡Úˆ ÛÙ· 250 Ì.Ã. Î·È ¤ÁÚ·„ ÙÔ ÔÁÎ҉˜ ¤ÚÁÔ ÙÔ˘ (13 ‚È‚Ï›·) Ù· “∞ÚÈıÌËÙÈο”, ¯ÚËÛÈÌÔÔÈ› ÚÒÙÔ˜ ÙÔ˘˜ ·ÚÓËÙÈÎÔ‡˜ ·ÚÈıÌÔ‡˜ ÛÙÔ˘˜ ÂӉȿÌÂÛÔ˘˜ ˘ÔÏÔÁÈÛÌÔ‡˜ ÙÔ˘, ÂÓÒ ˆ˜ χÛË ÂÓfi˜ ÚÔ‚Ï‹Ì·ÙÔ˜ ·Ó·˙ËÙ› ¿ÓÙ· ıÂÙÈÎfi ÚËÙfi ·ÚÈıÌfi.

- 116 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

¶·Ú¿ÛÙ·ÛË ÙˆÓ ÚËÙÒÓ ·ÚÈıÌÒÓ Ì ÛËÌ›· ÌÈ·˜ ¢ı›·˜ 䊉

∞Ó ıˆڋÛÔ˘Ì ·ÚÈÛÙÂÚ¿ Ù˘ ·Ú¯‹˜ √ ÙÔ˘ ËÌÈ¿ÍÔÓ· √x ÙˆÓ ·ÚÈıÌÒÓ, ÙÔÓ ·ÓÙÈΛÌÂÓÔ ·˘ÙÔ‡ ËÌÈ¿ÍÔÓ· Ox , ı· ¤¯Ô˘Ì ÙË ‰˘Ó·ÙfiÙËÙ·, Ì ·˘ÙfiÓ ÙÔÓ ÙÚfiÔ, Ó· ·Ú·ÛÙ‹ÛÔ˘Ì fiÏÔ˘˜ ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜.

x -3

䉴 䊉

O

-1,5

-2

-1

0,5

2,5

0

1

2

x 3

4

5

√ ¿ÍÔÓ·˜ x Ox ÂÚÈÏ·Ì‚¿ÓÂÈ fiÏÔ˘˜ ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ (·ÚÓËÙÈÎÔ‡˜, ıÂÙÈÎÔ‡˜ Î·È ÙÔ Ìˉ¤Ó).

∏ ı¤ÛË ÂÓfi˜ ÛËÌ›Ԣ ∞ Â¿Óˆ ÛÙËÓ Â˘ı›· ÔÚ›˙ÂÙ·È Ì ¤Ó·Ó ·ÚÈıÌfi Ô˘ ÔÓÔÌ¿˙ÂÙ·È ÙÂÙÌË̤ÓË ÙÔ˘ ÛËÌ›Ԣ.

B

x -3

-2

O -1

0

A 1

2

3

4

x 5

ΔÔ ÛËÌÂ›Ô ∞ ¤¯ÂÈ ÙÂÙÌË̤ÓË 4 Î·È ÙÔ ÛËÌÂ›Ô μ ¤¯ÂÈ ÙÂÙÌË̤ÓË –2.

¶∞ƒ∞¢∂π°ª∞ - ∂º∞ƒª√°∏ ¡· ÂÎÊÚ·ÛÙÔ‡Ó Ì ÙË ‚Ô‹ıÂÈ· ÙˆÓ ıÂÙÈÎÒÓ Î·È ·ÚÓËÙÈÎÒÓ ÚËÙÒÓ ·ÚÈıÌÒÓ: (·) 13,75 m οو ·fi ÙËÓ ÂÈÊ¿ÓÂÈ· Ù˘ ı¿Ï·ÛÛ·˜, (‚) 20Æ ∫ÂÏÛ›Ô˘ ¿Óˆ ·fi ÙÔ Ìˉ¤Ó, (Á) ΤډԘ 3.368,97 Q, (‰) ·‡ÍËÛË Î·Ù¿ 2.527,15 Q, (Â) Ì›ˆÛË Î·Ù¿ 50 ÌÔÓ¿‰Â˜ Î·È (ÛÙ) ¤ÎÙˆÛË 15% Â› Ù˘ ÙÈÌ‹˜.

§‡ÛË

(·) –13,75 m, (‚) +20Æ, (Á) +3.368,97 Q, (‰) +2.527,15 Q, (Â) –50, (ÛÙ) –15%.

¢ƒ∞™Δ∏ƒπ√Δ∏ΔA °π∞ Δ√ ™¶πΔπ A˜ ˘Ôı¤ÛÔ˘ÌÂ, fiÙÈ Ë ÛÙÈÁÌ‹ Ù˘ Á¤ÓÓËÛ‹˜ ÛÔ˘ Â›Ó·È ÙÔ Ìˉ¤Ó, ‰ËÏ·‰‹ Ë ·Ú¯‹ Ù˘ ̤ÙÚËÛ˘ ÙÔ˘ ¯ÚfiÓÔ˘. μÚ˜ ̤¯ÚÈ ‰¤Î· ¯ÚÔÓÔÏÔÁ›Â˜, Ô˘ ·ÊÔÚÔ‡Ó Ù· ÈÔ ÛËÌ·ÓÙÈο, ÁÈ· Û¤Ó·, ÚÔÛˆÈο Î·È ÔÈÎÔÁÂÓÂȷο ÁÂÁÔÓfiÙ· Î·È ÙÔÔı¤ÙËÛ¤ Ù· ÛÙËÓ ·Ú·Î¿Ùˆ ¢ı›·.

-30

-25

-20

-15

-10

-5

0

5

∏ Á¤ÓÓËÛ‹ ÛÔ˘

10

15

20

25

30

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 117 -

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

™˘ÌÏ‹ÚˆÛ ٷ ·Ú·Î¿Ùˆ ÎÂÓ¿: (·) √È ÚËÙÔ› Ô˘ ¤¯Ô˘Ó ÚfiÛËÌÔ “+” ϤÁÔÓÙ·È .......................................... ÂÓÒ ·˘ÙÔ› Ô˘ ¤¯Ô˘Ó ÚfiÛËÌÔ “–” ϤÁÔÓÙ·È .......................................... . (‚) √È ·ÚÈıÌÔ› Ì ÙÔ ›‰ÈÔ ÚfiÛËÌÔ Ï¤ÁÔÓÙ·È ................................................ ÂÓÒ ·˘ÙÔ› Ì ‰È·ÊÔÚÂÙÈÎfi ÚfiÛËÌÔ Ï¤ÁÔÓÙ·È .......................................... . (Á) ™ÙËÓ Â˘ı›· ÙˆÓ ·ÚÈıÌÒÓ, ‰ÂÍÈ¿ ÙÔ˘ ÌˉÂÓfi˜ ‚Ú›ÛÎÔÓÙ·È ÔÈ ........................... ÚËÙÔ› Î·È ·ÚÈÛÙÂÚ¿ ÙÔ˘ ÌˉÂÓfi˜ ÔÈ .......................................... ÚËÙÔ›.

2.

¡· ηٷٿÍÂȘ ÙÔ˘˜ ·Ú·Î¿Ùˆ ·ÚÈıÌÔ‡˜ Û ‰‡Ô ÔÌ¿‰Â˜, ÙÔ˘˜ ıÂÙÈÎÔ‡˜ Î·È ÙÔ˘˜ ·ÚÓËÙÈÎÔ‡˜: –3,1, +5, +8, –20, 7, –3, 18.

3.

ΔÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË (·) √È ·ÎfiÏÔ˘ıÔÈ ·ÚÈıÌÔ› Â›Ó·È fiÏÔÈ ıÂÙÈÎÔ›: +1, +5, +216, +3701 (‚) √È ·ÚÈıÌÔ› Ô˘ ·ÎÔÏÔ˘ıÔ‡Ó Â›Ó·È fiÏÔÈ ·ÚÓËÙÈÎÔ›: –3, –8, 7, –22

™ø™Δ√ §∞£√™

√È ÂfiÌÂÓ˜ ÙÚÂȘ ÂÚˆÙ‹ÛÂȘ ·Ó·Ê¤ÚÔÓÙ·È ÛÙÔ Û¯‹Ì· Ô˘ ·ÎÔÏÔ˘ı›. x

§ -4

(Á) (‰) (Â)

ª -3

-2

O -1

∫

0

+1

x

+2

+3

+4

∏ ÙÂÙÌË̤ÓË ÙÔ˘ ÛËÌ›Ԣ ∫ Â›Ó·È +2 ∏ ÙÂÙÌË̤ÓË ÙÔ˘ ÛËÌ›Ԣ § Â›Ó·È –4 Δ· ÛËÌ›· ∫ Î·È ª ¤¯Ô˘Ó ÙËÓ ›‰È· ÙÂÙÌË̤ÓË

4.

™Ù· ˙‡ÁË ·ÚÈıÌÒÓ Ô˘ ·ÎÔÏÔ˘ıÔ‡Ó Ó· ‚ÚÂȘ ÔÈÔÈ ·ÚÈıÌÔ› Â›Ó·È ÔÌfiÛËÌÔÈ Î·È ÔÈÔÈ Â›Ó·È ÂÙÂÚfiÛËÌÔÈ: (·) 3 Î·È +3, (‚) 0 Î·È 5, (Á) –2 Î·È –4, (‰) 7 Î·È +9, (Â) –2 Î·È 1, (ÛÙ) 17 Î·È –20, (˙) –9 Î·È –3,2, (Ë) –10,5 Î·È 11, (ı) 0 Î·È –100, (È) +6,7 Î·È +12,3.

5.

¡· ÂÎÊÚ¿ÛÂȘ Ì ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ ÙȘ ·Ú·Î¿Ùˆ ÚÔÙ¿ÛÂȘ: (·) ∫·Ù¿ıÂÛË 50.000 Q (‚) ∞Ó¿ÏË„Ë 78.000 Q, (Á) ∞‡ÍËÛË ÌÈÛıÔ‡ ηٿ 500 Q, (‰) ªÂ›ˆÛË ÂÈÙÔΛԢ ηٿ 1 ÌÔÓ¿‰·, (Â) 30 ̤ÙÚ· ·ÚÈÛÙÂÚ¿.

6.

μÚ˜ ÙË Ï¤ÍË Ô˘ Û¯ËÌ·Ù›˙ÂÙ·È ·fi Ù· ÁÚ¿ÌÌ·Ù· Ì ÙÂÙÌË̤Ó˜ –6, 10, 9, –9, 5, –5, 0 ÛÙÔ ·Ú·Î¿Ùˆ Û¯‹Ì·. ™ÙË Û˘Ó¤¯ÂÈ· ÁÚ¿„ ̒ ·˘Ùfi ÙÔÓ ÙÚfiÔ ¤Ó· fiÓÔÌ· Ô˘ ÛÔ˘ ·Ú¤ÛÂÈ. æ º Δ ƒ • ª ∫ £ ∑ ¢ μ √ ∞ ° ∂ ∏ π x -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1

7.

0

1

2

3

4

5

§ ¡ ¶ ™ À Ã ø 6

7

8 9 10 11 12

x

Δ· ÛËÌ›· ∞ Î·È μ ¤¯Ô˘Ó ÙÂÙÌË̤Ó˜ · Î·È ‚, ·ÓÙ›ÛÙÔȯ·. ¡· ‚ÚÂı› Ë ÙÂÙÌË̤ÓË ÙÔ˘ ̤ÛÔ˘ ª ÙÔ˘ ÙÌ‹Ì·ÙÔ˜ ∞μ fiÙ·Ó: (·) · = +5 Î·È ‚ = +8, (‚) · = –4 Î·È ‚ = –13.

- 118 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞.7.2. ∞fiÏ˘ÙË ÙÈÌ‹ ÚËÙÔ‡ - ∞ÓÙ›ıÂÙÔÈ ÚËÙÔ› - ™‡ÁÎÚÈÛË ÚËÙÒÓ

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 1 Ë μÚ˜ fiÛ˜ ÌÔÓ¿‰Â˜ ·¤¯Ô˘Ó ·fi ÙËÓ ·Ú¯‹ √ ÙÔ˘ ¿ÍÔÓ· Ù· ÛËÌ›· ∞, μ, ° Î·È ¢.

°

B

-5

-4

-3

-2

-1

O

A

0

1

¢ 2

3

4

5

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 2 Ë ™ÙËÓ ·Ú·Î¿Ùˆ ¢ı›· ‚Ú˜ ÙȘ ÙÂÙÌË̤Ó˜ ÙˆÓ ÛËÌ›ˆÓ ª Î·È ª.

M -8

-9

➣ ➣

M

O -7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

ΔÈ ·Ú·ÙËÚ›˜ ÁÈ· ÙȘ ÙÂÙÌË̤Ó˜ ÙˆÓ ÛËÌ›ˆÓ ª Î·È ª; ¶ÚÔÛ¿ıËÛ ӷ ÙÔÔıÂÙ‹ÛÂȘ ÛÙËÓ ·Ú·¿Óˆ ¢ı›· ÙˆÓ ÚËÙÒÓ Ù· ÛËÌ›· ∞ Î·È ∞ Ô˘ ·¤¯Ô˘Ó ·fi ÙËÓ ·Ú¯‹ √ ÙÔ˘ ¿ÍÔÓ· 3,5 ÌÔÓ¿‰Â˜. ∫¿Ó ÙÔ ›‰ÈÔ ÁÈ· Ù· ÛËÌ›· μ Î·È μ Ô˘ ·¤¯Ô˘Ó ·fi ÙËÓ ·Ú¯‹ √ ÙÔ˘ ¿ÍÔÓ· 6 ÌÔÓ¿‰Â˜.

➣

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì ∏ ·fiÏ˘ÙË ÙÈÌ‹ ÂÓfi˜ ÚËÙÔ‡ ·ÚÈıÌÔ‡ · ÂÎÊÚ¿˙ÂÈ ÙËÓ ·fiÛÙ·ÛË ÙÔ˘ ÛËÌ›Ԣ Ì ÙÂÙÌË̤ÓË · ·fi ÙËÓ ·Ú¯‹ √ ÙÔ˘ ¿ÍÔÓ· Î·È Û˘Ì‚ÔÏ›˙ÂÙ·È Ì · .

䊉

∞ÓÙ›ıÂÙÔÈ ÔÓÔÌ¿˙ÔÓÙ·È ‰‡Ô ·ÚÈıÌÔ› Ô˘ Â›Ó·È ÂÙÂÚfiÛËÌÔÈ Î·È ¤¯Ô˘Ó ÙËÓ ›‰È· ·fiÏ˘ÙË ÙÈÌ‹.

√ ·ÓÙ›ıÂÙÔ˜ ÙÔ˘ x Â›Ó·È Ô –x.

䉴 䉴 䉴

0 2

+2 2

–2 = +2 = 2 Ô ·ÓÙ›ıÂÙÔ˜ ÙÔ˘ +5,1 Â›Ó·È Ô –5,1. Ô ·ÓÙ›ıÂÙÔ˜ ÙÔ˘ –2,7 Â›Ó·È Ô –(–2,7)=+2,7

H ·fiÏ˘ÙË ÙÈÌ‹ ÂÓfi˜ ıÂÙÈÎÔ‡ ·ÚÈıÌÔ‡ Â›Ó·È Ô ›‰ÈÔ˜ Ô ·ÚÈıÌfi˜.

H ·fiÏ˘ÙË ÙÈÌ‹ ÂÓfi˜ ·ÚÓËÙÈÎÔ‡ ·ÚÈıÌÔ‡ Â›Ó·È Ô ·ÓÙ›ıÂÙfi˜ ÙÔ˘.

H ·fiÏ˘ÙË ÙÈÌ‹ ÙÔ˘ ÌˉÂÓfi˜ Â›Ó·È ÙÔ Ìˉ¤Ó.

¢‡Ô ÛËÌ›· Ô˘ ‚Ú›ÛÎÔÓÙ·È Û ›ÛË ·fiÛÙ·ÛË, ‰ÂÍÈ¿ Î·È ·ÚÈÛÙÂÚ¿ ·fi ÙËÓ ·Ú¯‹ ÙˆÓ ·ÍfiÓˆÓ, ¤¯Ô˘Ó ÙÂÙÌË̤Ó˜, ·ÓÙ›ıÂÙÔ˘˜ ·ÚÈıÌÔ‡˜.

.¯. +9,63 = 9,63 .¯. –8,4 = 8,4 = –(–8,4) .¯. 0 = 0 afiÛÙ·ÛË 3 afiÛÙ·ÛË 3 2 2

}

䉬

-2

}

䉬

䊉

-3 2

0

3 2

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 119 -

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 3 Ë ªÈ· ÎÚ‡· ̤ڷ ÙÔ˘ ¯ÂÈÌÒÓ· Ô ∫ÒÛÙ·˜ ÎÔÈÙÔ‡Û ÙË ıÂÚÌÔÎÚ·Û›· οı ‰‡Ô ÒÚ˜. √È ÂӉ›ÍÂȘ ÙÔ˘ ıÂÚÌÔ̤ÙÚÔ˘, Ô˘ ¤‚ÏÂÂ, Ê·›ÓÔÓÙ·È ·Ú·Î¿Ùˆ:

8:00

10:00

12:00

14:00

16:00

18:00

20:00

+5 +3 0

+1

0

0

0

0

0 –1

0 –2

-3

➣

ªÔÚ›˜ Ó· ηٷÁÚ¿„ÂȘ fiϘ ÙȘ ÂӉ›ÍÂȘ ÙÔ˘ ıÂÚÌÔ̤ÙÚÔ˘ Ì ·‡ÍÔ˘Û· ‹ Êı›ÓÔ˘Û· ÛÂÈÚ¿;

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì 䉴

√ ÌÂÁ·Ï‡ÙÂÚÔ˜ ·fi ‰‡Ô ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ Â›Ó·È ÂΛÓÔ˜ Ô˘ ‚Ú›ÛÎÂÙ·È ‰ÂÍÈfiÙÂÚ· ·fi ÙÔÓ ¿ÏÏÔ ¿Óˆ ÛÙÔÓ ¿ÍÔÓ·.

-4,75

-3,5

-3 2

-0,6

0 0,6

3 2

3,5

4,75

∫¿ı ıÂÙÈÎfi˜ ÚËÙfi˜ Â›Ó·È ÌÂÁ·Ï‡ÙÂÚÔ˜ ·fi οı ·ÚÓËÙÈÎfi ÚËÙfi ·ÚÈıÌfi.

–4,75 < –3,5 < –

3 3 < –0,6 < 0 < 0,6 < + < +3,5 < +4,75 2 2

䉬

ΔÔ Ìˉ¤Ó Â›Ó·È ÌÈÎÚfiÙÂÚÔ ·fi οı ıÂÙÈÎfi ·ÚÈıÌfi Î·È ÌÂÁ·Ï‡ÙÂÚÔ ·fi οı ·ÚÓËÙÈÎfi ·ÚÈıÌfi.

.¯. 0 < 2,9 Î·È –3,8 < 0

䉬

√ ÌÂÁ·Ï‡ÙÂÚÔ˜ ·fi ‰‡Ô ıÂÙÈÎÔ‡˜ ÚËÙÔ‡˜ Â›Ó·È ÂΛÓÔ˜ Ô˘ ¤¯ÂÈ ÙËÓ ÌÂÁ·Ï‡ÙÂÚË ·fiÏ˘ÙË ÙÈÌ‹, ‰ËÏ·‰‹ ·˘Ùfi˜ Ô˘ ‚Ú›ÛÎÂÙ·È ‰ÂÍÈfiÙÂÚ· ·fi ÙÔÓ ¿ÏÏÔ ¿Óˆ ÛÙÔÓ ¿ÍÔÓ·.

+2,67 < +5,89 ‰ÈfiÙÈ 2,67 < 5,89

䉬

√ ÌÂÁ·Ï‡ÙÂÚÔ˜ ·fi ‰‡Ô ·ÚÓËÙÈÎÔ‡˜ ÚËÙÔ‡˜ Â›Ó·È ÂΛÓÔ˜ Ô˘ ¤¯ÂÈ ÙËÓ ÌÈÎÚfiÙÂÚË ·fiÏ˘ÙË ÙÈÌ‹, ‰ËÏ·‰‹ ·˘Ùfi˜ Ô˘ ‚Ú›ÛÎÂÙ·È ‰ÂÍÈfiÙÂÚ· ·fi ÙÔÓ ¿ÏÏÔ ¿Óˆ ÛÙÔÓ ¿ÍÔÓ·.

–6,8 < –3,7 ‰ÈfiÙÈ –6,8 = 6,8 > 3,7 = –3,7

- 120 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

™ÙÔÓ ¿ÍÔÓ· ÙˆÓ ·ÚÈıÌÒÓ Ó· ÙÔÔıÂÙËıÔ‡Ó ÔÈ ·ÚÈıÌÔ›: 9 (·) –2,25, (‚) –3,33, (Á) –1,75, (‰) + , (Â) +4,75 Î·È (ÛÙ) +3,66.

4

§‡ÛË

9 4

-1,75 -3,33

0

-1

-2,25

3,66

4,75

1

9 –3,33 < –2,25 < –1,75 < + 4 < +3,66 < +4,75

2.

ΔÔ ÛËÌÂ›Ô ∫ ¤¯ÂÈ ÙÂÙÌË̤ÓË –7. ¡· ‚ÚÂı› ÙÔ ÛËÌÂ›Ô § Ì ·ÓÙ›ıÂÙË ÙÂÙÌË̤ÓË.

§‡ÛË K

§ √

-7

-6

-5

-4

-3

-2

-1

0

+1

+2

+3

+4

+5

+6

+7

3

4

¶¿Óˆ Û ¿ÍÔÓ· x Ox ‚Ú›ÛÎÔ˘Ì ÙÔ ÛËÌÂ›Ô K Ì ÙÂÙÌË̤ÓË –7. ΔfiÙ ÙÔ § ¤¯ÂÈ ÙÂÙÌË̤ÓË ÙÔÓ ·ÚÈıÌfi +7.

3.

∂¿Ó Ë ·fiÏ˘ÙË ÙÈÌ‹ ÙÔ˘ ·ÚÈıÌÔ‡ · Â›Ó·È 2, Ó· ‚ÚÂı› Ô ·ÚÈıÌfi˜ ·.

§‡ÛË ∂ÊfiÛÔÓ · =2 ÙfiÙÂ Ô ·ÚÈıÌfi˜ · ı· ›ӷÈ, ›Ù ÙÔ +2 ›Ù ÙÔ –2, ‰ÈfiÙÈ +2 = 2 Î·È –2 = 2 .

· = 2 ·=–2 -4

-3

-2

·=+2 -1

0

1

2

¶·Ú·ÙËÚԇ̠fiÙÈ ÔÈ ·ÚÈıÌÔ› –2 Î·È +2, ¤¯Ô˘Ó ÙËÓ ›‰È· ·fiÏ˘ÙË ÙÈÌ‹ ·ÏÏ¿ ·ÓÙ›ıÂÙÔ ÚfiÛËÌÔ.

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

™˘ÌÏ‹ÚˆÛ ٷ ·Ú·Î¿Ùˆ ÎÂÓ¿: (·) ∏ ·fiÛÙ·ÛË ÙÔ˘ ÛËÌ›Ԣ, Ì ÙÔ ÔÔ›Ô ·Ó··ÚÈÛÙ¿ÓÂÙ·È ¤Ó·˜ ÚËÙfi˜ ·ÚÈıÌfi˜, ·fi ÙËÓ ·Ú¯‹ ÙÔ˘ ¿ÍÔÓ· ϤÁÂÙ·È .................... ÙÔ˘ ·ÚÈıÌÔ‡ Î·È Â›Ó·È ¿ÓÙ· ................... ·ÚÈıÌfi˜. (‚) ¢‡Ô ÚËÙÔ› ·ÚÈıÌÔ› Ô˘ ¤¯Ô˘Ó ÙËÓ ›‰È· ·fiÏ˘ÙË ÙÈÌ‹ Î·È Â›Ó·È ÂÙÂÚfiÛËÌÔÈ Ï¤ÁÔÓÙ·È .......................................... . (Á) ∞Ó ¤Ó·˜ ·ÚÈıÌfi˜ · Â›Ó·È ıÂÙÈÎfi˜ Ô ·ÓÙ›ıÂÙfi˜ ÙÔ˘ Â›Ó·È ................................................ . ∞Ó Ë ·fiÏ˘ÙË ÙÈÌ‹ ÂÓfi˜ ·ÚÈıÌÔ‡ Â›Ó·È ›ÛË Ì 6 ÙfiÙÂ Ô ·ÚÈıÌfi˜ Â›Ó·È Ô ...... ‹ Ô ..... . (‰) ∞fi ‰‡Ô ıÂÙÈÎÔ‡˜ ÚËÙÔ‡˜ ÌÈÎÚfiÙÂÚÔ˜ Â›Ó·È ÂΛÓÔ˜ Ô˘ ¤¯ÂÈ ÙËÓ ............................... ·fiÏ˘ÙË ÙÈÌ‹. (Â) ∞fi ‰‡Ô ·ÚÓËÙÈÎÔ‡˜ ÚËÙÔ‡˜ ÌÂÁ·Ï‡ÙÂÚÔ˜ Â›Ó·È ÂΛÓÔ˜ Ô˘ ¤¯ÂÈ ÙËÓ ....................... ·fiÏ˘ÙË ÙÈÌ‹.

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

2.

- 121 -

¡· Û˘ÌÏËÚÒÛÂȘ ÙÔÓ ›Ó·Î· Ô˘ ·ÎÔÏÔ˘ı›: ∞ÚÈıÌfi˜

–2,73 +7,66 –1,05

0

+8,07

–8

AfiÛÙ·ÛË ÙÔ˘ ÛËÌ›Ԣ Ô˘ ·ÓÙÈÛÙÔȯ› ·fi ÙËÓ ·Ú¯‹ ÙÔ˘ ¿ÍÔÓ·

3.

ΔÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË (·) (‚) (Á) (‰) (Â)

4.

5.

6.

™ø™Δ√ §∞£√™

IÛ¯‡ÂÈ Ë ·ÓÈÛfiÙËÙ·: –5,7 < 5,7. πÛ¯‡ÂÈ Ë ·ÓÈÛfiÙËÙ·: –7,6 > –6,7. ™ÙËÓ ·ÓÈÛfiÙËÙ· 2,3 < x < 4,7 Ô x ÌÔÚ› Ó· ¿ÚÂÈ 2 ·Î¤Ú·È˜ ÙÈ̤˜. À¿Ú¯Ô˘Ó 5 ·ÎÚÈ‚Ò˜ ·Î¤Ú·ÈÔÈ Ô˘ ·ÏËıÂ‡Ô˘Ó ÙË Û¯¤ÛË: –2 x +2 ¢‡Ô ·Î¤Ú·ÈÔÈ Ì ·ÓÙ›ıÂÙÔ ÚfiÛËÌÔ Â›Ó·È ·ÓÙ›ıÂÙÔÈ.

μÚ˜ ÙËÓ ·fiÏ˘ÙË ÙÈÌ‹ ÙˆÓ ÚËÙÒÓ: (·) +7,25, (‚) –2,5, (Á) +16, (‰) –20,05, (Â) –58. μÚ˜ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ ¤¯Ô˘Ó ˆ˜ ·fiÏ˘ÙË ÙÈÌ‹: (·) 100, (‚) 21,7, (Á) 0, (‰) 7,03, (Â) 5,2. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

∞ÚÈıÌfi˜

1

∞ÓÙ›ıÂÙÔ˜ ∞fiÏ˘ÙË ÙÈÌ‹

7.

–19 –8 2

12 7

ΔÔÔı¤ÙËÛ ÛÙÔÓ ¿ÍÔÓ· x Ox Ù· ÛËÌ›· Ì ÙÂÙÌË̤Ó˜: –9, –5,5, +8, –3, –7,25, +1, +12, +3, +9. ¶ÔÈ· ·fi ·˘Ù¿ Â›Ó·È Û˘ÌÌÂÙÚÈο ˆ˜ ÚÔ˜ ÙËÓ ·Ú¯‹ ÙÔ˘ ¿ÍÔÓ·;

8.

™¯Â‰›·Û ÙÔÓ ¿ÍÔÓ· x Ox, Ì ηٿÏÏËÏË ÌÔÓ¿‰· ÁÈ· Ó· ·Ú·ÛÙ‹ÛÂȘ Ù· ÛËÌ›· Ì ÙÂÙÌË̤Ó˜ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: –20,5, +15, –39,75, –68,25, +70, +52,25,+43, –69.

9.

¡· Û˘ÁÎÚ›ÓÂȘ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: (·) +41 Î·È +38, (‚) 9 Î·È 11, (Á) –3 Î·È –2, (‰) –9 ηÈ

10.

¡· Û˘ÁÎÚ›ÓÂȘ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: (·) 11, –11 Î·È 11 , (‚) –3, +3 Î·È 3 . ΔÈ Û˘ÌÂÚ·›ÓÂȘ;

11.

12.

13.

–16, (Â) 7 Î·È –8, (ÛÙ) 0 Î·È –3, (˙) 0 Î·È +4.

¡· ÁÚ¿„ÂȘ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: –2, +7, +15, –3, 0, –4, +5, –8 Î·È –10 Û ·‡ÍÔ˘Û· ÛÂÈÚ¿. ¡· Û˘ÌÏËÚÒÛÂȘ Ì ÙÔ Î·Ù¿ÏÏËÏÔ Û‡Ì‚ÔÏÔ: <, > ‹ = Ù· ÎÂÓ¿, ÒÛÙ ӷ ÚÔ·„Ô˘Ó ·ÏËı›˜ Û¯¤ÛÂȘ: (·) –3 ... –8, (‚) –4 ... 10, (Á) 0 ... –1, (‰) +3 ... 0, (Â) –5 ... – –5 , (ÛÙ) –5 ... –(+5), (˙) +7 ... –7 , (Ë) –(–8) ... –8, (ı) +3 ... –(+4), (È) 0 ... – –4 . ΔÔ x ·ÚÈÛÙ¿ÓÂÈ ¤Ó·Ó ·Î¤Ú·ÈÔ ·ÚÈıÌfi. °È· ÔȘ ÙÈ̤˜ ÙÔ˘ x ı· ÈÛ¯‡Ô˘Ó ÔÈ Û¯¤ÛÂȘ: (·) –13 < x < –8, (‚) –4 > x > –5, (Á) –2 < x < 5.

- 122 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞.7.3. ¶ÚfiÛıÂÛË ÚËÙÒÓ ·ÚÈıÌÒÓ

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ ™Â οı ̛· ·fi ÙȘ ÂÚÈÙÒÛÂȘ Ô˘ ÂÚÈÁÚ¿ÊÔÓÙ·È ÛÙÔÓ ÚÒÙÔ ›Ó·Î·, Ó· ‚ÚÂȘ ÛÙÔÓ ‰Â‡ÙÂÚÔ ›Ó·Î· ÙËÓ ÚfiÛıÂÛË Ô˘ Ù˘ ·ÓÙÈÛÙÔȯ› Î·È Ù¤ÏÔ˜ ÙÔ ·ÓÙ›ÛÙÔÈ¯Ô ·ÔÙ¤ÏÂÛÌ· ÛÙÔÓ ÙÚ›ÙÔ ›Ó·Î·.

1

2

3

4

∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ ·˘Í‹ıËÎÂ Û˘Ó¯fiÌÂÓ· ‰‡Ô ÊÔÚ¤˜: ∏ ÚÒÙË ·‡ÍËÛË ‹Ù·Ó 8,5 Q Î·È Ë ‰Â‡ÙÂÚË 6,2 Q

∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ ÌÂÈÒıËÎÂ Û˘Ó¯fiÌÂÓ· ‰‡Ô ÊÔÚ¤˜: ∏ ÚÒÙË Ì›ˆÛË ‹Ù·Ó 8,5 Q Î·È Ë ‰Â‡ÙÂÚË 6,2 Q

∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ ·˘Í‹ıËΠηٿ 8,5 Q Î·È ÌÂÙ¿ ÌÂÈÒıËΠηٿ 6,2 Q

∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ ÌÂÈÒıËΠηٿ 8,5 Q Î·È ÌÂÙ¿ ·˘Í‹ıËΠηٿ 6,2 Q

· (+8,5) + (–6,2)

‚ (–8,5) + (+6,2)

Á (+8,5) + (+6,2)

‰ (–8,5) + (–6,2)

–14,7 ªÂÈÒıËΠηٿ 14,7 Q

I

+2,3 ∞˘Í‹ıËΠηٿ 2,3 Q

Iπ

–2,3 MÂÈÒıËΠηٿ 2,3 Q

Iππ

+14,7 ∞˘Í‹ıËΠηٿ 14,7 Q

IV

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì 䉬 °È· Ó· ÚÔÛı¤ÛÔ˘Ì ‰‡Ô ÔÌfiÛËÌÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜, ÚÔÛı¤ÙÔ˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜ Î·È ÛÙÔ ¿ıÚÔÈÛÌ· ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌfi ÙÔ˘˜.

+8,5 + +6,2 = +14,7

䉬 °È· Ó· ÚÔÛı¤ÛÔ˘Ì ‰‡Ô ÂÙÂÚfiÛËÌÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜, ·Ê·ÈÚԇ̠·fi ÙË ÌÂÁ·Ï‡ÙÂÚË ÙË ÌÈÎÚfiÙÂÚË ·fiÏ˘ÙË ÙÈÌ‹ Î·È ÛÙË ‰È·ÊÔÚ¿ ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌÔ ÙÔ˘ ÚËÙÔ‡ Ì ÙË ÌÂÁ·Ï‡ÙÂÚË ·fiÏ˘ÙË ÙÈÌ‹.

+8,5 + –6,2 = +2,3

–8,5 + –6,2 = –14,7

–8,5 + +6,2 = –2,3

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 123 -

π‰ÈfiÙËÙ˜ Ù˘ ÚfiÛıÂÛ˘ ¶·Ú·ÙËÚԇ̠fiÙÈ:

( +1,5 ) + ( –2,3 ) = –0,8

䉴

( –2,3 ) + ( +1,5 ) = –0,8

°ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ: ªÔÚԇ̠ӷ ·ÏÏ¿˙Ô˘Ì ÙË ÛÂÈÚ¿ ÙˆÓ ‰‡Ô ÚÔÛıÂÙ¤ˆÓ ÂÓfi˜ ·ıÚÔ›ÛÌ·ÙÔ˜. (∞ÓÙÈÌÂÙ·ıÂÙÈ΋ ȉÈfiÙËÙ·)

·+‚=‚+· –1,4 +( +2,7 + –3,1 )= –1,4 + –0,4 = –1,8

䉴

( –1,4 + +2,7 )+ –3,1 = +1,3 + –3,1 = –1,9

ªÔÚԇ̠ӷ ·ÓÙÈηıÈÛÙԇ̠ÚÔÛıÂÙ¤Ô˘˜ Ì ÙÔ ¿ıÚÔÈÛÌ¿ ÙÔ˘˜ ‹ Ó· ·Ó·Ï‡Ô˘Ì ¤Ó· ÚÔÛıÂÙ¤Ô Û ¿ıÚÔÈÛÌ·. (¶ÚÔÛÂÙ·ÈÚÈÛÙÈ΋ ȉÈfiÙËÙ·).

· + (‚+Á) = (·+‚) + Á

( +1,5 ) + 0 = +1,5

䉴

0 + ( –2,3 ) = –2,3

ΔÔ 0 fiÙ·Ó ÚÔÛÙÂı› Û ¤Ó· ÚËÙfi ‰ÂÓ ÙÔÓ ÌÂÙ·‚¿ÏÂÈ.

·+0=0+·=· (+

9 9 ) + (– ) = 0 4 4

(–

9 9 ) + (+ ) = 0 4 4

‹

䉴

ΔÔ ¿ıÚÔÈÛÌ· ‰‡Ô ·ÓÙ›ıÂÙˆÓ ·ÚÈıÌÒÓ Â›Ó·È Ìˉ¤Ó.

· + (–·) = (–·) + · = 0

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

™Â ÌÈ· fiÏË ·Ú·ÙËÚ‹ıËÎ·Ó ÔÈ ·Ú·Î¿Ùˆ ·˘ÍÔÌÂÈÒÛÂȘ Ù˘ ıÂÚÌÔÎÚ·Û›·˜: ∞Ú¯ÈΤ˜ ıÂÚÌÔÎڷۛ˜ ∞˘ÍÔÌÂÈÒÛÂȘ ıÂÚÌÔÎÚ·Û›·˜ (·) μÚ¿‰˘ +1ÆC ÙËÓ ÂfiÌÂÓË Ì¤Ú· ·˘Í‹ıËΠηٿ 4ÆC (‚) ªÂÛË̤ÚÈ –1ÆC ÙÔ ‚Ú¿‰˘ ÌÂÈÒıËΠηٿ 2ÆC (Á) μÚ¿‰˘ –2ÆC ÙËÓ ÂfiÌÂÓË Ì¤Ú· ·˘Í‹ıËΠηٿ 5ÆC (‰) ªÂÛË̤ÚÈ +5ÆC ÙÔ ‚Ú¿‰˘ ÌÂÈÒıËΠηٿ 7ÆC (Â) ªÂÛË̤ÚÈ –3ÆC ÙÔ ‚Ú¿‰˘ ÌÂÈÒıËΠηٿ 3ÆC ¶ÔÈ· ‹Ù·Ó Ë ÙÂÏÈ΋ ıÂÚÌÔÎÚ·Û›· Û οı ÂÚ›ÙˆÛË;

§‡ÛË (·)

+5

ΔËÓ ÂÔ̤ÓË Ë̤ڷ Ë ıÂÚÌÔÎÚ·Û›· ¤¯ÂÈ ·˘ÍËı› ηٿ 4ÆC, ‰ËÏ·‰‹ ¤¯ÂÈ ÌÂÙ·‚ÏËı› ηٿ +4ÆC. H ıÂÚÌÔÎÚ·Û›· ı· Â›Ó·È 5ÆC ¿Óˆ ·fi ÙÔ Ìˉ¤Ó, ‰ÈfiÙÈ:

+1

+4

0

0

0

0

(+1) + (+4) = +5 (‚)

∞fi –1ÆC Ë ıÂÚÌÔÎÚ·Û›· ÌÂÈÒıËΠηٿ 2ÆC, ¿Ú· ÌÂÙ·‚Ï‹ıËΠηٿ –2ÆC. H Ó¤· ıÂÚÌÔÎÚ·Û›· Â›Ó·È –3ÆC, ‰ÈfiÙÈ ¤¯Ô˘ÌÂ:

(–1) + (–2) = –3

–1

–3

–2

- 124 -

(Á)

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

™ÙËÓ ÂÚ›ÙˆÛË ·˘Ù‹ Ë ıÂÚÌÔÎÚ·Û›· ·fi –2ÆC, ·˘Í‹ıËΠηٿ 5ÆC, ‰ËÏ·‰‹ ¤¯Ô˘Ì ÌÈ· ÌÂÙ·‚ÔÏ‹ +5ÆC. ∏ ıÂÚÌÔÎÚ·Û›· ¤ÊÙ·Û ÛÙÔ˘˜ +3ÆC, ‰ÈfiÙÈ:

(–2) + (+5) = +3

+3

0

0

+5

–2

(‰)

∏ ·Ú¯È΋ ıÂÚÌÔÎÚ·Û›· ‹Ù·Ó +5ÆC Î·È ÌÂÈÒıËΠηٿ 7ÆC, ‰ËÏ·‰‹ ¤¯Ô˘Ì ÌÈ· ÌÂÙ·‚ÔÏ‹ ηٿ –7ÆC. H ıÂÚÌÔÎÚ·Û›· ¤ÁÈÓÂ, ÙÂÏÈο –2ÆC, ‰ÈfiÙÈ:

+5

–7 0

0 –2

(+5) + (–7) = –2

(Â)

∞fi 3ÆC Ë ıÂÚÌÔÎÚ·Û›· ÌÂÈÒıËΠηٿ 3ÆC, ‰ËÏ·‰‹ ÌÂÙ·‚Ï‹ıËΠηٿ –3ÆC. ∏ ıÂÚÌÔÎÚ·Û›· ¤ÁÈÓ ÙÂÏÈο 0ÆC, ‰ÈfiÙÈ:

+3

0

(+3) + (–3) = 0

2.

¡· ˘ÔÏÔÁÈÛÙÔ‡Ó Ù· ·Ú·Î¿Ùˆ ·ıÚÔ›ÛÌ·Ù·: (·) (+5,6) + (+8,7) + (–3,2) + (–6,9) + (+3,2) + (–7,4) Î·È (‚) (–1,8) + (+4,8) + (+9,7) + (–4,8) + (–3,4) + (+1,5)

§‡ÛË (·)

( +5,6 ) + ( +8,7 ) + ( –3,2 ) + ( –6,9 ) + ( +3,2 ) + ( –7,4 ) = = ( +5,6 ) + ( +8,7 ) + ( +3,2 ) + ( –3,2 ) + ( –6,9 ) + ( –7,4 ) = (¯ˆÚ›˙Ô˘Ì ÙÔ˘˜ ·ÚÓËÙÈÎÔ‡˜ ·fi ÙÔ˘˜ ıÂÙÈÎÔ‡˜)

= ( +17,5 )

+ ( –17,5 ) = 0

(ÚÔÛı¤ÙÔ˘Ì ¯ˆÚÈÛÙ¿ ÙÔ˘˜ ·ÚÓËÙÈÎÔ‡˜ Î·È ÙÔ˘˜ ıÂÙÈÎÔ‡˜)

(‚)

( –1,8 ) + ( +4,8 ) + ( +9,7 ) + ( –4,8 ) + ( –3,4 ) + ( +1,5 ) = = ( –1,8 ) + ( –4,8 ) + ( –3,4 ) + ( +4,8 ) + ( +9,7 ) + ( +1,5 ) = = ( –10 )

+ ( +16 ) = +6

0

–3

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 125 -

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

ΔÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË (·) (‚) (Á) (‰) (Â)

2.

3.

4.

™ø™Δ√ §∞£√™

™ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ Ë ÚfiÛıÂÛË ÛËÌ·›ÓÂÈ ¿ÓÙ· ·‡ÍËÛË ∞Ó ÙÔ ¿ıÚÔÈÛÌ· ‰‡Ô ÚËÙÒÓ Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜, ÙfiÙÂ Î·È ÔÈ ‰‡Ô ÚËÙÔ› Â›Ó·È ·ÚÓËÙÈÎÔ› ·ÚÈıÌÔ› ∞Ó · + ‚ = 0, ÙfiÙ ÔÈ · Î·È ‚ Â›Ó·È ·ÓÙ›ıÂÙÔÈ ÚËÙÔ› ·ÚÈıÌÔ› ∞Ó ÙÔ ¿ıÚÔÈÛÌ· ‰‡Ô ÚËÙÒÓ Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜, ÙfiÙÂ Î·È ÔÈ ‰‡Ô ÚËÙÔ› Â›Ó·È ıÂÙÈÎÔ› ·ÚÈıÌÔ›. ΔÔ ¿ıÚÔÈÛÌ· ÂÓfi˜ ÚËÙÔ‡ Î·È ÙÔ˘ ·ÓÙ›ıÂÙÔ˘ ·˘ÙÔ‡ Â›Ó·È ¿ÓÙ· Ìˉ¤Ó.

ÀÔÏfiÁÈÛ ٷ ·ıÚÔ›ÛÌ·Ù·: (·) (+4,05) + (+6,15), (‚) (+5,03) + (+4,07), (Á) (+2,7) + (+97,3), (‰) (+2,6) + (+11,4), (Â) (+7,25) + (+8,75), (ÛÙ) (–3,5) + (–2,5), (˙) (–1,3) + (–5,2), (Ë) (–7,15) + (–4,85), (ı) (–5,25) + (–9,75), (È) (–13,7) + (–6,3) ÀÔÏfiÁÈÛ ٷ ·ıÚÔ›ÛÌ·Ù·: (·) (+4,05) + (–6,15), (‚) (+5,03) + (–4,07), (Á) (–2,7) + (+97,3), (‰) (–2,6) + (+11,4), (Â) (+7,25) + (–8,75), (ÛÙ) (+3,5) + (–2,5), (˙) (–1,3) + (+5,2), (Ë) (+7,15) + (–4,85), (ı) (–5,25) + (+9,75), (È) (+13,7) + (–6,3) ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

+ –5

+4

–8

–11 +17

+9 –4 –21

5.

ToÔı¤ÙËÛ ÛÙ· ÎÂÓ¿ Ù· ηٿÏÏËÏ· ÚfiÛËÌ·, ÒÛÙ ӷ ÚÔ·„Ô˘Ó ·ÏËı›˜ ÈÛfiÙËÙ˜: (·) (....6) + (–8) = –2, (‚) (+5) + (....5) = 0, (Á) (+7) +(....9) = +16, (‰) (....9) + (....8) = –17, (Â) (....6) + (....5) = +11

6.

∂ͤٷÛ ·Ó Â›Ó·È Ì·ÁÈο Ù· ÙÂÙÚ¿ÁˆÓ·: (ª·ÁÈο ÙÂÙÚ¿ÁˆÓ· Â›Ó·È ·˘Ù¿ ÛÙ· ÔÔ›· Ë ÚfiÛıÂÛË ÙˆÓ ·ÚÈıÌÒÓ Î¿ı ÛÙ‹Ï˘ ‹ ÁÚ·ÌÌ‹˜, ηıÒ˜ Î·È ÙˆÓ ‰È·ÁˆÓ›ˆÓ ÙÔ˘˜, ‰›ÓÔ˘Ó ÙÔ ›‰ÈÔ ·ÎÚÈ‚Ò˜ ¿ıÚÔÈÛÌ·).

7.

8.

-1

+4

-3

+1,1 +2,4 -2,5

-2

0

+2

-0,1 +3,5 -2,4

+3

-4

+1

ÀÔÏfiÁÈÛ ٷ ·ıÚÔ›ÛÌ·Ù·: (·) (–3,8) + (+2,8) + (–5,4) + (+8,2) Î·È (‚) (–3,5) + (–9,99) + (+2,5) + (–15,75) + (+20,75) + (+9,99) ÀÔÏfiÁÈÛ ٷ ·ıÚÔ›ÛÌ·Ù·: 9 5 2 5 7 20 (·) (+ ) + (– ) + (+ ) + (– ) + (+ ) + (– ) Î·È 4 4 3 3 13 13 (‚) (+

1 5 3 1 ) + (– ) + (+ ) + (– ) 7 7 5 35

0

-4,9 +5,9

- 126 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞.7.4. ∞Ê·›ÚÂÛË ÚËÙÒÓ ·ÚÈıÌÒÓ

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ ™ÙÔ Û¯‹Ì· ‚ϤÔ˘Ì ÙË Ì¤ÛË ıÂÚÌÔÎÚ·Û›· ÌÈ·˜ ÂÚÈÔ¯‹˜ ÁÈ· ÙÔ˘˜ 12 Ì‹Ó˜ ÙÔ˘ ¯ÚfiÓÔ˘ ÛÂ Û˘ÁÎÂÎÚÈ̤ÓË ÒÚ· Ù˘ Ë̤ڷ˜. £ÂÚÌÔÎÚ·Û›·

+40 +30 +20 +20 ª‹Ó˜

0 -10

I·Ó.

ºÂ‚.

M·Ú. ∞Ú. ª¿ÈÔ˜ πÔ˘Ó. πÔ˘Ï. ∞˘Á. ™ÂÙ. √ÎÙ.

¡ÔÂÌ. ¢ÂÎ.

➣ ➣ ➣

¶ÔÈÔ˜ Â›Ó·È Ô ÈÔ ˙ÂÛÙfi˜ Ì‹Ó·˜ ÙÔ˘ ¤ÙÔ˘˜ Î·È ÔÈÔ˜ Ô ÈÔ ÎÚ‡Ô˜; ¶ÔÈ· Â›Ó·È Ë ‰È·ÊÔÚ¿ ıÂÚÌÔÎÚ·Û›·˜ ÌÂٷ͇ ·˘ÙÒÓ ÙˆÓ ÌËÓÒÓ; ¶ÔÈ· Â›Ó·È Ë ‰È·ÊÔÚ¿ ıÂÚÌÔÎÚ·Û›·˜ ÌÂٷ͇ οı ‰‡Ô ‰È·‰Ô¯ÈÎÒÓ ÌËÓÒÓ;

䉬

°È· Ó· ·Ê·ÈÚ¤ÛÔ˘Ì ·fi ÙÔÓ ·ÚÈıÌfi ( +8,5 ) – ( +6,2 ) = ( +8,5 ) + ( –6,2 ) = · ÙÔÓ ·ÚÈıÌfi ‚, ÚÔÛı¤ÙÔ˘Ì ÛÙÔÓ = 8,5 - 6,2 = 2,3 · ÙÔÓ ·ÓÙ›ıÂÙÔ ÙÔ˘ ‚. ( +8,5 ) – ( –6,2 ) = ( +8,5 ) + ( +6,2 ) =

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘ÌÂ

· – ‚ = · + (–‚)

䉬

= 8,5 + 6,2 = 14,7 ™ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ Ë ·Ê·›ÚÂÛË ÌÂÙ·ÙÚ¤ÂÙ·È Û ÚfiÛıÂÛË Î·È ÂÔ̤ӈ˜ Â›Ó·È ¿ÓÙ· ‰˘Ó·Ù‹ (‰ËÏ·‰‹, ‰ÂÓ ··ÈÙÂ›Ù·È Ó· Â›Ó·È Ô ÌÂȈ٤Ԙ ¿ÓÙ· ÌÂÁ·Ï‡ÙÂÚÔ˜ ·fi ÙÔÓ ·Ê·ÈÚÂÙ¤Ô, fiˆ˜ ›Û¯˘Â ̤¯ÚÈ ÙÒÚ·).

∞·ÏÔÈÊ‹ ·ÚÂÓı¤ÛÂˆÓ ™Â ·ÚÎÂÙ¤˜ ÂÚÈÙÒÛÂȘ ·ÚÈıÌËÙÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ ÂÌÊ·Ó›˙ÔÓÙ·È ÂÚÈÛÛfiÙÂÚÔÈ ÙÔ˘ ÂÓfi˜ ·ÚÈıÌÔ› Ì ٷ ÚfiÛËÌ¿ ÙÔ˘˜ ̤۷ Û ·ÚÂÓı¤ÛÂȘ, ÌÚÔÛÙ¿ ·fi ÙȘ Ôԛ˜ ÌÔÚ› Ó· ˘¿Ú¯Ô˘Ó Ù· ÚfiÛËÌ· + ‹ – . °È· Ó· ··Ï›„Ô˘Ì ÙȘ ·ÚÂÓı¤ÛÂȘ ÂÚÁ·˙fiÌ·ÛÙ ˆ˜ ÂÍ‹˜: 䉬

䉬

ŸÙ·Ó ÌÈ· ·Ú¤ÓıÂÛË ¤¯ÂÈ ÌÚÔÛÙ¿ Ù˘ ÙÔ + (+5) + (-7) = +5 - 7 = -2 (‹ ‰ÂÓ ¤¯ÂÈ ÚfiÛËÌÔ), ÌÔÚԇ̠ӷ ÙËÓ (9,1–6,2+3,4) + (–7,5+10–8,3) = ··Ï›„Ô˘Ì ̷˙› Ì ÙÔ + (·Ó ¤¯ÂÈ) Î·È Ó· ÁÚ¿„Ô˘Ì ÙÔ˘˜ fiÚÔ˘˜ Ô˘ ÂÚȤ¯ÂÈ Ì ٷ = 9,1–6,2 + 3,4–7,5 + 10–8,3 ÚfiÛËÌ¿ ÙÔ˘˜.

(-5) - (-7) = -5 + 7 = +2 ŸÙ·Ó ÌÈ· ·Ú¤ÓıÂÛË ¤¯ÂÈ ÌÚÔÛÙ¿ Ù˘ ÙÔ –, ÌÔÚԇ̠ӷ ÙËÓ ··Ï›„Ô˘Ì ̷˙› Ì ÙÔ – Î·È –(9,1–6,2+3,4)–(–7,5+10–8,3) = Ó· ÁÚ¿„Ô˘Ì ÙÔ˘˜ fiÚÔ˘˜ Ô˘ ÂÚȤ¯ÂÈ Ì = –9,1+6,2–3,4+7,5–10+8,3 ·ÓÙ›ıÂÙ· ÚfiÛËÌ·.

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 127 -

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

ŒÓ· ‚Ú¿‰˘ ÙÔ ıÂÚÌfiÌÂÙÚÔ ÛÙÔ Ì·ÏÎfiÓÈ ÂÓfi˜ ÛÈÙÈÔ‡ ¤‰ÂȯÓ –3Æ C Î·È Ì¤Û· ÛÙÔ Û›ÙÈ 18ÆC. ¶fiÛË ‹Ù·Ó Ë ‰È·ÊÔÚ¿ ıÂÚÌÔÎÚ·Û›·˜;

§‡ÛË

To Úfi‚ÏËÌ· ˙ËÙ¿ÂÈ Ó· ˘ÔÏÔÁ›ÛÔ˘Ì ÙË ‰È·ÊÔÚ¿ ÙˆÓ ıÂÚÌÔÎÚ·ÛÈÒÓ, ‰ËÏ·‰‹ ÙË ‰È·ÊÔÚ¿ (+18) – (–3) . ∞Ó ·Ú·ÙËÚ‹ÛÔ˘Ì ÙÔ Û¯‹Ì· ı· ‰Ô‡Ì fiÙÈ Ë ‰È·ÊÔÚ¿ ıÂÚÌÔÎÚ·Û›·˜ ÌÂٷ͇ ÙÔ˘ ÂÛˆÙÂÚÈÎÔ‡ ÙÔ˘ ÛÈÙÈÔ‡ Î·È ÙÔ˘ Â͈ÙÂÚÈÎÔ‡ ÙÔ˘ ‹Ù·Ó +21ÆC. ™‡Ìʈӷ Ì ÙÔÓ ÔÚÈÛÌfi Ù˘ ·Ê·›ÚÂÛ˘ ÚËÙÒÓ ı· ¤¯Ô˘ÌÂ:

(+18) – (–3) = (+18) + (+3) = (+21)

2.

+20

+20 +18

+10

+10

+21 0

0

–3

–3

–10

–10

ŒÓ·˜ ¤ÌÔÚÔ˜ ¯ÚˆÛÙ¿ÂÈ ÛÙÔÓ ÚÔÌËıÂ˘Ù‹ ÙÔ˘ 897,56 Q Î·È ÙÔ˘ ÔÊ›ÏÂÈ ¤Ó·˜ ÂÏ¿Ù˘ 527,42 Q. ¶fiÛ· Q Ú¤ÂÈ Ó· ¤¯ÂÈ ÛÙÔ Ù·ÌÂ›Ô ÁÈ· Ó· ͯÚÂÒÛÂÈ;

§‡ÛË

∞Ó x Â›Ó·È ÙÔ ÔÛfi ÙˆÓ ¯ÚËÌ¿ÙˆÓ Ô˘ ¯ÚÂÈ¿˙ÂÙ·È, ı· ›ӷÈ:

x + (+527,42) = +897,56 . °ÓˆÚ›˙Ô˘Ì fiÙÈ: x = (+897,56) – (+527,42) . ™‡Ìʈӷ Ì ÙÔÓ Î·ÓfiÓ· Ù˘ ·Ê·›ÚÂÛ˘ ÚËÙÒÓ, ¤¯Ô˘Ì fiÙÈ:

x = (+897,56) + (–527,42) . ÕÚ·, x = +(897,56 – 527,42) ‹ x = +370,14 Q

3.

N· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: (·) x + (+3) = (–9), (‚) (–8) – x = +7

§‡ÛË (·)

∞Ó Â›Ó·È: x + (+3) = (–9) ÙfiÙ x = (–9) – (+3) ‹ x = (–9) + (–3) ‹ x = (–12). ¢ËÏ·‰‹,

(‚)

x = –12.

∂Ê’ fiÛÔÓ (–8) – x = +7 ı· ÈÛ¯‡ÂÈ fiÙÈ: (–8) = (+7) + x Î·È Â›Û˘:

x = (–8) – (+7) ‹ x = (–8) + (–7) ‰ËÏ·‰‹ x = –15.

4.

N· ‚ÚÂı› Ë ÙÈÌ‹ Ù˘ ·Ú¿ÛÙ·Û˘: –13 – (0,38 – 11 – 13) + (0,38 – 11).

§‡ÛË Œ¯Ô˘ÌÂ:

–13 – (0,38 – 11 – 13) + (0,38 – 11) = = –13 – 0,38 + 11 + 13 + 0,38 – 11 = = –13 + 13 – 0,38 + 0,38 – 11 + 11 = =0+0+0=0

- 128 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

ΔÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË ™ø™Δ√ §∞£√™ (·) ™ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ Ë ·Ê·›ÚÂÛË ÛËÌ·›ÓÂÈ ¿ÓÙ· ÂÏ¿ÙÙˆÛË (‚) ∞Ó Ë ‰È·ÊÔÚ¿ ‰‡Ô ÚËÙÒÓ Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜, ÙfiÙÂ Î·È ÔÈ ‰‡Ô ÚËÙÔ› Â›Ó·È ·ÚÓËÙÈÎÔ› ·ÚÈıÌÔ›. (Á) πÛ¯‡ÂÈ ÛÙËÓ ·Ê·›ÚÂÛË Ë ·ÓÙÈÌÂÙ·ıÂÙÈ΋ ȉÈfiÙËÙ·: · – ‚ = ‚ – · (‰) πÛ¯‡ÂÈ fiÙÈ: 6 – (+8) + (+5) + (–3) + (2) + (–1) = 0 (Â) §‡ÛË Ù˘ Â͛ۈÛ˘ x + (–3) = –2 Â›Ó·È Ô ·ÚÈıÌfi˜ +1 (ÛÙ) √È ÂÍÈÛÒÛÂȘ x+(–2)=+5 Î·È x–(+7)=–10+(+5) ¤¯Ô˘Ó ÙËÓ ›‰È· χÛË. (˙) §‡ÛË Ù˘ Â͛ۈÛ˘ x – (–2) = –8 + (+7) – (–4) Â›Ó·È Ô ·ÚÈıÌfi˜ +1

2.

ÀÔÏfiÁÈÛ ÙȘ ‰È·ÊÔÚ¤˜: 2 2 (·) 5 – (–7), (‚) –8 – (+8), (Á) –2 – (–15,2), (‰) 14,55 – 18,45, (Â) – – (– ).

7 7

3.

∫¿Ó ÙȘ Ú¿ÍÂȘ: (·) +3 + –2 + –9 , (‚) –20 + –10 – +10 , (Á) –3 – –2 + –5 – +6 .

4.

∫¿Ó ÙȘ Ú¿ÍÂȘ: (·) (+5) – (+3) + (+8), (‚) (–25) + (–4) – (–10), (Á) (+12) + (+2) – (–8).

5.

6.

7.

8.

9.

· +3

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î· Ì ÙÔ˘˜ ηٿÏÏËÏÔ˘˜ ·ÚÈıÌÔ‡˜:

‚

·+‚ –5 +10

–8 –5

–2 –9

·–‚

+6

5 7 ¡· χÛÂȘ ÙȘ ÂÍÈÛÒÛÂȘ: (·) x + (–8) = –18, (‚) x + 12 = –14, (Á) x+ = , 4 8 5 (‰) x– =2.

4 · ‚ ·–‚ ‚–· 7 3 ™˘ÌÏ‹ÚˆÛ ÙȘ ‰‡Ô ÙÂÏÂ˘Ù·›Â˜ ÛًϘ ÙÔ˘ ›Ó·Î·: ΔÈ Û˘ÌÂÚ·›ÓÂȘ ÁÈ· ÙÔ˘˜ ·ÚÈıÌÔ‡˜ ÙˆÓ ‰‡Ô ·˘ÙÒÓ ÛÙËÏÒÓ;

2H 33 –5,55 –2,45 3 –2,1

ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛÂˆÓ Ì ‰‡Ô ÙÚfiÔ˘˜: (·) 11–(12–2)+(10–5)–(8+5), (‚) –(13,7–2,6)+14,8–(–8,7+5), (Á) 1 –( 3 – 5 )–( 7 + 5 ) 6 4 4 12 6

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

x

3,5

y z

–1,5

x+y+z x–y–z

0

1,89 4,3 –2,3

3,11 0,22

0

3 –3 –

1

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 129 -

∞.7.5. ¶ÔÏÏ·Ï·ÛÈ·ÛÌfi˜ ÚËÙÒÓ ·ÚÈıÌÒÓ

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ ŒÓ·˜ ¤ÌÔÚÔ˜ ‰È·›ÛÙˆÛÂ, fiÙÈ Î¿ı Ë̤ڷ ÙÔ˘ ÙÂÏÂ˘Ù·›Ô˘ ‰Âη‹ÌÂÚÔ˘ ÙˆÓ ÂÎÙÒÛÂˆÓ ¤‚Á·˙ ΤډԘ 524,5Q.

+524,5

i

–26 5,4

ΔÔ ÂfiÌÂÓÔ, fï˜, ‰Âη‹ÌÂÚÔ Â›¯Â ηıËÌÂÚÈÓ‹ ˙ËÌÈ¿ 265,4Q. ∂›Ó·È ÁÓˆÛÙfi, fiÙÈ ÛÙ· ÏÔÁÈÛÙÈο ‚È‚Ï›· ÙÔ Î¤Ú‰Ô˜ ηٷ¯ˆÚÂ›Ù·È ˆ˜ ıÂÙÈ΋ ÂÁÁÚ·Ê‹ Î·È Ë ˙ËÌÈ¿ ˆ˜ ·ÚÓËÙÈ΋. ¢ËÏ·‰‹, ÙÔ Û˘ÓÔÏÈÎfi ΤډԘ ÁÈ· ÙÔ ‰Âη‹ÌÂÚÔ ÙˆÓ ÂÎÙÒÛÂˆÓ ı· Â›Ó·È (+524,5Q) (+10 Ë̤Ú˜) Î·È ÁÈ· ÙÔ ÂfiÌÂÓÔ ‰Âη‹ÌÂÚÔ Ë Û˘ÓÔÏÈ΋ ˙ËÌÈ¿ ı· Â›Ó·È (–265,4Q) (+10 Ë̤Ú˜)

➣ ➣

¶ÚÔÛ¿ıËÛ ӷ ‚ÚÂȘ ÙÔ ·ÔÙ¤ÏÂÛÌ· ÙˆÓ ·Ú·¿Óˆ Ú¿ÍÂˆÓ ¯ˆÚ›˜ Ó· οÓÂȘ ÙÔ˘˜ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡˜. ΔÈ ·Ú·ÙËÚ›˜ ÁÈ· ÙÔ ÚfiÛËÌÔ ÙˆÓ ·ÔÙÂÏÂÛÌ¿ÙˆÓ;

¢È·ÈÛÙÒÓÔ˘ÌÂ, ÏÔÈfiÓ, fiÙÈ: 䉴 ΔÔ ÁÈÓfiÌÂÓÔ ‰‡Ô ıÂÙÈÎÒÓ ÚËÙÒÓ Â›Ó·È ıÂÙÈÎfi˜ ÚËÙfi˜

䉴

ΔÔ ÁÈÓfiÌÂÓÔ ÂÓfi˜ ıÂÙÈÎÔ‡ Î·È ÂÓfi˜ ·ÚÓËÙÈÎÔ‡ ÚËÙÔ‡ Â›Ó·È ·ÚÓËÙÈÎfi˜ ÚËÙfi˜

∞˜ ‰Ô‡Ì ÙÒÚ· Ò˜ ‚Ú›ÛÎÔ˘Ì ÙÔ ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÚÓËÙÈÎÒÓ ·ÎÂÚ·›ˆÓ.

(–10) (+9) = –90 ™ËÌÂÈÒÓÔ˘Ì ÙÔ˘˜ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡˜ ‰‡Ô ·Ú·ÁfiÓÙˆÓ, (–10) (+8) = –80 (–10) (+7) = –70 ·fi ÙÔ˘˜ ÔÔ›Ô˘˜ Ô ¤Ó·˜ ̤ÓÂÈ ÛÙ·ıÂÚfi˜, ÙÔ –10, Î·È Ô ¿ÏÏÔ˜ ÌÂÈÒÓÂÙ·È ‰È·‰Ô¯Èο ηٿ 1 οı ÊÔÚ¿.

(–10) (+6) = –60 (–10) (+5) = –50 (–10) (+4) = –40 ¶·Ú·ÙËÚԇ̠fiÙÈ Ù· ÁÈÓfiÌÂÓ· ·˘Í¿ÓÔÓÙ·È ‰È·‰Ô¯Èο ηٿ 10 (–10) (+3) = –30 (–10) (+2) = –20 ∞Ó ˘Ôı¤ÛÔ˘Ì fiÙÈ Î·È ÌÂÙ¿ ÙÔ ÌˉÂÓÈÛÌfi ÙÔ˘ ‰Â‡ÙÂÚÔ˘ ·Ú¿ÁÔÓÙ· Ù· ÁÈÓfiÌÂÓ· Û˘Ó¯›˙Ô˘Ó Ó· ·˘Í¿ÓÔÓÙ·È Ì ÙÔÓ (–10) (+1) = –10 = 0 ›‰ÈÔ ÙÚfiÔ, Ú¤ÂÈ Ó· ÔÚ›ÛÔ˘Ì fiÙÈ: (–10) 0 (–10) (–1) = +10 = +(10 1) (–10) (–1) = ; (–10) (–2) = ; (–10) (–2) = +20 = +(10 2) (–10) (–3) = +30 = +(10 3) (–10) (–3) = ; (–10) (–4) = +40 = +(10 4) (–10) (–4) = ; ......................... ............................................... ¢È·ÈÛÙÒÓÔ˘Ì ÂÔ̤ӈ˜ fiÙÈ: 䉴 ΔÔ ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÚÓËÙÈÎÒÓ ·ÎÂÚ·›ˆÓ Â›Ó·È ıÂÙÈÎfi˜ ·Î¤Ú·ÈÔ˜ °ÂÓÈÎfiÙÂÚ·: 䉴 ΔÔ ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÚÓËÙÈÎÒÓ ÚËÙÒÓ Â›Ó·È ıÂÙÈÎfi˜ ÚËÙfi˜.

- 130 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì 䊉

°È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰‡Ô ÔÌfiÛËÌÔ˘˜ ÚËÙÔ‡˜ ( +1,5 ) ·ÚÈıÌÔ‡˜, ÔÏÏ·Ï·ÛÈ¿˙Ô˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜ Î·È ÛÙÔ ÁÈÓfiÌÂÓÔ ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌÔ «+». ( –1,5 ) ¢ËÏ·‰‹: + + = +

䊉

ηÈ

( +2,2 ) = ( +3,3 ) ( –2,2 ) = ( +3,3 )

– –=+

°È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰‡Ô ÂÙÂÚfiÛËÌÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜, ÔÏÏ·Ï·ÛÈ¿˙Ô˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜ Î·È ÛÙÔ ÁÈÓfiÌÂÓÔ ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌÔ « – ». ¢ËÏ·‰‹: + – = – Î·È – + = –

( +1,5 )

( –2,2 ) = ( –3,3 )

( –1,5 )

( +2,2 ) = ( –3,3 )

√ ¢ÈfiÊ·ÓÙÔ˜ ÚÒÙÔ˜ ÂÈÛ¿ÁÂÈ ÙËÓ ¤ÓÓÔÈ· «§∂πæπ™» (·ÚÓËÙÈÎfi˜) ‰È·Ù˘ÒÓÔÓÙ·˜ ÙÔ˘˜ ηÓfiÓ˜ Ù˘ Ú¿Í˘ ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ Ì ÙËÓ ¤ÎÊÚ·ÛË: «§∂πæπ™ ∂π §∂πæπ¡ ¶√π∂π À¶∞ƒ•IN, §∂πæπ™ ∂¶π À¶∞ƒ•IN ¶√π∂π §∂πæIN»

π‰ÈfiÙËÙ˜ Ùo˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ °ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:

¶·Ú·ÙËÚԇ̠fiÙÈ:

( +1,5 )

( –2,2 ) =

–3,3

( –2,2 )

( +1,5 ) =

–3,3

䉴 ªÔÚԇ̠ӷ ·ÏÏ¿˙Ô˘Ì ÙË ÛÂÈÚ¿ ‰‡Ô ·Ú·ÁfiÓÙˆÓ ÂÓfi˜ ÁÈÓÔ̤ÓÔ˘ (∞ÓÙÈÌÂÙ·ıÂÙÈ΋ ȉÈfiÙËÙ·).

· ‚=‚ ·

–0,5 ( +2,2 –3,5 )= –0,5 –7,7 = +3,85

( –0,5 +2,2 ) –3,5 )= –1,1 –3,5 = +3,85 1

( +1,5 )

1

(

–2,2

= +1,5

1 =

+1,5

) = –2,2

1 =

–2,2

0,15 (–5) + 1,85 (–5) = (–0,75) + (–9,25) = –10 ( 0 , 1 5 + 1 , 8 5 ) ( – 5 ) = 2 ( – 5 ) = – 1 0 (+3) (+ 1 ) = +(3 1 ) = 1 3

3 (– 2 ) (– 3 ) = + ( 2 3 ) = 1 3 2 3 2

(–0,25) (–4) = +(0,25 4) = 1 (–1,3) 0 = 0 ‹ 0 (+ 2 ) = 0 3

䉴 ªÔÚԇ̠ӷ ·ÓÙÈηıÈÛÙԇ̠·Ú¿ÁÔÓÙ˜ Ì ÙÔ

ÁÈÓfiÌÂÓfi ÙÔ˘˜ ‹ Ó· ·Ó·Ï‡Ô˘Ì ¤Ó· ·Ú¿ÁÔÓÙ· Û ÁÈÓfiÌÂÓÔ (¶ÚÔÛÂÙ·ÈÚÈÛÙÈ΋ ȉÈfiÙËÙ·).

· (‚ Á) = (· ‚) Á

䉴 ŸÙ·Ó ¤Ó·˜ ÚËÙfi˜ ÔÏÏ·Ï·ÛÈ¿˙ÂÙ·È Ì ÙÔÓ ·ÚÈıÌfi 1 ‰ÂÓ ÌÂÙ·‚¿ÏÏÂÙ·È.

1 · = · 1 = ·

䉴 ∂ÈÌÂÚÈÛÙÈ΋ ȉÈfiÙËÙ· ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ ˆ˜ ÚÔ˜ ÙËÓ ÚfiÛıÂÛË Î·È ÙËÓ ·Ê·›ÚÂÛË: · (‚+Á)=· ‚+· Á Î·È · (‚–Á)=· ‚–· Á

䊉 √È ÚËÙÔ› ·ÚÈıÌÔ› · Î·È ‚ ϤÁÔÓÙ·È ·ÓÙ›ÛÙÚÔÊÔÈ, fiÙ·Ó Â›Ó·È ‰È¿ÊÔÚÔÈ ÙÔ˘ ÌˉÂÓfi˜ Î·È ÙÔ ÁÈÓfiÌÂÓfi ÙÔ˘˜ Â›Ó·È ›ÛÔ Ì ÙË ÌÔÓ¿‰·: · ‚ = 1

√ ηı¤Ó·˜ ·fi ÙÔ˘˜ · Î·È ‚ Â›Ó·È ·ÓÙ›ÛÙÚÔÊÔ˜ ÙÔ˘ ¿ÏÏÔ˘. 䉬 ŸÙ·Ó ¤Ó·˜ ÚËÙfi˜ ÔÏÏ·Ï·ÛÈ¿˙ÂÙ·È Ì ÙÔ 0 ÌˉÂÓ›˙ÂÙ·È. 0 · = · 0 = 0

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 131 -

°ÈÓfiÌÂÓÔ ÔÏÏÒÓ ·Ú·ÁfiÓÙˆÓ ¶Ò˜ ÂÚÁ·˙fiÌ·ÛÙ fiÙ·Ó ¤¯Ô˘Ì ӷ ˘ÔÏÔÁ›ÛÔ˘Ì ¤Ó· ÁÈÓfiÌÂÓÔ Ì ÂÚÈÛÛfiÙÂÚÔ˘˜ ·fi ‰‡Ô ·Ú¿ÁÔÓÙ˜; °ÓˆÚ›˙Ô˘Ì fiÙÈ ÙÔ ÁÈÓfiÌÂÓÔ ıÂÙÈÎÒÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· ıÂÙÈÎfi. ∞Ó ˘¿Ú¯ÂÈ ¤Ó·˜ ·Ú¿ÁÔÓÙ·˜ Ô˘ Â›Ó·È ·ÚÓËÙÈÎfi˜ ÌÂÙ·ÙÚ¤ÂÈ ÙÔ ÁÈÓfiÌÂÓÔ Û ·ÚÓËÙÈÎfi. ™ÙËÓ ÂÚ›ÙˆÛË Ô˘ ˘¿Ú¯ÂÈ Î·È ‰Â‡ÙÂÚÔ˜ ·ÚÓËÙÈÎfi˜ ·Ú¿ÁÔÓÙ·˜ Í·Ó·ÌÂÙ·ÙÚ¤ÂÈ ÙÔ ÁÈÓfiÌÂÓÔ Û ıÂÙÈÎfi Î.Ô.Î. ÕÚ·: 䉬 °È· Ó· ˘ÔÏÔÁ›ÛÔ˘Ì ¤Ó· ÁÈÓfiÌÂÓÔ ÔÏÏÒÓ ·Ú·ÁfiÓÙˆÓ (Ô˘ ηӤӷ˜ ‰ÂÓ Â›Ó·È Ìˉ¤Ó), ÔÏÏ·Ï·ÛÈ¿˙Ô˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜ Î·È ÛÙÔ ÁÈÓfiÌÂÓÔ ‚¿˙Ô˘ÌÂ: ñ ΔÔ ÚfiÛËÌÔ +, ·Ó ÙÔ Ï‹ıÔ˜ ÙˆÓ ·ÚÓËÙÈÎÒÓ ·Ú·ÁfiÓÙˆÓ Â›Ó·È ¿ÚÙÈÔ (˙˘Áfi). ñ ΔÔ ÚfiÛËÌÔ –, ·Ó ÙÔ Ï‹ıÔ˜ ÙˆÓ ·ÚÓËÙÈÎÒÓ ·Ú·ÁfiÓÙˆÓ Â›Ó·È ÂÚÈÙÙfi (ÌÔÓfi). 䉬 AÓ ÙÔ˘Ï¿¯ÈÛÙÔÓ ¤Ó·˜ ·Ú¿ÁÔÓÙ·˜ Â›Ó·È Ìˉ¤Ó, ÙfiÙÂ Î·È ÙÔ ÁÈÓfiÌÂÓÔ Â›Ó·È ›ÛÔ Ì Ìˉ¤Ó.

ΔÔ ÛËÌÂ›Ô ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ « » ÌÂٷ͇ ÙˆÓ ÁÚ·ÌÌ¿ÙˆÓ Î·È ÙˆÓ ·ÚÂÓı¤ÛÂˆÓ ·Ú·Ï›ÂÙ·È.

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

N· ˘ÔÏÔÁÈÛÙÔ‡Ó Ù· ÁÈÓfiÌÂÓ·: (·) (–1,4) 5, (‚) (+

2 ) (–2,1), (Á) (–10) (–0,7) 3

§‡ÛË (–1,4) 5 = –(1,4 5) = –7 2 2 (‚) (+ 3 ) (–2,1 ) = –(3 2,1) = –1,4 (·)

(Á)

2.

(–10) (–0,7) = +(10 0,7) = +7

N· ˘ÔÏÔÁÈÛÙ› ÙÔ ÁÈÓfiÌÂÓÔ (–1)·, fiÙ·Ó ÙÔ · ·›ÚÓÂÈ ÙȘ ÙÈ̤˜: +3, –1,2, +

2 , –2.

3

§‡ÛË °È· · = +3

2

›ӷÈ: (–1)(+3) = –3

2

2

°È· · =+ 3 ›ӷÈ: ( – 1 ) ( + 3 ) = – 3

3.

°È· · = –1,2 ›ӷÈ: (–1)(–1,2) = +1,2 °È· · = –2 ›ӷÈ: (–1)(–2) = +2

¡· ‰Âȯı› fiÙÈ: (·+‚)(Á+‰) = ·Á + ·‰ + ‚Á + ‚‰

§‡ÛË ™‡Ìʈӷ Ì ÙËÓ ÂÈÌÂÚÈÛÙÈ΋ ȉÈfiÙËÙ·, ¤¯Ô˘ÌÂ:

(·+‚)(Á+‰) = (·+‚)Á+(·+‚)‰ = ·Á+‚Á+·‰+‚‰

4.

¡· ˘ÔÏÔÁÈÛÙ› Ë ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: (–1)(–20)(+

§‡ÛË

2 (–1)(–20)(+ 3 )(–3)(–0,25)=

2 )(–3)(–0,25).

3

(Ï‹ıÔ˜ ·ÚÓËÙÈÎÒÓ ·Ú·ÁfiÓÙˆÓ 4)

2 = +(1 20 3 3 0,25) = +(20 2 0,25) = +(40 0,25) = +10

- 132 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

2.

3.

4.

¡· Û˘ÌÏËÚˆıÔ‡Ó Ù· ·Ú·Î¿Ùˆ ÎÂÓ¿: (·) ΔÔ ÚfiÛËÌÔ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ‰‡Ô ÔÌfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· .................................. . (‚) ΔÔ ÚfiÛËÌÔ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ‰‡Ô ÂÙÂÚfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· ............................ . (Á) ŒÓ·˜ ÚËÙfi˜ fiÙ·Ó ÔÏÏ·Ï·ÛÈ¿˙ÂÙ·È Ì ÙÔ 1 ‰ÂÓ .......................................................... . (‰) ΔÔ ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÓÙ›ÛÙÚÔÊˆÓ ·ÚÈıÌÒÓ Â›Ó·È ¿ÓÙ· ›ÛÔ Ì ....................................... . (Â) ΔÔ ÚfiÛËÌÔ ÁÈÓÔ̤ÓÔ˘ ÔÏÏÒÓ ·Ú·ÁfiÓÙˆÓ ÂÍ·ÚÙ¿Ù·È ·fi ÙÔ Ï‹ıÔ˜ ÙˆÓ .......................................... ·Ú·ÁfiÓÙˆÓ. ÀÔÏfiÁÈÛ ٷ ÁÈÓfiÌÂÓ·: (·) (–1)(–1), (‚) –3(–10), (Á) –1,2(–0,5), (‰) 0(–10589), 12 15 (Â) 1(–20015), (ÛÙ) –0,725(+1000), (˙) (– ).

25 24 ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛÂˆÓ Ì ÙȘ ÏÈÁfiÙÂÚ˜ ‰˘Ó·Ù¤˜ Ú¿ÍÂȘ: 6 6 (–10)+(– )(+3) (·) –5 27 + 2 27, (‚) 10,35(–25) + 9,65(–25), (Á) – 7 7 ™˘ÌÏ‹ÚˆÛ ÙÔÓ ‰ÈÏ·Ófi ›Ó·Î·:

–1

ñ

–2 –3,2

–1

0

+2

+3

+G +10 1 1 1 1 1 + – ), (Á)–10–6( – ) 4 2 8 2 3

5.

∫¿Ó ÙȘ Ú¿ÍÂȘ: (·) –7(–8+10–5), (‚) (0,25–0,05)(–

6.

∫¿Ó ÙȘ Ú¿ÍÂȘ: (·) (5+·)(2+‚), (‚) (·+7)(·–7), (Á) (·–3)(‚–3), (‰) (Á+8)(‰+5).

7.

ÀÔÏfiÁÈÛ ٷ ÁÈÓfiÌÂÓ·: (·) (–1)(–1), (‚) (–1)(–1)(–1), (Á) (–1)(–1)(–1)(–1)

8.

ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: ∞ = (·–1)(·+1)(·–2)(·+2), B = ‚(‚–3)(‚+3)(‚–5)(‚+5), ° = Á(2Á–1)(3Á+1)(4Á–2)(Á+2)(Á–2),

9.

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

fiÙ·Ó · = 3 fiÙ·Ó ‚ = 2 fiÙ·Ó Á=0,5

x y z ˆ –2 0,5 +1 –3 – 1 +6 –4 –0,3 –2 + G 0,2 –7

∞=xyz B=yxˆ °=x∞–μ ∞μ+°

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 133 -

∞.7.6. ¢È·›ÚÂÛË ÚËÙÒÓ ·ÚÈıÌÒÓ

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì 䊉 °È· Ó· ‰È·ÈÚ¤ÛÔ˘Ì ‰‡Ô ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜, ‰È·ÈÚԇ̠ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜ Î·È ÛÙÔ ËÏ›ÎÔ ‚¿˙Ô˘ÌÂ:

䉴 ÙÔ ÚfiÛËÌÔ +, ·Ó Â›Ó·È ÔÌfiÛËÌÔÈ. ¢ËÏ·‰‹: + : + = + Î·È – : – = +

䉴 ÙÔ ÚfiÛËÌÔ –, ·Ó Â›Ó·È ÂÙÂÚfiÛËÌÔÈ. ¢ËÏ·‰‹:

( +11,22 ) : ( +2,2 ) = ( +5,1 ) ( –11,22 ) : ( –2,2 ) = ( +5,1 ) ( +11,22 ) : ( –2,2 ) = ( –5,1 ) ( –11,22 ) : ( +2,2 ) = ( –5,1 )

+ : – = – Î·È – : + = – · ‚

√ ÏfiÁÔ˜ ÙÔ˘ –20 ÚÔ˜ ÙÔ 4 ›ӷÈ: -20 ϤÁÂÙ·È ÏfiÁÔ˜ ÙÔ˘ · ÚÔ˜ ÙÔ ‚ Î·È (–20) : (+4) = +4 = –5 ‰ÈfiÙÈ (+4) (–5) = (-20) ÔÚ›˙ÂÙ·È ˆ˜ Ë ÌÔÓ·‰È΋ χÛË Ù˘ √ ÏfiÁÔ˜ ÙÔ˘ –7 ÚÔ˜ ÙÔ –2 ›ӷÈ: ‚ x=·. Â͛ۈÛ˘ -7 7 7 (–7) : (–2) = –2 = 2 ‰ÈfiÙÈ (-2) 2 = (-7)

䊉

ΔÔ ËÏ›ÎÔ Ù˘ ‰È·›ÚÂÛ˘ ·:‚ ‹

䉬

∏ ‰È·›ÚÂÛË

· 1 ÌÔÚ› Î·È Ó· ÁÚ·Ê› · , ÂÔ̤ӈ˜ ‚ ‚

-3 1 (–3) : (–4) = -4 = –3 (– 4 )

ÁÈ· Ó· ‰È·ÈÚ¤ÛÔ˘Ì ‰‡Ô ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜, ·ÚΛ Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ÙÔ ‰È·ÈÚÂÙ¤Ô Ì ÙÔÓ ·ÓÙ›ÛÙÚÔÊÔ ÙÔ˘ ‰È·ÈÚ¤ÙË.

6 1 6 : (–7) = -7 = 6 (– 7 )

· 1 =· ‚ ‚ 䉬

¢È·›ÚÂÛË Ì ‰È·ÈÚ¤ÙË ÙÔ Ìˉ¤Ó ‰ÂÓ ÔÚ›˙ÂÙ·È.

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

¡· ˘ÔÏÔÁÈÛÙÔ‡Ó Ù· Ëϛη: 2 7 (·) (+1,5):(+5), (‚) (+ ):(– ), (Á) (–0,45):(–0,15).

3 5

§‡ÛË (·) (+1,5) : (+5) = +(1,5 : 5) = +0,3

2

7

2

7

2

5

10

(‚) (+ 3 ) : (– 5 ) = –( 3 : 5 ) = –( 3 7 ) = – 21 . (Á) (–0,45) : (–0,15) = +(0,45 : 0,15) = +3

-5 1 (–5) : (+2) = 2 = –5 ( 2 )

- 134 -

2.

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

¡· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: (·) –6x = –24, (‚) –3x = +15, (Á) x : (–2) = –3

§‡ÛË (·) –6x = –24

(‚)

x = (–24) : (–6) x = +(24 : 6) x = +4

3.

–3x = +15 x = (+15) : (–3) x = –(15 : 3) x = –5

N· ‚ÚÂı› Ë ÙÈÌ‹ Ù˘ ·Ú¿ÛÙ·Û˘: [

(Á)

x : (–2) = –3 x = (–3) (–2) x = +(3 2) x = +6

2 (–3)–(–2)(–9)] : [0,4(–10)–(–0,2)(–5)]+7.

3

§‡ÛË 2 [ 3 (–3)–(–2)(–9)] : [0,4(–10)–(–0,2)(–5)]+7 = 2 = [–( 3 3)–(2 9)] : [–(0,4 10)–(0,2 5)]+7 = = = = = =

(–2–18) : (–4–1) + 7 = (–20) : (–5) + 7 = +(20 : 5) + 7 = +4 +7 = +11

(οÓÔ˘Ì ÙȘ Ú¿ÍÂȘ ̤۷ ÛÙȘ ·ÚÂÓı¤ÛÂȘ)

(οÓÔ˘Ì ÙÔ˘˜ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡˜ Î·È ÙȘ ‰È·ÈÚ¤ÛÂȘ) (οÓÔ˘Ì ÙȘ ÚÔÛı¤ÛÂȘ Î·È ÙȘ ·Ê·ÈÚ¤ÛÂȘ)

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

™˘ÌÏ‹ÚˆÛ ٷ ·Ú·Î¿Ùˆ ÎÂÓ¿: (·) ΔÔ ÚfiÛËÌÔ ÙÔ˘ ËÏ›ÎÔ˘ ‰‡Ô ÔÌfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· .................................. . (‚) ΔÔ ÚfiÛËÌÔ ÙÔ˘ ËÏ›ÎÔ˘ ‰‡Ô ÂÙÂÚfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· ............................... . (Á) °È· Ó· ‰È·ÈÚ¤ÛÔ˘Ì ‰‡Ô ÚËÙÔ‡˜, ·ÚΛ Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ÙÔ ............................. Ì ÙÔÓ ·ÓÙ›ÛÙÚÔÊÔ ÙÔ˘ ................................ . (‰) ŒÓ· ËÏ›ÎÔ · : ‚ ϤÁÂÙ·È Î·È ................................ ÙÔ˘ · ÚÔ˜ ÙÔ ‚.

2.

∫¿Ó ÙȘ ‰È·ÈÚ¤ÛÂȘ: (·) (+15,15) : (+3), (‚) (–4,5) : (–1,5), (Á) (–81) : (+0,9), (‰) 49 : (–7)

3.

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

x

y

-7 3

5 -6

x+y

x–y

xy

x:y

1,7 2,3 – K –1 10 –0,75 –120 1 2 , (‚) , (Á) , (‰) (–3 ):(–2 ).

0,25 –0,5 (–12) + (–8) 5 3

4.

YÔÏfiÁÈÛ ٷ Ëϛη: (·)

5.

6.

7.

§‡Û ÙȘ ÂÍÈÛÒÛÂȘ: (·) –3x = 74, (‚) –0,14x = –49, (Á) x(–2) = 12, (‰) K¿Ó ÙȘ Ú¿ÍÂȘ: (·)

2 4 x=– .

3 6

–1 2 12 (–2)(–5)(–1) –7 5 3 + – , (‚) – , (Á) ( – ):(– ).

3 –6 –15 –10 3 –3 2

ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ Ù˘ ·Ú¿ÛÙ·Û˘: [(–8)(

–7 9 )–(–15) : (–8)](–8)+(–27) : (– ).

64 8

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 135 -

∞.7.7. ¢Âη‰È΋ ÌÔÚÊ‹ ÚËÙÒÓ ·ÚÈıÌÒÓ

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 1 Ë ŒÓ·˜ ·Ù¤Ú·˜ Á˘ÚÓÒÓÙ·˜ ÛÙÔ Û›ÙÈ ·fi ÙË ‰Ô˘ÏÂÈ¿ ÙÔ˘ ¤ÊÂÚ ¤ÓÙ ÛÔÎÔÏ¿Ù˜ ÁÈ· Ù· ‰‡Ô ·È‰È¿ ÙÔ˘. ŸÙ·Ó ¤ÊÙ·Û ÛÙÔ Û›ÙÈ, ‰È·›ÛÙˆÛ fiÙÈ Ì·˙› Ì ٷ ‰‡Ô ·È‰È¿ ÙÔ˘, ‹Ù·Ó Î·È ¤Ó·˜ Ê›ÏÔ˜ ÙÔ˘˜.

➣

°È·Ù› ‰ÂÓ ÌÔÚÔ‡Ó Ó· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ÔÈ ‰‡Ô ÛÔÎÔÏ¿Ù˜ ÛÙ· ÙÚ›· ·È‰È¿;

™ÎÂÊÙfiÌ·ÛÙÂ

5,0000...

20 20 20 .. ..

3 1,666...

∞Ó ‰ÂÓ ˘‹Ú¯Â Ô Ê›ÏÔ˜ ÙˆÓ ·È‰ÈÒÓ, ı· ¤ÙÚˆÁ ηı¤Ó· ·fi Ù· ‰‡Ô ·È‰È¿ 5:2 =2,5 ÛÔÎÔÏ¿Ù˜. ΔÒÚ·, fï˜, Ú¤ÂÈ Ó· ‰Ô‡Ì ÔÈÔ Â›Ó·È ÙÔ ·ÎÚÈ‚¤˜ ·ÔÙ¤ÏÂÛÌ· Ù˘ ‰È·›ÚÂÛ˘ 5 : 3. ¶·Ú·ÙËÚÔ‡ÌÂ, fiÙÈ Ë ‰È·›ÚÂÛË ‰ÂÓ Â›Ó·È Ù¤ÏÂÈ·. ¢›ÓÂÈ ËÏ›ÎÔ 1 Î·È ·Ê‹ÓÂÈ ˘fiÏÔÈÔ 2. ∞Ó Û˘Ó¯›ÛÔ˘Ì ÙË ‰È·›ÚÂÛË, ı· ¿ÚÔ˘Ì ËÏ›ÎÔ ÙÔ ‰Âη‰ÈÎfi ·ÚÈıÌfi: 1,666... ∂Âȉ‹, fï˜, ÙÔ ˘fiÏÔÈÔ Ù˘ ‰È·›ÚÂÛ˘ Â›Ó·È ÙÔ ›‰ÈÔ ¿ÓÙ·, Ù· ‰Âη‰Èο „ËÊ›· Â·Ó·Ï·Ì‚¿ÓÔÓÙ·È Î·È Â›Ó·È fiÏ· ›Û· Ì 6. ÕÚ·, ‰ÂÓ ÌÔÚÔ‡Ó Ó· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ‰‡Ô ÛÔÎÔÏ¿Ù˜ Û ÙÚ›· ·È‰È¿.

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ 2 Ë ∂Ù¿ Ô‰ÔÛÊ·ÈÚÈΤ˜ ÔÌ¿‰Â˜ Ú¤ÂÈ Ó· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ÌÈ· ÂȯÔÚ‹ÁËÛË 1.000.000 Q. ➣ μÚ˜, Ì fiÛÔ ÌÂÁ·Ï‡ÙÂÚË ·ÎÚ›‚ÂÈ· ÌÔÚ›˜, ÙÔ ÔÛfi Ô˘ ·ÓÙÈÛÙÔȯ› Û οı ÔÌ¿‰·.

™ÎÂÊÙfiÌ·ÛÙÂ

∂›Û˘, Î·È ÛÙÔ ‰Â‡ÙÂÚÔ ·Ú¿‰ÂÈÁÌ· 1 . 0 0 0 . 0 0 0 , 0 0 0 . . . ‚ϤÔ˘Ì fiÙÈ Ë ‰È·›ÚÂÛË 1.000.000 : 7 3 0 20 ‰ÂÓ Â›Ó·È Ù¤ÏÂÈ·. ¢›ÓÂÈ ËÏ›ÎÔ 142.857 60 Î·È ˘fiÏÔÈÔ 1. ∞Ó Û˘Ó¯›ÛÔ˘Ì ÙË 40 ‰È·›ÚÂÛË ı· ‚Úԇ̠ÙÔ ‰Âη‰ÈÎfi 50 ·ÚÈıÌfi 142.857, 142857 142857... Ì 10 ¿ÂÈÚ· ‰Âη‰Èο „ËÊ›·, Ù¤ÙÔÈ· ÒÛÙÂ, 30 Ó· Â·Ó·Ï·Ì‚¿ÓÔÓÙ·È Û˘Ó¯Ҙ Ù· ›‰È· 20 ¤ÍÈ „ËÊ›· 142857. ..

ª·ı·›ÓÔ˘ÌÂ

7 142857,142857 142857

..

䊉 ΔÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ ‚ڋηÌ ·Ú·¿Óˆ ÙÔ˘˜ ÔÓÔÌ¿˙Ô˘Ì ÂÚÈÔ‰ÈÎÔ‡˜ ‰Âη‰ÈÎÔ‡˜ ·ÚÈıÌÔ‡˜. 䊉 ΔÔ Ï‹ıÔ˜ ÙˆÓ Â·Ó·Ï·Ì‚·ÓÔÌ¤ÓˆÓ ‰Âη‰ÈÎÒÓ „ËÊ›ˆÓ οı ÂÚÈÔ‰ÈÎÔ‡ ·ÚÈıÌÔ‡ ÔÓÔÌ¿˙ÂÙ·È ÂÚ›Ô‰Ô˜.

°ÂÓÈÎfiÙÂÚ·, ÏÔÈfiÓ, ÌÔÚԇ̠ӷ ԇ̠fiÙÈ: 䉴 ∫¿ı ÚËÙfi˜ ·ÚÈıÌfi˜ ÌÔÚ› Ó· ¤¯ÂÈ ÙË ÌÔÚÊ‹ ‰Âη‰ÈÎÔ‡ ‹ ÂÚÈÔ‰ÈÎÔ‡ ‰Âη‰ÈÎÔ‡ ·ÚÈıÌÔ‡ Î·È Û˘Ì‚ÔÏ›˙ÂÙ·È fiˆ˜ Ê·›ÓÂÙ·È ÛÙ· ·Ú·‰Â›ÁÌ·Ù·.

5 1.000.000 .¯. = 1,6 Î·È = 142857,142857.

3 7

- 136 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

¶ÚÔËÁÔ˘Ì¤Óˆ˜, ›‰·Ì Ì ÔÈÔÓ ÙÚfiÔ ¤Ó·˜ ÚËÙfi˜ ·ÚÈıÌfi˜ ÌÔÚ› Ó· ÁÚ·Ê› Ì ÙË ÌÔÚÊ‹ ÂÚÈÔ‰ÈÎÔ‡ ‰Âη‰ÈÎÔ‡ ·ÚÈıÌÔ‡. °ÂÓÓȤٷÈ, fï˜, ÙÔ ÂÚÒÙËÌ· ·Ó ÌÔÚԇ̠ӷ οÓÔ˘ÌÂ Î·È ÙÔ ·ÓÙ›ÛÙÚÔÊÔ. ¢ËÏ·‰‹, ·Ó ÌÔÚԇ̠¤Ó· ÂÚÈÔ‰ÈÎfi ‰Âη‰ÈÎfi ·ÚÈıÌfi Ó· ÙÔÓ ÁÚ¿„Ô˘Ì Ì ÌÔÚÊ‹ ÚËÙÔ‡.

¶∞ƒ∞¢∂π°ª∞ - ∂º∞ƒª√°∏ N· ÁÚ·ÊÔ‡Ó Ì ÎÏ·ÛÌ·ÙÈ΋ ÌÔÚÊ‹ ÔÈ ‰Âη‰ÈÎÔ› ÂÚÈÔ‰ÈÎÔ› ·ÚÈıÌÔ›: (·) 0,2 Î·È (‚) 1,64.

§‡ÛË (·)

£¤ÙÔ˘ÌÂ

x x 10x 10x 10x (10–1)x 9x

= = = = = = =

0,2 Î·È ¤¯Ô˘Ì ‰È·‰Ô¯Èο: 0,222...

2,222...

2+0,222...

2+x 2 2 2 2 x = 9 ¢ËÏ·‰‹: 0,2= 9

x x 100x 100x 100x (100–1)x 99x

(‚) ∞Ó

= = = = = = =

1 ,64 ¤¯Ô˘Ì 1,646464...

164,646464...

164+0,646464...

164+x–1 163 163 163 163 x = 99 ¢ËÏ·‰‹: 1 ,64= 99

™˘ÌÂÚ·›ÓÔ˘Ì fiÙÈ: 䉴 ∫¿ı ÂÚÈÔ‰ÈÎfi˜ ‰Âη‰ÈÎfi˜ ·ÚÈıÌfi˜ ÌÔÚ› Ó· ¤¯ÂÈ ÙË ÌÔÚÊ‹ ÎÏ·ÛÌ·ÙÈÎÔ‡ ÚËÙÔ‡.

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

2.

3.

μÚ˜ ÙË ‰Âη‰È΋ ÌÔÚÊ‹ ÙˆÓ ÚËÙÒÓ: (·) –

15 5 13 20 32 , (‚) , (Á) , (‰) , (Â) .

10 8 14 11 31

μÚ˜ ÙËÓ ÎÏ·ÛÌ·ÙÈ΋ ÌÔÚÊ‹ ÙˆÓ ·ÚÈıÌÒÓ: (·) 57,92, (‚) 2,8, (Á) 3,83, (‰) 7,4561, (Â) 15,399.

BÚ˜ ÌÈ· ¿ÏÏË ‰Âη‰È΋ ÌÔÚÊ‹ ÙˆÓ ·ÚÈıÌÒÓ: (·) 2,9, (‚) 7,69, (Á) 7,3259.

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ °π∞ Δ√ ™¶πΔπ √ ·Ú¯·›Ô˜ ÊÈÏfiÛÔÊÔ˜ ∑‹ÓˆÓ·˜, Ô˘ ¤˙ËÛ ÛÙË ªÂÁ¿ÏË ∂ÏÏ¿‰· ÙÔ 490 - 430 .Ã. ‰È·Ù‡ˆÛÂ, ÌÂٷ͇ ¿ÏψÓ, Î·È ÙÔ ·Ú·Î¿Ùˆ ·Ú¿‰ÔÍÔ ÙÔ˘ ∞¯ÈÏϤ· Ì ÙË ¯ÂÏÒÓ·: “√ ∞¯ÈÏϤ·˜ ‚·‰›˙ÂÈ 10 ÊÔÚ¤˜ ÈÔ ÁÚ‹ÁÔÚ· ·fi ÙË ¯ÂÏÒÓ·. ¢Â ı· ÌÔÚ¤ÛÂÈ ÔÙ¤ Ó· ÙË ÊÙ¿ÛÂÈ, ·Ó Ë ¯ÂÏÒÓ· ÚÔËÁÂ›Ù·È ¤Ó· ÛÙ¿‰ÈÔ (192 ̤ÙÚ· ÂÚ›Ô˘) ·’ ·˘ÙfiÓ”. ∂Ú‡ÓËÛÂ Î·È ÚÔÛ¿ıËÛ ӷ ÂȂ‚·ÈÒÛÂȘ ‹ Ó· ·ÔÚÚ›„ÂȘ ÙÔ ÏfiÁÔ ÁÈ· ÙÔÓ ÔÔ›Ô Ô ∑‹ÓˆÓ·˜ ÈÛ¯˘Ú›˙ÂÙ·È Î¿ÙÈ Ù¤ÙÔÈÔ.

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 137 -

∞.7.8. ¢˘Ó¿ÌÂȘ ÚËÙÒÓ ·ÚÈıÌÒÓ Ì ÂÎı¤ÙË Ê˘ÛÈÎfi

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ ŒÓ·˜ ˘ÔÏÔÁÈÛÙ‹˜ ÌÔχÓıËΠ·fi οÔÈÔ Èfi, Ô ÔÔ›Ô˜ ›¯Â ÙËÓ È‰ÈfiÙËÙ· Ó· ηٷÛÙÚ¤ÊÂÈ Ù· ËÏÂÎÙÚÔÓÈο ·Ú¯Â›· Ì ÙÔÓ ÂÍ‹˜ ÙÚfiÔ: ∫¿ı ÌÔÏ˘Ṳ̂ÓÔ ·Ú¯Â›Ô ÌfiÏ˘ÓÂ, Ì ÙË ÛÂÈÚ¿ ÙÔ˘, ÙÚ›· ¿ÏÏ· ·Ú¯Â›· ̤۷ Û ̛· ÒÚ· ÏÂÈÙÔ˘ÚÁ›·˜ ÙÔ˘ ˘ÔÏÔÁÈÛÙ‹.

➣

¶ÚÔÛ¿ıËÛ ӷ ‚ÚÂȘ, fiÛ· ·Ú¯Â›· ı· ¤¯Ô˘Ó ÌÔÏ˘Óı› Û ¤ÓÙ ÒÚ˜.

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì ™˘Ì‚ÔÏÈÛÌÔ›

}

Ó ·Ú¿ÁÔÓÙ˜ 䊉 ΔÔ ÁÈÓfiÌÂÓÔ · · · ... · (›ÙÂ Ô · Â›Ó·È ıÂÙÈÎfi˜ ›Ù ·ÚÓËÙÈÎfi˜ ÚËÙfi˜), Û˘Ì‚ÔÏ›˙ÂÙ·È Ì ÙÔ ·Ó Î·È Ï¤ÁÂÙ·È ‰‡Ó·ÌË Ì ‚¿ÛË ÙÔ · Î·È ÂÎı¤ÙË ÙÔ Ê˘ÛÈÎfi Ó>1.

ÂÎı¤Ù˘

}

· Ó = · · · . . . · ‚¿ÛË

Ó ·Ú¿ÁÔÓÙ˜

䊉 °È· Ó = 1, ÁÚ¿ÊÔ˘Ì ·1 = · 䊉 ∏ ‰‡Ó·ÌË ·Ó ‰È·‚¿˙ÂÙ·È Î·È ÓÈÔÛÙ‹ ‰‡Ó·ÌË ÙÔ˘ ·. 䊉 ∏ ‰‡Ó·ÌË ·2 ϤÁÂÙ·È Î·È ÙÂÙÚ¿ÁˆÓÔ ÙÔ˘ · ‹ · ÛÙÔ ÙÂÙÚ¿ÁˆÓÔ. 䊉 ∏ ‰‡Ó·ÌË ·3 ϤÁÂÙ·È Î‡‚Ô˜ ÙÔ˘ · ‹ · ÛÙÔÓ Î‡‚Ô.

¶ÚfiÛËÌÔ ‰‡Ó·Ì˘ ¶·Ú·ÙËÚԇ̠fiÙÈ: 5

(+2) = (+2)(+2)(+2)(+2)(+2) = +32 > 0

䉴

¢‡Ó·ÌË Ì ‚¿ÛË ıÂÙÈÎfi ·ÚÈıÌfi Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜. ∞Ó · > 0, ÙfiÙ ·Ó > 0

䉴

¢‡Ó·ÌË Ì ‚¿ÛË ·ÚÓËÙÈÎfi ·ÚÈıÌfi Î·È ÂÎı¤ÙË ¿ÚÙÈÔ Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜. ∞Ó · < 0 Î·È Ó ¿ÚÙÈÔ˜, ÙfiÙ ·Ó > 0

䉴

¢‡Ó·ÌË Ì ‚¿ÛË ·ÚÓËÙÈÎfi ·ÚÈıÌfi Î·È ÂÎı¤ÙË ÂÚÈÙÙfi Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜. ∞Ó · < 0 Î·È Ó ÂÚÈÙÙfi˜, ÙfiÙ ·Ó < 0

}

¿ÚÙÈÔ Ï‹ıÔ˜

°ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:

4

(–2) = (–2)(–2)(–2)(–2) = +16 > 0

}

ÂÚÈÙÙfi Ï‹ıÔ˜

5

(–2) = (–2)(–2)(–2)(–2)(–2) = –32 < 0

- 138 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

π‰ÈfiÙËÙ˜ ‰˘Ó¿ÌÂˆÓ ÚËÙÒÓ Ì ÂÎı¤ÙË Ê˘ÛÈÎfi °ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:

¶·Ú·ÙËÚԇ̠fiÙÈ:

䉴

(–3) 3 (–3) 5 =

3 ·Ú¿ÁÔÓÙ˜

5 ·Ú¿ÁÔÓÙ˜

= (–3)(–3)(–3)(–3)(–3)(–3)(–3)(–3) = 8 ·Ú¿ÁÔÓÙ˜

·Ì ·Ó = ·Ì+Ó

= (–3) 8 = (–3) 3+5

䉴 78 : 73 =

78 7 7 7 7 7 7 7 7 = 3 = 7 7 7 7

= 7 7 7 7 7 = 7 5 = 7 8–3

6

䉴

=(2 2 2 2 2 2) (7 7 7 7 7 7)= = 26 76

2 2 2 2 2 = 9 9 9 9 9 = =

(8 3 ) 7

=

2 2 2 2 2 9 9 9 9 9

25 = 5 9

8 3 8 3 8 3 8 3 8 3 8 3 8 3

= 8 3+3+3+3+3+3+3 = = 8 7 3 = 8 21

°È· Ó· ˘„ÒÛÔ˘Ì ¤Ó· ÁÈÓfiÌÂÓÔ Û ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì οı ·Ú¿ÁÔÓÙ· ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ÛÙÔÓ ÂÎı¤ÙË ·˘Ùfi.

(· ‚)Ó = ·Ó ‚Ó

䉴

( 29 )

°È· Ó· ‰È·ÈÚ¤ÛÔ˘Ì ‰˘Ó¿ÌÂȘ Ì ÙËÓ ›‰È· ‚¿ÛË, ·Ê‹ÓÔ˘Ì ÙËÓ ›‰È· ‚¿ÛË Î·È ‚¿˙Ô˘Ì ÂÎı¤ÙË ÙË ‰È·ÊÔÚ¿ ÙÔ˘ ÂÎı¤ÙË ÙÔ˘ ‰È·ÈÚ¤ÙË ·fi ÙÔÓ ÂÎı¤ÙË ÙÔ˘ ‰È·ÈÚÂÙ¤Ô˘.

·Ì : ·Ó = ·Ì–Ó

(2 7) = (2 7)(2 7)(2 7)(2 7)(2 7)(2 7)

5

°È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰˘Ó¿ÌÂȘ Ì ÙËÓ ›‰È· ‚¿ÛË, ·Ê‹ÓÔ˘Ì ÙËÓ ›‰È· ‚¿ÛË Î·È ‚¿˙Ô˘Ì ÂÎı¤ÙË ÙÔ ¿ıÚÔÈÛÌ· ÙˆÓ ÂÎıÂÙÒÓ.

°È· Ó· ˘„ÒÛÔ˘Ì ¤Ó· ËÏ›ÎÔ Û ¤Ó·Ó ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ηı¤Ó· ·fi ÙÔ˘˜ fiÚÔ˘˜ ÙÔ˘ ËÏ›ÎÔ˘ ÛÙÔÓ ÂÎı¤ÙË ·˘Ùfi.

· Ó ·Ó = Ó ‚ ‚

( ) 䉴 =

°È· Ó· ˘„ÒÛÔ˘Ì ̛· ‰‡Ó·ÌË Û ¤Ó·Ó ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ÙË ‚¿ÛË Ù˘ ‰‡Ó·Ì˘ ÛÙÔ ÁÈÓfiÌÂÓÔ ÙˆÓ ÂÎıÂÙÒÓ.

(·Ì)Ó = ·ÌÓ

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 139 -

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

N· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: (·) –33, (‚) (–3)3, (Á) –34 ,(‰) (–3)4. ΔÈ ·Ú·ÙËÚ›ÙÂ;

§‡ÛË

(·) (‚) (Á) (‰)

2.

∏ ·Ú¿ÛÙ·ÛË ı· ›ӷÈ: –33 = -3 3 3 = –27 ∂Âȉ‹ Ô ÂÎı¤Ù˘ Â›Ó·È ÂÚÈÙÙfi˜, Ë ‰‡Ó·ÌË ı· Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜. ÕÚ·, ı· ›ӷÈ: (–3)3 = (-3) (-3) (-3) = –33 = –27. ∏ ·Ú¿ÛÙ·ÛË ı· ›ӷÈ: –34 = -3 3 3 3 = –81 ∂Âȉ‹ Ô ÂÎı¤Ù˘ Â›Ó·È ¿ÚÙÈÔ˜, Ë ‰‡Ó·ÌË ı· Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜. ÕÚ·, ı· ›ӷÈ: (–3)4 = (-3) (-3) (-3) (-3) = +34 = +81

N· ˘ÔÏÔÁÈÛÙ› Ë ÙÈÌ‹ Ù˘ ·Ú¿ÛÙ·Û˘: ¶=(–2)3 3–34+(–2)4:16+[–1–(–1)7 8]

§‡ÛË ∏ ÛÂÈÚ¿ ÙˆÓ Ú¿ÍÂˆÓ Â›Ó·È Ë ÂÍ‹˜: 1Ô ¢˘Ó¿ÌÂȘ, 2Ô ¶ÔÏÏ·Ï·ÛÈ·ÛÌÔ› Î·È ‰È·ÈÚ¤ÛÂȘ, 3Ô ¶ÚÔÛı¤ÛÂȘ Î·È ·Ê·ÈÚ¤ÛÂȘ. ∞Ó ˘¿Ú¯Ô˘Ó ·ÚÂÓı¤ÛÂȘ, ÚÔËÁÔ‡ÓÙ·È ÔÈ Ú¿ÍÂȘ ̤۷ Û’ ·˘Ù¤˜ Ì ÙËÓ ›‰È· ÛÂÈÚ¿. ÕÚ·: ¶ = (–2) 3 3–3 4 +(–2) 4 :16+[–1–(–1) 7 8] = (–8) 3–81+(+16):16+[–1+8] =

= –24–81+1+7 = –97

1.

2.

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞ ™˘ÌÏ‹ÚˆÛ ٷ ·Ú·Î¿Ùˆ ÎÂÓ¿: (·) ¢‡Ó·ÌË Ì ‚¿ÛË ıÂÙÈÎfi ·ÚÈıÌfi Â›Ó·È .................... ·ÚÈıÌfi˜. (‚) ¢‡Ó·ÌË Ì ‚¿ÛË ·ÚÓËÙÈÎfi ·ÚÈıÌfi Î·È ÂÎı¤ÙË .................... Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜. (Á) ¢‡Ó·ÌË Ì ‚¿ÛË ................ ·ÚÈıÌfi Î·È ÂÎı¤ÙË ÂÚÈÙÙfi Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜. (‰) °È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰˘Ó¿ÌÂȘ Ì ÙËÓ ›‰È· ‚¿ÛË, ·Ê‹ÓÔ˘Ì ÙËÓ ›‰È· ‚¿ÛË Î·È ‚¿˙Ô˘Ì ÂÎı¤ÙË ÙÔ ................................ ÙˆÓ ÂÎıÂÙÒÓ. (Â) °È· Ó· ‰È·ÈÚ¤ÛÔ˘Ì ‰˘Ó¿ÌÂȘ Ì ÙËÓ ›‰È· ‚¿ÛË, ·Ê‹ÓÔ˘Ì ÙËÓ ›‰È· ‚¿ÛË Î·È ‚¿˙Ô˘Ì ÂÎı¤ÙË ................................................................ . (ÛÙ) °È· Ó· ˘„ÒÛÔ˘Ì ¤Ó· ÁÈÓfiÌÂÓÔ Û ¤Ó·Ó ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ....................................... ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ÛÙÔÓ ÂÎı¤ÙË ·˘Ùfi. (˙) °È· Ó· ˘„ÒÛÔ˘Ì ¤Ó· ËÏ›ÎÔ Û ¤Ó·Ó ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ........................................... ÙÔ˘ ËÏ›ÎÔ˘ ÛÙÔÓ ÂÎı¤ÙË ·˘Ùfi. (Ë) °È· Ó· ˘„ÒÛÔ˘Ì ÌÈ· ‰‡Ó·ÌË Û ¤Ó·Ó ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ÙË ‚¿ÛË Ù˘ ‰‡Ó·Ì˘ ÛÙÔ ............................................... ÙˆÓ ÂÎıÂÙÒÓ. μÚ˜ Ì ÔÈÔ ÛÙÔÈ¯Â›Ô Ù˘ 2˘ Î·È Ù˘ 3˘ ÁÚ·ÌÌ‹˜ ·ÓÙ›ÛÙÔȯ· Â›Ó·È ›ÛÔ Î¿ı ÛÙÔÈ¯Â›Ô Ù˘ 1˘ ÁÚ·ÌÌ‹˜ ÙÔ˘ ·Ú·Î¿Ùˆ ›Ó·Î·.

2 32 3 + 52 (3 + 5)2 3 52 (3 5)2 3 – 52 (3 – 5)2 35 5 ΔÂÙÚ¿ÁˆÓÔ ΔÂÙÚ¿ÁˆÓÔ ΔÂÙÚ¿ÁˆÓÔ ΔÂÙÚ¿ÁˆÓÔ ¢È·ÊÔÚ¿ ÕıÚÔÈÛÌ· °ÈÓfiÌÂÓÔ ¶ËÏ›ÎÔ Ù˘ ‰È·ÊÔÚ¿˜ ÙÔ˘ ËÏ›ÎÔ˘ ÙÔ˘ ·ıÚÔ›ÛÌ·ÙÔ˜ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ÙˆÓ 3 Î·È 5 2 ÙˆÓ 3 Î·È 5 2 ÙˆÓ 3 Î·È 5 2 ÙˆÓ 3 2 Î·È 5 3 Â› 5 3 ÏËÓ 5 3 ‰È· 5 3 Î·È 5

75

3.

4

28

64

0,36

225

1,8

–22

YÔÏfiÁÈÛ ÙȘ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: ∞ = (–1)1 +(–1)2 +(–1)3 + (–1)4 + (–1)5, μ = 32 54 – 25 45 + 87,5 43,

°=–

(– 6)5 103 84 – + 5 3 (–5)3 (–4)4

- 140 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞.7.9. ¢˘Ó¿ÌÂȘ ÚËÙÒÓ ·ÚÈıÌÒÓ Ì ÂÎı¤ÙË ·Î¤Ú·ÈÔ

£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘Ì ™‡Ìʈӷ Ì ÙÔÓ Î·ÓfiÓ· Ù˘ ‰È·›ÚÂÛ˘ ÙˆÓ ‰˘Ó¿ÌÂˆÓ Ì ÙËÓ ›‰È· ‚¿ÛË, Ô˘ Ì¿ı·Ì ÛÙËÓ ÚÔËÁÔ‡ÌÂÓË ·Ú¿ÁÚ·ÊÔ, ›ӷÈ: 57 57 = 57–7 = 50, ÁÓˆÚ›˙Ô˘Ì fiÙÈ Â›Ó·È Î·È 7 = 1 ÂÔ̤ӈ˜, 50 = 1. °ÂÓÈο ÈÛ¯‡ÂÈ: 7 5 5 䊉 ∏ ‰‡Ó·ÌË Î¿ı ·ÚÈıÌÔ‡, ‰È¿ÊÔÚÔ˘ ÙÔ˘ ÌˉÂÓfi˜

Ì ÂÎı¤ÙË ÙÔ Ìˉ¤Ó Â›Ó·È ›ÛË Ì ÌÔÓ¿‰·.

·0 = 1

∂›Û˘, ı· ›ӷÈ: 7 57 7–8 = 5–1, ÁÓˆÚ›˙Ô˘Ì fiÙÈ Â›Ó·È Î·È 5 = 5 5 5 5 5 5 5 = 1 ,¿Ú· 5–1 = 5 57 58 5 5 5 5 5 5 5 5 5

=

1 5

6 56 5 5 5 5 5 5 = 1 ,¿Ú· 5–2 = 1 6–8 = 5–2, ÁÓˆÚ›˙Ô˘Ì fiÙÈ Â›Ó·È Î·È 5 = = 5 52 58 58 5 5 5 5 5 5 5 5 52

Î.Ô.Î. ™˘ÌÂÚ·›ÓÔ˘ÌÂ, ÏÔÈfiÓ fiÙÈ:

䊉

∏ ‰‡Ó·ÌË Î¿ı ·ÚÈıÌÔ‡, ‰È¿ÊÔÚÔ˘ ÙÔ˘ ÌˉÂÓfi˜, Ì ÂÎı¤ÙË ·ÚÓËÙÈÎfi Â›Ó·È ›ÛË Ì ÎÏ¿ÛÌ· Ô˘ ¤¯ÂÈ ·ÚÈıÌËÙ‹ ÙË ÌÔÓ¿‰· Î·È ·ÚÔÓÔÌ·ÛÙ‹ ÙË ‰‡Ó·ÌË

1 1 Ó ·–Ó = = ( ) Ó · ·

ÙÔ˘ ·ÚÈıÌÔ‡ ·˘ÙÔ‡ Ì ·ÓÙ›ıÂÙÔ ÂÎı¤ÙË. ∂Âȉ‹ Ù·

· ‚ Î·È Â›Ó·È ·ÓÙ›ÛÙÚÔÊÔÈ ·ÚÈıÌÔ›, · ‚

fiˆ˜ Î·È Ù· · ηÈ

1 ÛÙËÓ ÚÔËÁÔ‡ÌÂÓË Û¯¤ÛË, ·

ÂÍ¿ÁÔ˘Ì ÙÔ Û˘Ì¤Ú·ÛÌ· fiÙÈ ÈÛ¯‡ÂÈ:

䉬

( ·‚ )–Ó= ( ‚· )Ó

OÈ È‰ÈfiÙËÙ˜ ÙˆÓ ‰˘Ó¿ÌÂˆÓ Ì ÂÎı¤ÙË Ê˘ÛÈÎfi, Ô˘ Ì¿ı·Ì ÛÙËÓ ÚÔËÁÔ‡ÌÂÓË ·Ú¿ÁÚ·ÊÔ, ÈÛ¯‡Ô˘Ó Î·È ÁÈ· ÙȘ ‰˘Ó¿ÌÂȘ Ì ÂÎı¤ÙË ·Î¤Ú·ÈÔ.

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 141 -

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°E™

1.

¡· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ‰˘Ó¿ÌÂȘ: (·) (–2)–5, (‚) –3–3, (Á) (–234567)0.

§‡ÛË (–2) –5 =

(·)

2.

1 1 1 1 1 = =– , (‚) –3 –3 =– 3 =– , (Á) (–234567) 0 =1 5 (–2) –32 32 3 27

¡· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: (·)

[(–3)3]2, (‚) 33 : 3–2, (Á) (–2)4 (–2)6, (‰)

§‡ÛË (·)

[ ( – 3 ) 3 ] 2 = ( – 3 ) 3 2 = ( – 3 ) 6 = 7 2 9

(‚)

3 3 : 3 –2 = 3 3–(–2) = 3 3+2 = 3 5 = 2 4 3

(Á)

( – 2 ) 4 ( – 2 ) 6 = ( – 2 ) 4+6 = ( – 2 ) 10 = 1 0 2 4

(‰)

3.

12-3 .

3-3

12–3 = 3–3

(

12 3

–3

) = (

4 1

–3

) = (

1 4

3

) =

1 1 = 3 4 64

¡· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ‰˘Ó¿ÌÂȘ: 10–1, 10–2, 10–3, 10–4, 10–5, 10–6, 10–7.

§‡ÛË 10 –1 =

1 1 = =0,1 1 10 10

10 –2 =

1 1 = =0,01 2 10 100

10 –3 =

1 1 = =0,001 3 10 1000

10 –4 =

1 1 = =0,0001 4 10 10000

10 –5 =

1 1 = =0,00001 5 10 100000

10 –6 =

1 1 = =0,000001 106 1000000

10 –7 =

1 1 = =0,0000001 107 10000000

- 142 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

·

‚

1

–2

–1

–

Á

(·+‚)2 (·‚)2

( ‚· )

2

(–·)–2 (Á‚)–1

4

–

1 G

10 –10 0,01

2.

ÀÔÏfiÁÈÛ ÙȘ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: ∞ = (–1)–3+(–1)–2+(–1)–1+(–1)0+(–1)1+(–1)2 μ = [(–2)2]5[(–3)2]–2+[(–23,5)2(23,5)–2]5,

3.

μÚ˜ ÔÈÔ˜ ·fi ÙÔ˘˜ ·ÚÈıÌÔ‡˜:

4.

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·: x

0,001

0,01

0,1

°=

(–6)–5 16–4 5–3 + – 12–5 (–32)–4 (–10)–3

1 1 ,103 5 2, 3 ,103+102, ‰ÂÓ Â›Ó·È ‰‡Ó·ÌË ÙÔ˘ 10.

10 10

–10

–100

2 104

5 10–3

1

G

–4

x–3 x3 x–1

5.

™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:

10–3 10–2 10–1 10 0 101 102 103

10–3 10–2 10–1

10 0

101

102

103

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 143 -

∞.7.10. T˘ÔÔÈË̤ÓË ÌÔÚÊ‹ ÌÂÁ¿ÏˆÓ Î·È ÌÈÎÚÒÓ ·ÚÈıÌÒÓ

¢ƒ∞™Δ∏ƒπ√Δ∏Δ∞ ∏ ‰È¿ÌÂÙÚÔ˜ ÂÓfi˜ ·ÙfiÌÔ˘ ˘‰ÚÔÁfiÓÔ˘ Â›Ó·È 0,00000000016 cm. ➣ ªÔÚ›˜ Ó· ‰È·‚¿ÛÂȘ Î·È Ó· ı˘ÌËı›˜ ‡ÎÔÏ· ·˘ÙfiÓ ÙÔÓ ·ÚÈıÌfi; ¶·Ú·ÙËÚԇ̠fiÙÈ ˘¿Ú¯ÂÈ, ·ÚÎÂÙ‹ ‰˘ÛÎÔÏ›· ÛÙË ÁÚ·Ê‹ Î·È ÙˆÓ ·ÚÈıÌÒÓ Ô˘ ÂÎÊÚ¿˙Ô˘Ó Ôχ ÌÈÎÚ¿ ÌÂÁ¤ıË. ŸÌˆ˜, Ë Î·Ù¿ÏÏËÏ· ÚÔÛ·ÚÌÔṲ̂ÓË ¯Ú‹ÛË Ù˘ “Ù˘ÔÔÈËÌÂÓ˘ ÌÔÚÊ‹˜” ÙˆÓ ·ÚÈıÌÒÓ, Ô˘ Ì¿ı·Ì ÛÙËÓ ·Ú¿ÁÚ·ÊÔ 3.4. , ÁÈ· ÙË ÁÚ·Ê‹ ÙˆÓ Ôχ ÌÂÁ¿ÏˆÓ ·ÚÈıÌÒÓ, ÌÔÚ› Ó· ‚ÔËı‹ÛÂÈ ÛÙËÓ ·ÓÙÈÌÂÙÒÈÛË Î·È Ù˘ ÁÚ·Ê‹˜ ÙˆÓ Ôχ ÌÈÎÚÒÓ ·ÚÈıÌÒÓ.

ª·ı·›ÓÔ˘Ì 䊉

Ÿˆ˜ ÔÈ Ôχ ÌÂÁ¿ÏÔÈ, ¤ÙÛÈ Î·È ÔÈ Ôχ ÌÈÎÚÔ› ·ÚÈıÌÔ› ÌÔÚÔ‡Ó Ó· ÁÚ·ÊÔ‡Ó ÛÂ Ù˘ÔÈË̤ÓË ÌÔÚÊ‹ Î·È Û˘ÁÎÂÎÚÈ̤ӷ ÛÙË ÌÔÚÊ‹: · 1 0 - Ó , fiÔ˘ · Â›Ó·È ¤Ó·˜ ‰Âη‰ÈÎfi˜ ·ÚÈıÌfi˜ Ì ·Î¤Ú·ÈÔ Ì¤ÚÔ˜ ÌÂÁ·Ï‡ÙÂÚÔ ‹ ›ÛÔ ÙÔ˘ 1 Î·È ÌÈÎÚfiÙÂÚÔ ÙÔ˘ 10 Î·È Ó Ê˘ÛÈÎfi ·ÚÈıÌfi.

¶∞ƒ∞¢∂π°ª∞Δ∞ - ∂º∞ƒª√°∂™

1.

¡· ÂÎÊÚ·ÛÙ› ÌÂ Ù˘ÔÔÈË̤ÓË ÌÔÚÊ‹ ÙÔ ‚¿ÚÔ˜ ÂÓfi˜ ÌÔÚ›Ô˘ ÓÂÚÔ‡, Ô˘ ›ӷÈ: 0,00000000000000000000003 gr.

§‡ÛË

°È· Ó· ÂÎÊÚ¿ÛÔ˘Ì ÙÔ ‚¿ÚÔ˜ ÂÓfi˜ ÌÔÚ›Ô˘ ÓÂÚÔ‡ Ì ÙËÓ Ù˘ÔÔÈË̤ÓË ÌÔÚÊ‹ Ú¤ÂÈ Ó· ‚Úԇ̠ÂΛÓË ÙË ‰‡Ó·ÌË ÙÔ˘ 10 Ô˘, fiÙ·Ó ÔÏÏ·Ï·ÛÈ¿ÛÂÈ ¤Ó· ‰Âη‰ÈÎfi ·ÚÈıÌfi Ì ¤Ó· ÌfiÓÔ ·Î¤Ú·ÈÔ „ËÊ›Ô, ‰›ÓÂÈ Í·Ó¿ ÙÔ ·Ú·¿Óˆ ‚¿ÚÔ˜. ¢ËÏ·‰‹: 0,00000000000000000000003 gr=3 10-23 23 ı¤ÛÂȘ

2.

°È· Ó· ‚Úԇ̠ÙÔÓ Î·Ù¿ÏÏËÏÔ ·Î¤Ú·ÈÔ ÂÎı¤ÙË Ù˘ ‰‡Ó·Ì˘ ÙÔ˘ 10 ÌÂÙÚ¿Ì ÙȘ ‰Âη‰ÈΤ˜ ı¤ÛÂȘ ÌÂÙ¿ ÙËÓ ˘ԉȷÛÙÔÏ‹. ¡· ÂÎÊÚ·ÛÙÔ‡Ó ÌÂ Ù˘ÔÔÈË̤ÓË ÌÔÚÊ‹ ÔÈ ·ÚÈıÌÔ›: (·) 0,123456789, (‚) 0,00000003598, (Á) 0,000008:1000000

§‡ÛË

(·) 0,123456789=1,23456789 10-1, (‚) 0,00000003598=3,598 10-8, (Á) 0,000008:1000000=0,000000000008=8 10-12

∞™∫∏™∂π™ ∫∞𠶃√μ§∏ª∞Δ∞

1.

2.

3.

°Ú¿„ ÌÂ Ù˘ÔÔÈË̤ÓË ÌÔÚÊ‹ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: (·) ∏ ·fiÛÙ·ÛË °Ë˜ - ™ÂÏ‹Ó˘ Â›Ó·È 384.400.000 m. (‚) ∏ ËÏÈΛ· Ù˘ °Ë˜ Â›Ó·È 4.500.000.000 ¤ÙË. (Á) ∏ ·fiÛÙ·ÛË °Ë˜ - ◊ÏÈÔ˘ Â›Ó·È 149.600.000 km.

H Ì¿˙· ÙÔ˘ ·ÙfiÌÔ˘ ÙÔ˘ ˘‰ÚÔÁfiÓÔ˘ Â›Ó·È 1,67 10–27 gr. ¡· ‚ÚÂȘ fiÛ· ¿ÙÔÌ· ÂÚȤ¯ÂÈ 1 gr ˘‰ÚÔÁfiÓÔ˘. °Ú¿„ ÌÂ Ù˘ÔÔÈË̤ÓË ÌÔÚÊ‹ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: (·) ∏ ‰È¿ÌÂÙÚÔ˜ ÂÓfi˜ ˘Ú‹Ó· ·ÙfiÌÔ˘ Â›Ó·È 0,00000000000001 cm. (‚) ΔÔ ‚¿ÚÔ˜ ÂÓfi˜ ÌÔÚ›Ô˘ ·Ï·ÙÈÔ‡ Â›Ó·È 0,000000000000000000000097 gr.

- 144 -

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

AÓ·ÎÂÊ·Ï·›ˆÛË AΤڷÈÔÈ ·ÚÈıÌÔ›:

...., –4, –3, –2, –1, 0, 1, 2, 3, 4,.....

ƒËÙÔ› ·ÚÈıÌÔ›:

º˘ÛÈÎÔ›, ∫Ï¿ÛÌ·Ù·, ¢Âη‰ÈÎÔ› (£ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ›)

√ÌfiÛËÌÔÈ ÚËÙÔ› ·ÚÈıÌÔ›:

Œ¯Ô˘Ó ÙÔ ›‰ÈÔ ÚfiÛËÌÔ

∂ÙÂÚfiÛËÌÔÈ ÚËÙÔ› ·ÚÈıÌÔ›:

Œ¯Ô˘Ó ·ÓÙ›ıÂÙÔ ÚfiÛËÌÔ

∞fiÏ˘ÙË ÙÈÌ‹ ÚËÙÔ‡ · :

∂ÎÊÚ¿˙ÂÈ ÙËÓ ·fiÛÙ·ÛË ÛËÌ›Ԣ Ì ÙÂÙÌË̤ÓË · ·fi ÙËÓ ·Ú¯‹ √ ÙÔ˘ ¿ÍÔÓ· ÙˆÓ ÚËÙÒÓ

∞ÓÙ›ıÂÙÔÈ ÚËÙÔ› ·ÚÈıÌÔ›:

√È ÂÙÂÚfiÛËÌÔÈ Ì ›‰È· ·fiÏ˘ÙË ÙÈÌ‹

∞Ó · > Ô, ÙfiÙ · =· Î·È ·Ó · < Ô, ÙfiÙ · = –·

¶Ú¿ÍÂȘ ÌÂٷ͇ ÚËÙÒÓ ·ÚÈıÌÒÓ ¶ÔÏÏ·Ï·ÛÈ·ÛÌfi˜

¶ÚfiÛıÂÛË

· ‚= · ‚ ñ ·, ‚>0 ‹ ·,‚<0 (ÔÌfiÛËÌÔÈ) ·+‚ = +( · + ‚ ) ñ ·<0<‚ ‹ ‚<0<· (ÂÙÂÚfiÛËÌÔÈ) · ‚=– · ‚ ·+‚ = –( · + ‚ ) ·+‚ = –( · – ‚ ) ·Ó · > ‚ π‰ÈfiÙËÙ˜ ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡: ·<0<‚ ·+‚ = +( ‚ – · ) ·Ó · < ‚ ñ · ‚ = ‚ · (∞ÓÙÈÌÂÙ·ıÂÙÈ΋) ñ · (‚ Á)=(· ‚) Á

(¶ÚÔÛÂÙ·ÈÚÈÛÙÈ΋) π‰ÈfiÙËÙ˜ Ù˘ ÚfiÛıÂÛ˘: ·+‚ = ‚+· (∞ÓÙÈÌÂÙ·ıÂÙÈ΋) ñ · 1=1 ·=· ·+(‚+Á)=(·+‚)+Á (¶ÚÔÛÂÙ·ÈÚÈÛÙÈ΋) ñ · 1 = 1 ·=1 (· Î·È 1 ·ÓÙ›ÛÙÚÔÊÔÈ) · · · ·+0=0+·=· ·+(–·)=(–·)+·=0 (· Î·È –·, ·ÓÙ›ıÂÙÔÈ) ñ · 0=0

ñ ·, ‚ > 0 ñ ·, ‚ < 0 ñ

ñ ñ ñ ñ

{

ñ ·—‚=·+(–‚)

¢È·›ÚÂÛË 1 ñ·:‚=· =· ‚ ‚

∞Ê·›ÚÂÛË

∂¶πª∂ƒπ™Δπ∫∏ π¢π√Δ∏Δ∞

· ( ‚ + Á ) = · ‚+· Á · (‚–Á)=· ‚–· Á

ΔÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ ˆ˜ ÚÔ˜ ÙËÓ ÚfiÛıÂÛË: ΔÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ ˆ˜ ÚÔ˜ ÙËÓ ·Ê·›ÚÂÛË:

¶ÚÔÙÂÚ·ÈfiÙËÙ· ¶Ú¿ÍˆÓ

➊ ¢˘Ó¿ÌÂȘ ➜ ➋ ¶ÔÏÏ·Ï·ÛÈ·ÛÌÔ› & ¢È·ÈÚ¤ÛÂȘ ➜ ➌ ¶ÚÔÛı¤ÛÂȘ & ∞Ê·ÈÚ¤ÛÂȘ √È Ú¿ÍÂȘ ̤۷ ÛÙȘ ·ÚÂÓı¤ÛÂȘ ÚÔËÁÔ‡ÓÙ·È Î·È Á›ÓÔÓÙ·È Ì ÙËÓ ·Ú·¿Óˆ ÛÂÈÚ¿

¢À¡∞ª∂π™

√ÚÈÛÌÔ›

·Ó

= · · · ... · (Ó ÊÔÚ¤˜)

ΔÔ · ϤÁÂÙ·È ‚¿ÛË Î·È ÙÔ Ó ÂÎı¤Ù˘

·0 = 1 Î·È ·1 = · 1 1 ·–Ó = Ó ‹ · –Ó = ( · · –Ó Ó ( ·‚ ) = ( ·‚ )

)

Ó

π‰ÈfiÙËÙ˜ ÙˆÓ ‰˘Ó¿ÌˆÓ

ñ ·Ì·Ó = · Ì+Ó ñ ·Ì:· Ó = · Ì–Ó ñ (·‚) Ó = · Ó ‚Ó ·Ó · Ó ñ(‚) = Ó ‚ ñ (· Ì)Ó =· ÌÓ

(fiÔ˘: ·, ‚ 0 Î·È Ì, Ó Ê˘ÛÈÎÔ› ·ÚÈıÌÔ›)

ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›

- 145 -

∂·Ó·ÏËÙÈΤ˜ ∂ÚˆÙ‹ÛÂȘ ∞˘ÙÔ·ÍÈÔÏfiÁËÛ˘ ∞.

∞Û΋ÛÂȘ ™ˆÛÙÔ‡ ‹ §¿ıÔ˘˜ ΔÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË 1. 7,2 + (–5) = 2,2 2. –1,2 – 0,2 = –1 3. –2,2 + 2,2 = –4,4 4. 7,8 – 8 = 0,2 5. 3,5 – 9 = –5,5 6. 3,5 – 4,5 = –1 7. 6 – 15 = –11 8. 3 – 8,4 = –5,4 9. 6 – 17 = –9 10. 59 – 64 = –5

™ø™Δ√ §∞£√™

B. ∞ Û Î ‹ Û Â È ˜ ™ ˘ Ì  Ï ‹ Ú ˆ Û Ë ˜ Î Â Ó Ô ‡

1. ™˘ÌÏ‹ÚˆÛ ٷ ÎÂÓ¿ ÛÙȘ ·Ú·Î¿Ùˆ ÈÛfiÙËÙ˜: (·) (...8)+(...3)+(...6)+(...5)=+4 (‚) (...8)+(...3)+(...6)+(...5)=–10 (Á)(...3,7)+(....14,8)+(...5,2)+(...16,3)=0 (‰) (...3,7)+(...14,8)+(...5,2)+(...16,3)=–10,4 2. μÚ˜ ÔÈÔ ·Ô Ù· ∞, μ, °, ¢ Î·È ∂ Â›Ó·È ÙÔ ÌÂÁ·Ï‡ÙÂÚÔ, ·Ó ÁÓˆÚ›˙ÂȘ fiÙÈ: ∞ + (–1) = μ + 3 = ° + (–3) = ¢ + 4 = ∂ + (–5) 3. μÚ˜ Ù· ·ıÚÔ›ÛÌ·Ù·: (·) 1+(–2)+3+(–4)+ ... +49+(–50), (‚) 1+(–2)+3+(–4)+ ... +(–198)+199 4. μ¿Ï ٷ ÁÚ¿ÌÌ·Ù· ∞, ∂, π, ∫, √, ¶, ƒ, À Î·È ø Ì ·‡ÍÔ˘Û· ÛÂÈÚ¿ Î·È ÁÚ¿„ ÙË Ï¤ÍË Ô˘ ‚ڋΘ, fiÙ·Ó: ∞ = 4+(–1,5), ∂ = –0,8+(–4,8), π = –0,8+4,8, ∫ = 4+1,5, √ = 0,8+4,8, ¶ = 0,8+(–0,8), ƒ = 0,8+(–4,8), À = –4+(–1,5), ø = –4+1,5 5. ¶ÔÏÏ·Ï·Û›·Û ·Ó¿ ‰‡Ô ÙÔ˘˜ ÙÚÂȘ ÚËÙÔ‡˜ –6,5, 3,5 Î·È –4,5 Ì fiÏÔ˘˜ ÙÔ˘˜ ‰˘Ó·ÙÔ‡˜ ÙÚfiÔ˘˜. (·) ¶fiÛÔÈ ÙÚfiÔÈ ˘¿Ú¯Ô˘Ó; (‚) ¶ÔÈÔ˜ ·fi ÙÔ˘˜ Ù¤ÛÛÂÚȘ ÚËÙÔ‡˜ 29,25, –15,75, –22,75 Î·È 15,75 ˆ˜ ·ÔÙ¤ÏÂÛÌ· ÙˆÓ ÔÏÏ·Ï·ÛÈ·ÛÌÒÓ ·˘ÙÒÓ Â›Ó·È Ï¿ıÔ˜; 6. ™˘ÌÏ‹ÚˆÛ ٷ ÎÂÓ¿ ÛÙÔ Û¯‹Ì·: ∞Ó ÁÓˆÚ›˙ÂȘ fiÙÈ: · ‚ 90

·

‚

15 4

–3

7. ™˘ÌÏ‹ÚˆÛ ٷ ÎÂÓ¿ ÛÙ· Û¯‹Ì·Ù·, ·Ó ÁÓˆÚ›˙ÂȘ fiÙÈ: 2 -3

1

-1 2

- 2 0 ,5 - 8

-3 5 -0,2 -1

-6

°. (·) (·) (·) (·) (·)

∞Û΋ÛÂȘ ∞ÓÙÈÛÙÔ›¯ÈÛ˘

∞ÓÙÈÛÙÔ›¯ÈÛ οı ÛÙÔÈ¯Â›Ô Ù˘ ÚÒÙ˘ ÛÙ‹Ï˘ ÛÙÔ ÛÙÔÈ¯Â›Ô Ù˘ ‰Â‡ÙÂÚ˘ ÛÙ‹Ï˘ Ô˘ ‚Á¿˙ÂÈ ÙÔ ›‰ÈÔ ·ÔÙ¤ÏÂÛÌ·. (+14)+(–17)

(+3)+(–23)

(–12)+(–8)

(+11)+(–11)

(+11)+(–9)

(–22)+(+19)

(–5)+(+25)

(–19)+(+21)

(–16)+(+16)

(+37)+(–17)

(‚) (+13)–(–18) (‚) (+11)–(+3) (‚) (–5)–(+25) (‚) (–16)–(–16) (‚) (–12)–(–8)

(+3)–(+7)

(Á) (–2) 0,5 9 10 (+13)–(+13) (Á) 2 5 (–0,9) (–10) (–37)–(–7) (Á) 2 (–5) (–9) (–10) (+17)–(+9) (Á) –2 5 9 (–10) (–2)–(–33) (Á) 0,2 (–5) (–0,9) 10

900 –900 9 –90 90