121 146
- 1. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:26 ™ÂÏ›‰·121
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 121 -
2. ¡· Û˘ÌÏËÚÒÛÂȘ ÙÔÓ ›Ó·Î· Ô˘ ·ÎÔÏÔ˘ı›:
∞ÚÈıÌfi˜ –2,73 +7,66 –1,05 0 +8,07 –8
AfiÛÙ·ÛË ÙÔ˘ ÛËÌ›Ԣ Ô˘ ·ÓÙÈÛÙÔȯ›
·fi ÙËÓ ·Ú¯‹ ÙÔ˘ ¿ÍÔÓ·
3. ∆ÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË ™ø™∆√ §∞£√™
(·) IÛ¯‡ÂÈ Ë ·ÓÈÛfiÙËÙ·: –5,7 < 5,7.
(‚) πÛ¯‡ÂÈ Ë ·ÓÈÛfiÙËÙ·: –7,6 > –6,7.
(Á) ™ÙËÓ ·ÓÈÛfiÙËÙ· 2,3 < x < 4,7 Ô x ÌÔÚ› 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
(Â) ¢‡Ô ·Î¤Ú·ÈÔÈ Ì ·ÓÙ›ıÂÙÔ ÚfiÛËÌÔ Â›Ó·È ·ÓÙ›ıÂÙÔÈ.
4. µÚ˜ ÙËÓ ·fiÏ˘ÙË ÙÈÌ‹ ÙˆÓ ÚËÙÒÓ: (·) +7,25, (‚) –2,5, (Á) +16, (‰) –20,05, (Â) –58.
5. µÚ˜ ÙÔ˘˜ ·ÚÈıÌÔ‡˜ Ô˘ ¤¯Ô˘Ó ˆ˜ ·fiÏ˘ÙË ÙÈÌ‹: (·) 100, (‚) 21,7, (Á) 0, (‰) 7,03, (Â) 5,2.
6. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·: ∞ÚÈıÌfi˜ 1 –19
∞ÓÙ›ıÂÙÔ˜ –8 12
∞fiÏ˘ÙË ÙÈÌ‹ 2 7
7. ∆ÔÔı¤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›· ÌÂ
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
–16, (Â) 7 Î·È –8, (ÛÙ) 0 Î·È –3, (˙) 0 Î·È +4.
10. ¡· Û˘ÁÎÚ›ÓÂȘ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: (·) 11, –11 Î·È 11 , (‚) –3, +3 Î·È 3 . ∆È Û˘ÌÂÚ·›ÓÂȘ;
11. ¡· ÁÚ¿„ÂȘ ÙÔ˘˜ ·ÚÈıÌÔ‡˜: –2, +7, +15, –3, 0, –4, +5, –8 Î·È –10 Û ·‡ÍÔ˘Û· ÛÂÈÚ¿.
12. ¡· Û˘ÌÏËÚÒÛÂȘ 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 .
13. ∆Ô 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.
- 2. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:26 ™ÂÏ›‰·122
- 122 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
∞.7.3. ¶ Ú fi Û ı Â Û Ë Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó
¢ƒ∞™∆∏ƒπ√∆∏∆∞
™Â οı ̛· ·fi ÙȘ ÂÚÈÙÒÛÂȘ 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›ÛÙÔȯÔ
·ÔÙ¤ÏÂÛÌ· ÛÙÔÓ ÙÚ›ÙÔ ›Ó·Î·.
∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜
–14,7
·˘Í‹ıËÎÂ Û˘Ó¯fiÌÂÓ· ·
‰‡Ô ÊÔÚ¤˜: ∏ ÚÒÙË ªÂÈÒıËÎÂ
·‡ÍËÛË ‹Ù·Ó 8,5 Q ηÈ
(+8,5) + (–6,2) I
ηٿ 14,7 Q
Ë ‰Â‡ÙÂÚË 6,2 Q
1
∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ +2,3
ÌÂÈÒıËÎÂ Û˘Ó¯fiÌÂÓ· ‚
∞˘Í‹ıËÎÂ
‰‡Ô ÊÔÚ¤˜: ∏ ÚÒÙË (–8,5) + (+6,2) ηٿ 2,3 Q Iπ
Ì›ˆÛË ‹Ù·Ó 8,5 Q ηÈ
Ë ‰Â‡ÙÂÚË 6,2 Q
2
∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ Á –2,3
·˘Í‹ıËΠηٿ 8,5 Q MÂÈÒıËÎÂ
Î·È ÌÂÙ¿ ÌÂÈÒıËÎÂ
(+8,5) + (+6,2)
ηٿ 2,3 Q Iππ
ηٿ 6,2 Q
3
∏ ÙÈÌ‹ ÂÓfi˜ ÚÔ˚fiÓÙÔ˜ ‰
ÌÂÈÒıËΠηٿ 8,5 Q
+14,7
Î·È ÌÂÙ¿ ·˘Í‹ıËΠ(–8,5) + (–6,2)
ηٿ 6,2 Q ∞˘Í‹ıËΠIV
4 ηٿ 14,7 Q
£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘ÌÂ
°È· Ó· ÚÔÛı¤ÛÔ˘Ì ‰‡Ô ÔÌfiÛËÌÔ˘˜ ÚËÙÔ‡˜ +8,5 + +6,2 = +14,7
·ÚÈıÌÔ‡˜, ÚÔÛı¤ÙÔ˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜
–8,5 + –6,2 = –14,7
Î·È ÛÙÔ ¿ıÚÔÈÛÌ· ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌfi ÙÔ˘˜.
°È· Ó· ÚÔÛı¤ÛÔ˘Ì ‰‡Ô ÂÙÂÚfiÛËÌÔ˘˜ ÚËÙÔ‡˜ +8,5 + –6,2 = +2,3
·ÚÈıÌÔ‡˜, ·Ê·ÈÚԇ̠·fi ÙË ÌÂÁ·Ï‡ÙÂÚË ÙË
–8,5 + +6,2 = –2,3
ÌÈÎÚfiÙÂÚË ·fiÏ˘ÙË ÙÈÌ‹ Î·È ÛÙË ‰È·ÊÔÚ¿ ‚¿˙Ô˘ÌÂ
ÙÔ ÚfiÛËÌÔ ÙÔ˘ ÚËÙÔ‡ Ì ÙË ÌÂÁ·Ï‡ÙÂÚË ·fiÏ˘ÙË ÙÈÌ‹.
- 3. ∫∂º∞§∞π√-7(113-146)-(20,5 à 28) 3-12-06 14:48 ™ÂÏ›‰·123
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 123 -
π‰ÈfiÙËÙ˜ Ù˘ ÚfiÛıÂÛ˘
¶·Ú·ÙËÚԇ̠fiÙÈ: °ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:
( +1,5 ) + ( –2,3 ) = –0,8 ªÔÚԇ̠ӷ ·ÏÏ¿˙Ô˘Ì ÙË ÛÂÈÚ¿ ÙˆÓ
‰‡Ô ÚÔÛıÂÙ¤ˆÓ ÂÓfi˜ ·ıÚÔ›ÛÌ·ÙÔ˜.
( –2,3 ) + ( +1,5 ) = –0,8 (∞ÓÙÈÌÂÙ·ıÂÙÈ΋ ȉÈfiÙËÙ·)
·+‚=‚+·
ªÔÚԇ̠ӷ ·ÓÙÈηıÈÛÙԇ̠ÚÔÛıÂÙ¤-
–1,4 +( +2,7 + –3,1 )= –1,4 + –0,4 = –1,8
Ô˘˜ Ì ÙÔ ¿ıÚÔÈÛÌ¿ ÙÔ˘˜ ‹ Ó· ·Ó·Ï‡Ô˘ÌÂ
¤Ó· ÚÔÛıÂÙ¤Ô ÛÂ ¿ıÚÔÈÛÌ·.
( –1,4 + +2,7 )+ –3,1 = +1,3 + –3,1 = –1,9
(¶ÚÔÛÂÙ·ÈÚÈÛÙÈ΋ ȉÈfiÙËÙ·) .
· + (‚+Á) = (·+‚) + Á
( +1,5 ) + 0 = +1,5 ∆Ô 0 fiÙ·Ó ÚÔÛÙÂı› Û ¤Ó· ÚËÙfi ‰ÂÓ
ÙÔÓ ÌÂÙ·‚¿ÏÂÈ.
0 + ( –2,3 ) = –2,3
·+0=0+·=·
9 9 ∆Ô ¿ıÚÔÈÛÌ· ‰‡Ô ·ÓÙ›ıÂÙˆÓ ·ÚÈıÌÒÓ
(+ ) + (– ) = 0 ‹
4 4 Â›Ó·È Ìˉ¤Ó.
9 9
(– ) + (+ ) = 0 · + (–·) = (–·) + · = 0
4 4
¶∞ƒ∞¢∂π°ª∞∆∞ - ∂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 ÂfiÌÂÓË Ì¤Ú· ·˘Í‹ıËΠηٿ 4ÆC
(‚) ªÂÛË̤ÚÈ –1ÆC ÙÔ ‚Ú¿‰˘ ÌÂÈÒıËΠηٿ 2ÆC
(Á) µÚ¿‰˘ –2ÆC ÙËÓ ÂfiÌÂÓË 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
(·) ∆ËÓ ÂÔ̤ÓË Ë̤ڷ Ë ıÂÚÌÔÎÚ·Û›· ¤¯ÂÈ ·˘ÍËı› ηٿ +4
+1
4ÆC, ‰ËÏ·‰‹ ¤¯ÂÈ ÌÂÙ·‚ÏËı› ηٿ +4ÆC. 0 0
H ıÂÚÌÔÎÚ·Û›· ı· Â›Ó·È 5ÆC ¿Óˆ ·fi ÙÔ Ìˉ¤Ó, ‰ÈfiÙÈ:
(+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: 0
–1
0
(–1) + (–2) = –3 –3
–2
- 4. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 24-11-06 12:23 ™ÂÏ›‰·124
- 124 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
(Á) ™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: +3
(–2) + (+5) = +3 0 0 +5
–2
+5
(‰) ∏ ·Ú¯È΋ ıÂÚÌÔÎÚ·Û›· ‹Ù·Ó +5ÆC Î·È ÌÂÈÒıËΠηٿ
–7
7ÆC, ‰ËÏ·‰‹ ¤¯Ô˘Ì ÌÈ· ÌÂÙ·‚ÔÏ‹ ηٿ –7ÆC. 0 0
H ıÂÚÌÔÎÚ·Û›· ¤ÁÈÓÂ, ÙÂÏÈο –2ÆC, ‰ÈfiÙÈ: –2
(+5) + (–7) = –2
+3
(Â) ∞fi 3ÆC Ë ıÂÚÌÔÎÚ·Û›· ÌÂÈÒıËΠηٿ 3ÆC, ‰ËÏ·‰‹ –3
0 0
ÌÂÙ·‚Ï‹ıËΠηٿ –3ÆC.
∏ ıÂÚÌÔÎÚ·Û›· ¤ÁÈÓ ÙÂÏÈο 0ÆC, ‰ÈfiÙÈ:
(+3) + (–3) = 0
¡· ˘ÔÏÔÁÈÛÙÔ‡Ó Ù· ·Ú·Î¿Ùˆ ·ıÚÔ›ÛÌ·Ù·:
2. (·) (+5,6) + (+8,7) + (–3,2) + (–6,9) + (+3,2) + (–7,4) ηÈ
(‚) (–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
- 5. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:26 ™ÂÏ›‰·125
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 125 -
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
1. ∆ÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË ™ø™∆√ §∞£√™
(·) ™ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ Ë ÚfiÛıÂÛË ÛËÌ·›ÓÂÈ ¿ÓÙ· ·‡ÍËÛË
(‚) ∞Ó ÙÔ ¿ıÚÔÈÛÌ· ‰‡Ô ÚËÙÒÓ Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜, ÙfiÙ ηÈ
ÔÈ ‰‡Ô ÚËÙÔ› Â›Ó·È ·ÚÓËÙÈÎÔ› ·ÚÈıÌÔ›
(Á) ∞Ó · + ‚ = 0, ÙfiÙ ÔÈ · Î·È ‚ Â›Ó·È ·ÓÙ›ıÂÙÔÈ ÚËÙÔ› ·ÚÈıÌÔ›
(‰) ∞Ó ÙÔ ¿ıÚÔÈÛÌ· ‰‡Ô ÚËÙÒÓ Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜, ÙfiÙ ηÈ
ÔÈ ‰‡Ô ÚËÙÔ› Â›Ó·È ıÂÙÈÎÔ› ·ÚÈıÌÔ›.
(Â) ∆Ô ¿ıÚÔÈÛÌ· ÂÓfi˜ ÚËÙÔ‡ Î·È ÙÔ˘ ·ÓÙ›ıÂÙÔ˘ ·˘ÙÔ‡ Â›Ó·È ¿ÓÙ· Ìˉ¤Ó.
2. ÀÔÏfiÁÈÛ 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)
3. ÀÔÏfiÁÈÛ 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)
4. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·: + +4 –8 –11 +17
–5
+9
–4
–21
5. ToÔı¤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·: -1 +4 -3 +1,1 +2,4 -2,5
(ª·ÁÈο ÙÂÙÚ¿ÁˆÓ· Â›Ó·È ·˘Ù¿ ÛÙ· ÔÔ›·
Ë ÚfiÛıÂÛË ÙˆÓ ·ÚÈıÌÒÓ Î¿ı ÛÙ‹Ï˘ ‹ -2 0 +2 -0,1 +3,5 -2,4
ÁÚ·ÌÌ‹˜, ηıÒ˜ Î·È ÙˆÓ ‰È·ÁˆÓ›ˆÓ +3 -4 +1 0 -4,9 +5,9
ÙÔ˘˜, ‰›ÓÔ˘Ó ÙÔ ›‰ÈÔ ·ÎÚÈ‚Ò˜ ¿ıÚÔÈÛÌ·).
7. ÀÔÏfiÁÈÛ ٷ ·ıÚÔ›ÛÌ·Ù·:
(·) (–3,8) + (+2,8) + (–5,4) + (+8,2) ηÈ
(‚) (–3,5) + (–9,99) + (+2,5) + (–15,75) + (+20,75) + (+9,99)
8. ÀÔÏfiÁÈÛ ٷ ·ıÚÔ›ÛÌ·Ù·:
9 5 2 5 7 20
(·) (+ ) + (– ) + (+ ) + (– ) + (+ ) + (– ) ηÈ
4 4 3 3 13 13
1 5 3 1
(‚) (+ ) + (– ) + (+ ) + (– )
7 7 5 35
- 6. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:26 ™ÂÏ›‰·126
- 126 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
∞.7.4. ∞ Ê · › Ú Â Û Ë Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó
¢ƒ∞™∆∏ƒπ√∆∏∆∞
™ÙÔ Û¯‹Ì· ‚Ï¤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
M·Ú. ∞Ú. ª¿ÈÔ˜ πÔ˘Ó. πÔ˘Ï. ∞˘Á. ™ÂÙ. √ÎÙ.
ºÂ‚. ¡ÔÂÌ.
-10 I·Ó.
¢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;
£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘ÌÂ
°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 ) =
· – ‚ = · + (–‚)
= 8,5 + 6,2 = 14,7
™ÙÔ˘˜ ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜ Ë ·Ê·›ÚÂÛË ÌÂÙ·ÙÚ¤ÂÙ·È Û ÚfiÛıÂÛË Î·È ÂÔ̤ӈ˜
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, fiˆ˜ ›Û¯˘Â ̤¯ÚÈ ÙÒÚ·).
∞·ÏÔÈÊ‹ ·ÚÂÓı¤ÛˆÓ
™Â ·ÚÎÂÙ¤˜ ÂÚÈÙÒÛÂȘ ·ÚÈıÌËÙÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ ÂÌÊ·Ó›˙ÔÓÙ·È ÂÚÈÛÛfiÙÂÚÔÈ ÙÔ˘ ÂÓfi˜ ·ÚÈıÌÔ›
Ì ٷ ÚfiÛËÌ¿ ÙÔ˘˜ ̤۷ Û ·ÚÂÓı¤ÛÂȘ, ÌÚÔÛÙ¿ ·fi ÙȘ Ôԛ˜ ÌÔÚ› Ó·
˘¿Ú¯Ô˘Ó Ù· ÚfiÛËÌ· + ‹ – . °È· Ó· ··Ï›„Ô˘Ì ÙȘ ·ÚÂÓı¤ÛÂȘ ÂÚÁ·˙fiÌ·ÛÙ ˆ˜ ÂÍ‹˜:
ŸÙ·Ó ÌÈ· ·Ú¤ÓıÂÛË ¤¯ÂÈ ÌÚÔÛÙ¿ Ù˘ ÙÔ + (+5) + (-7) = +5 - 7 = -2
(‹ ‰ÂÓ ¤¯ÂÈ ÚfiÛËÌÔ), ÌÔÚԇ̠ӷ 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˘˜.
ŸÙ·Ó ÌÈ· ·Ú¤ÓıÂÛË ¤¯ÂÈ ÌÚÔÛÙ¿ Ù˘ ÙÔ –, (-5) - (-7) = -5 + 7 = +2
ÌÔÚԇ̠ӷ 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 ÌÂ
= –9,1+6,2–3,4+7,5–10+8,3
·ÓÙ›ıÂÙ· ÚfiÛËÌ·.
- 7. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·127
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 127 -
¶∞ƒ∞¢∂π°ª∞∆∞ - ∂º∞ƒª√°∂™
1. ŒÓ· ‚Ú¿‰˘ ÙÔ ıÂÚÌfiÌÂÙÚÔ ÛÙÔ Ì·ÏÎfiÓÈ ÂÓfi˜ ÛÈÙÈÔ‡ ¤‰ÂȯÓ –3Æ C Î·È Ì¤Û· ÛÙÔ
Û›ÙÈ 18ÆC. ¶fiÛË ‹Ù·Ó Ë ‰È·ÊÔÚ¿ ıÂÚÌÔÎÚ·Û›·˜;
§‡ÛË +20 +20
+18
To Úfi‚ÏËÌ· ˙ËÙ¿ÂÈ Ó· ˘ÔÏÔÁ›ÛÔ˘Ì ÙË ‰È·ÊÔÚ¿ ÙˆÓ
ıÂÚÌÔÎÚ·ÛÈÒÓ, ‰ËÏ·‰‹ ÙË ‰È·ÊÔÚ¿ (+18) – (–3) .
+10 +10
∞Ó ·Ú·ÙËÚ‹ÛÔ˘Ì ÙÔ Û¯‹Ì· ı· ‰Ô‡Ì fiÙÈ Ë ‰È·ÊÔÚ¿ +21
ıÂÚÌÔÎÚ·Û›·˜ ÌÂٷ͇ ÙÔ˘ ÂÛˆÙÂÚÈÎÔ‡ ÙÔ˘ ÛÈÙÈÔ‡ Î·È ÙÔ˘
0 0
Â͈ÙÂÚÈÎÔ‡ ÙÔ˘ ‹Ù·Ó +21ÆC.
™‡Ìʈӷ Ì ÙÔÓ ÔÚÈÛÌfi Ù˘ ·Ê·›ÚÂÛ˘ ÚËÙÒÓ ı· ¤¯Ô˘ÌÂ: –3 –3
(+18) – (–3) = (+18) + (+3) = (+21) –10 –10
ŒÓ·˜ ¤ÌÔÚÔ˜ ¯ÚˆÛÙ¿ÂÈ ÛÙÔÓ ÚÔÌËıÂ˘Ù‹ ÙÔ˘ 897,56 Q Î·È ÙÔ˘ ÔÊ›ÏÂÈ ¤Ó·˜ ÂÏ¿-
2. 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
- 8. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·128
- 128 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
1. ∆ÔÔı¤ÙËÛ ¤Ó· “x” ÛÙËÓ ·ÓÙ›ÛÙÔÈ¯Ë ı¤ÛË ™ø™∆√ §∞£√™
(·) ™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. ÀÔÏfiÁÈÛ 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).
· ‚ ·+‚ ·–‚
+3 –5
5. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î· Ì ÙÔ˘˜
ηٿÏÏËÏÔ˘˜ ·ÚÈıÌÔ‡˜: –8 +10
–2 –5
–9 +6
5 7
6. ¡· χÛÂȘ ÙȘ ÂÍÈÛÒÛÂȘ: (·) x + (–8) = –18, (‚) x + 12 = –14, (Á) x+ = ,
5
4 8
(‰) x– =2.
4
· ‚ ·–‚ ‚–·
7 3
7. ™˘ÌÏ‹ÚˆÛ ÙȘ ‰‡Ô ÙÂÏÂ˘Ù·›Â˜ ÛًϘ ÙÔ˘ ›Ó·Î·:
2H 33
∆È Û˘ÌÂÚ·›ÓÂȘ ÁÈ· ÙÔ˘˜ ·ÚÈıÌÔ‡˜ ÙˆÓ ‰‡Ô ·˘ÙÒÓ
ÛÙËÏÒÓ; –5,55 –2,45
3 –2,1
8. ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛÂˆÓ Ì ‰‡Ô ÙÚfiÔ˘˜: (·) 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
x 3,5 1,89 –3
y –1,5 4,3 –3
9. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·: z –2,3 3,11
x+y+z 0 0,22 1
x–y–z 0
- 9. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·129
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 129 -
∞.7.5. ¶ Ô Ï Ï · Ï · Û È · Û Ì fi ˜ Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó
¢ƒ∞™∆∏ƒπ√∆∏∆∞
ŒÓ·˜ ¤ÌÔÚÔ˜ ‰È·›ÛÙˆÛÂ, fiÙÈ Î¿ı Ë̤ڷ ÙÔ˘ ÙÂÏÂ˘Ù·›Ô˘
+524,5 i –26
5,4
‰Âη‹ÌÂÚÔ˘ ÙˆÓ ÂÎÙÒÛÂˆÓ ¤‚Á·˙ ΤډԘ 524,5Q.
∆Ô ÂfiÌÂÓÔ, fï˜, ‰Âη‹ÌÂÚÔ Â›¯Â ηıËÌÂÚÈÓ‹ ˙ËÌÈ¿ 265,4Q.
∂›Ó·È ÁÓˆÛÙfi, fiÙÈ ÛÙ· ÏÔÁÈÛÙÈο ‚È‚Ï›· ÙÔ Î¤Ú‰Ô˜ ηٷ¯ˆ-
ÚÂ›Ù·È ˆ˜ ıÂÙÈ΋ ÂÁÁÚ·Ê‹ Î·È Ë ˙ËÌÈ¿ ˆ˜ ·ÚÓËÙÈ΋.
¢ËÏ·‰‹, ÙÔ Û˘ÓÔÏÈÎfi ΤډԘ ÁÈ· ÙÔ ‰Âη‹ÌÂÚÔ ÙˆÓ
ÂÎÙÒÛÂˆÓ ı· Â›Ó·È (+524,5Q) (+10 Ë̤Ú˜) Î·È ÁÈ· ÙÔ
ÂfiÌÂÓÔ ‰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‡˜.
➣ ∆È ·Ú·ÙËÚ›˜ ÁÈ· ÙÔ ÚfiÛËÌÔ ÙˆÓ ·ÔÙÂÏÂÛÌ¿ÙˆÓ;
¢È·ÈÛÙÒÓÔ˘ÌÂ, ÏÔÈfiÓ, fiÙÈ:
∆Ô ÁÈÓfiÌÂÓÔ ‰‡Ô ıÂÙÈÎÒÓ ÚËÙÒÓ Â›Ó·È ıÂÙÈÎfi˜ ÚËÙfi˜
∆Ô 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
(–10) (+8) = –80 ™ËÌÂÈÒÓÔ˘Ì ÙÔ˘˜ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡˜ ‰‡Ô ·Ú·ÁfiÓÙˆÓ,
(–10) (+7) = –70 ·fi ÙÔ˘˜ ÔÔ›Ô˘˜ Ô ¤Ó·˜ ̤ÓÂÈ ÛÙ·ıÂÚfi˜, ÙÔ –10, Î·È Ô
(–10) (+6) = –60 ¿ÏÏÔ˜ ÌÂÈÒÓÂÙ·È ‰È·‰Ô¯Èο ηٿ 1 οı ÊÔÚ¿.
(–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˘
(–10) (+1) = –10 ·Ú¿ÁÔÓÙ· Ù· ÁÈÓfiÌÂÓ· Û˘Ó¯›˙Ô˘Ó Ó· ·˘Í¿ÓÔÓÙ·È Ì ÙÔÓ
(–10) 0 = 0 ›‰ÈÔ ÙÚfiÔ, Ú¤ÂÈ Ó· ÔÚ›ÛÔ˘Ì fiÙÈ:
(–10) (–1) = ; (–10) (–1) = +10 = +(10 1)
(–10) (–2) = ; (–10) (–2) = +20 = +(10 2)
(–10) (–3) = ; (–10) (–3) = +30 = +(10 3)
(–10) (–4) = ; (–10) (–4) = +40 = +(10 4)
......................... ...............................................
¢È·ÈÛÙÒÓÔ˘Ì ÂÔ̤ӈ˜ fiÙÈ:
∆Ô ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÚÓËÙÈÎÒÓ ·ÎÂÚ·›ˆÓ Â›Ó·È ıÂÙÈÎfi˜ ·Î¤Ú·ÈÔ˜
°ÂÓÈÎfiÙÂÚ·:
∆Ô ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÚÓËÙÈÎÒÓ ÚËÙÒÓ Â›Ó·È ıÂÙÈÎfi˜ ÚËÙfi˜.
- 10. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 24-11-06 12:28 ™ÂÏ›‰·130
- 130 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘ÌÂ
°È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰‡Ô ÔÌfiÛËÌÔ˘˜ ÚËÙÔ‡˜ ( +1,5 ) ( +2,2 ) = ( +3,3 )
·ÚÈıÌÔ‡˜, ÔÏÏ·Ï·ÛÈ¿˙Ô˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜
ÙÔ˘˜ Î·È ÛÙÔ ÁÈÓfiÌÂÓÔ ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌÔ «+». ( –1,5 )
+ ( –2,2 ) = ( +3,3 )
¢ËÏ·‰‹: + +=+ Î·È – –=+
°È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰‡Ô ÂÙÂÚfiÛËÌÔ˘˜ ÚËÙÔ‡˜
·ÚÈıÌÔ‡˜, ÔÏÏ·Ï·ÛÈ¿˙Ô˘Ì ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ( +1,5 ) ( –2,2 ) = ( –3,3 )
ÙÔ˘˜ Î·È ÛÙÔ ÁÈÓfiÌÂÓÔ ‚¿˙Ô˘Ì ÙÔ ÚfiÛËÌÔ « – ». ( –1,5 ) ( +2,2 ) = ( –3,3 )
¢ËÏ·‰‹: + – = – Î·È – + = –
√ ¢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‡
¶·Ú·ÙËÚԇ̠fiÙÈ: °ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:
( +1,5 ) ( –2,2 ) = –3,3 ªÔÚԇ̠ӷ ·ÏÏ¿˙Ô˘Ì ÙË ÛÂÈÚ¿ ‰‡Ô ·Ú·Áfi-
ÓÙˆÓ ÂÓfi˜ ÁÈÓÔ̤ÓÔ˘ (∞ÓÙÈÌÂÙ·ıÂÙÈ΋ ȉÈfiÙËÙ·).
( –2,2 ) ( +1,5 ) = –3,3 · ‚=‚ ·
–0,5 ( +2,2 –3,5 )= –0,5 –7,7 = +3,85 ªÔÚԇ̠ӷ ·ÓÙÈηıÈÛÙԇ̠·Ú¿ÁÔÓÙ˜ Ì ÙÔ
ÁÈÓfiÌÂÓfi ÙÔ˘˜ ‹ Ó· ·Ó·Ï‡Ô˘Ì ¤Ó· ·Ú¿ÁÔÓÙ·
( –0,5 +2,2 ) –3,5 )= –1,1 –3,5 = +3,85 Û ÁÈÓfiÌÂÓÔ (¶ÚÔÛÂÙ·ÈÚÈÛÙÈ΋ ȉÈfiÙËÙ·).
· (‚ Á) = (· ‚) Á
1 ( +1,5 ) = +1,5 1 = +1,5 ŸÙ·Ó ¤Ó·˜ ÚËÙfi˜ ÔÏÏ·Ï·ÛÈ¿˙ÂÙ·È Ì ÙÔÓ
·ÚÈıÌfi 1 ‰ÂÓ ÌÂÙ·‚¿ÏÏÂÙ·È.
1 ( –2,2 ) = –2,2 1 = –2,2
1 · = · 1= ·
0,15 (–5) + 1,85 (–5) = ∂ÈÌÂÚÈÛÙÈ΋ ȉÈfiÙËÙ· ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡
ˆ˜ ÚÔ˜ ÙËÓ ÚfiÛıÂÛË Î·È ÙËÓ ·Ê·›ÚÂÛË:
(–0,75) + (–9,25) = –10
· ( ‚ + Á ) = · ‚ + · Á Î·È · ( ‚ – Á ) = · ‚ – · Á
(0,15+1,85) (–5)=2 (–5)= –10
√È ÚËÙÔ› ·ÚÈıÌÔ› · Î·È ‚ ϤÁÔÓÙ·È ·ÓÙ›ÛÙÚÔÊÔÈ,
(+3) (+ 1 ) = +(3 1 ) = 1
3 3 fiÙ·Ó Â›Ó·È ‰È¿ÊÔÚÔÈ ÙÔ˘ ÌˉÂÓfi˜ Î·È ÙÔ ÁÈÓfiÌÂÓfi
(– 2 ) (– 3 ) = + ( 2 3 ) = 1 ÙÔ˘˜ Â›Ó·È ›ÛÔ Ì ÙË ÌÔÓ¿‰·: · ‚ = 1
3 2 3 2
(–0,25) (–4) = +(0,25 4) = 1 √ ηı¤Ó·˜ ·fi ÙÔ˘˜ · Î·È ‚ Â›Ó·È ·ÓÙ›ÛÙÚÔÊÔ˜
ÙÔ˘ ¿ÏÏÔ˘.
(–1,3) 0 = 0 ‹ 0 (+ 2 ) = 0 ŸÙ·Ó ¤Ó·˜ ÚËÙfi˜ ÔÏÏ·Ï·ÛÈ¿˙ÂÙ·È Ì ÙÔ 0
3
ÌˉÂÓ›˙ÂÙ·È . 0 · = · 0 = 0
- 11. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·131
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·:
°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 ∆Ô ÚfiÛËÌÔ +, ·Ó ÙÔ Ï‹ıÔ˜ ÙˆÓ ·ÚÓËÙÈÎÒÓ ·Ú·ÁfiÓÙˆÓ Â›Ó·È ¿ÚÙÈÔ (˙˘Áfi).
ñ ∆Ô ÚfiÛËÌÔ –, ·Ó ÙÔ Ï‹ıÔ˜ ÙˆÓ ·ÚÓËÙÈÎÒÓ ·Ú·ÁfiÓÙˆÓ Â›Ó·È ÂÚÈÙÙfi (ÌÔÓfi).
AÓ ÙÔ˘Ï¿¯ÈÛÙÔÓ ¤Ó·˜ ·Ú¿ÁÔÓÙ·˜ Â›Ó·È Ìˉ¤Ó, ÙfiÙÂ Î·È ÙÔ ÁÈÓfiÌÂÓÔ Â›Ó·È ›ÛÔ Ì Ìˉ¤Ó.
∆Ô ÛËÌÂ›Ô ÙÔ˘ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡ « » ÌÂٷ͇ ÙˆÓ ÁÚ·ÌÌ¿ÙˆÓ Î·È ÙˆÓ ·ÚÂÓı¤ÛˆÓ
·Ú·Ï›ÂÙ·È.
¶∞ƒ∞¢∂π°ª∞∆∞ - ∂º∞ƒª√°∂™
2
1. N· ˘ÔÏÔÁÈÛÙÔ‡Ó Ù· ÁÈÓfiÌÂÓ·: (·) (–1,4) 5, (‚) (+
3
) (–2,1), (Á) (–10) (–0,7)
§‡ÛË
(·) (–1,4) 5 = –(1,4 5) = –7
2 2
(‚) (+ 3 ) (–2,1 ) = –(3 2,1) = –1,4
(Á) (–10) (–0,7) = +(10 0,7) = +7
2
2. N· ˘ÔÏÔÁÈÛÙ› ÙÔ ÁÈÓfiÌÂÓÔ (–1)·, fiÙ·Ó ÙÔ · ·›ÚÓÂÈ ÙȘ ÙÈ̤˜: +3, –1,2, +
3
.
, –2.
§‡ÛË
°È· · = +3 ›ӷÈ: (–1)(+3) = –3 °È· · = –1,2 ›ӷÈ: (–1)(–1,2) = +1,2
2 2 2
°È· · =+ 3 ›ӷÈ: ( – 1 ) ( + 3 ) = – 3 °È· · = –2 ›ӷÈ: (–1)(–2) = +2
3. ¡· ‰Âȯı› fiÙÈ: (·+‚)(Á+‰) = ·Á + ·‰ + ‚Á + ‚‰
§‡ÛË
™‡Ìʈӷ Ì ÙËÓ ÂÈÌÂÚÈÛÙÈ΋ ȉÈfiÙËÙ·, ¤¯Ô˘ÌÂ:
(·+‚)(Á+‰) = (·+‚)Á+(·+‚)‰ = ·Á+‚Á+·‰+‚‰
2
4. ¡· ˘ÔÏÔÁÈÛÙ› Ë ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: (–1)(–20)(+
3
)(–3)(–0,25).
§‡ÛË
2
(–1)(–20)(+ 3 )(–3)(–0,25)= (Ï‹ıÔ˜ ·ÚÓËÙÈÎÒÓ ·Ú·ÁfiÓÙˆÓ 4)
2
= +(1 20 3 3 0,25) = +(20 2 0,25) = +(40 0,25) = +10
- 12. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·132
- 132 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
1. ¡· Û˘ÌÏËÚˆıÔ‡Ó Ù· ·Ú·Î¿Ùˆ ÎÂÓ¿:
(·) ∆Ô ÚfiÛËÌÔ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ‰‡Ô ÔÌfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· .................................. .
(‚) ∆Ô ÚfiÛËÌÔ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ‰‡Ô ÂÙÂÚfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· ............................ .
(Á) ŒÓ·˜ ÚËÙfi˜ fiÙ·Ó ÔÏÏ·Ï·ÛÈ¿˙ÂÙ·È Ì ÙÔ 1 ‰ÂÓ .......................................................... .
(‰) ∆Ô ÁÈÓfiÌÂÓÔ ‰‡Ô ·ÓÙ›ÛÙÚÔÊˆÓ ·ÚÈıÌÒÓ Â›Ó·È ¿ÓÙ· ›ÛÔ Ì ....................................... .
(Â) ∆Ô ÚfiÛËÌÔ ÁÈÓÔ̤ÓÔ˘ ÔÏÏÒÓ ·Ú·ÁfiÓÙˆÓ ÂÍ·ÚÙ¿Ù·È ·fi ÙÔ Ï‹ıÔ˜ ÙˆÓ
.......................................... ·Ú·ÁfiÓÙˆÓ.
2. ÀÔÏfiÁÈÛ ٷ ÁÈÓfiÌÂÓ·: (·) (–1)(–1), (‚) –3(–10), (Á) –1,2(–0,5), (‰) 0(–10589),
12 15
(Â) 1(–20015), (ÛÙ) –0,725(+1000), (˙) (– ).
25 24
3. ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛÂˆÓ Ì ÙȘ ÏÈÁfiÙÂÚ˜ ‰˘Ó·Ù¤˜ Ú¿ÍÂȘ:
6 6
(·) –5 27 + 2 27, (‚) 10,35(–25) + 9,65(–25), (Á) – (–10)+(– )(+3)
7 7
4. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ‰ÈÏ·Ófi ›Ó·Î·: ñ –1 –1 0 +2 +3
–2
–3,2
+G
+10
1 1 1 1 1
5. ∫¿Ó ÙȘ Ú¿ÍÂȘ: (·) –7(–8+10–5), (‚) (0,25–0,05)(– + – ), (Á)–10–6( – )
4 2 8 2 3
6. ∫¿Ó ÙȘ Ú¿ÍÂȘ: (·) (5+·)(2+‚), (‚) (·+7)(·–7), (Á) (·–3)(‚–3), (‰) (Á+8)(‰+5).
7. ÀÔÏfiÁÈÛ ٷ ÁÈÓfiÌÂÓ·: (·) (–1)(–1), (‚) (–1)(–1)(–1), (Á) (–1)(–1)(–1)(–1)
8. ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ:
∞ = (·–1)(·+1)(·–2)(·+2), fiÙ·Ó · = 3
B = ‚(‚–3)(‚+3)(‚–5)(‚+5), fiÙ·Ó ‚ = 2
° = Á(2Á–1)(3Á+1)(4Á–2)(Á+2)(Á–2), fiÙ·Ó Á=0,5
x y z ˆ ∞=xyz B=yxˆ °=x∞–µ ∞µ+°
–2 0,5 +1 –3
9. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:
– 1 +6 –4 –0,3
–2 + G 0,2 –7
- 13. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·133
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 133 -
∞.7.6. ¢ È · › Ú Â Û Ë Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó
£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘ÌÂ
°È· Ó· ‰ È · È Ú ¤ Û Ô ˘ Ì Â ‰ ‡ Ô Ú Ë Ù Ô ‡ ˜ · Ú È ı Ì Ô ‡ ˜ , ( +11,22 ) : ( +2,2 ) = ( +5,1 )
‰È·ÈÚԇ̠ÙȘ ·fiÏ˘Ù˜ ÙÈ̤˜ ÙÔ˘˜ Î·È ÛÙÔ ËÏ›ÎÔ
( –11,22 ) : ( –2,2 ) = ( +5,1 )
‚¿˙Ô˘ÌÂ:
ÙÔ ÚfiÛËÌÔ +, ·Ó Â›Ó·È ÔÌfiÛËÌÔÈ. ¢ËÏ·‰‹:
( +11,22 ) : ( –2,2 ) = ( –5,1 )
+ : + = + Î·È – : – = +
( –11,22 ) : ( +2,2 ) = ( –5,1 )
ÙÔ ÚfiÛËÌÔ –, ·Ó Â›Ó·È ÂÙÂÚfiÛËÌÔÈ. ¢ËÏ·‰‹:
+ : – = – Î·È – : + = –
·
√ ÏfiÁÔ˜ ÙÔ˘ –20 ÚÔ˜ ÙÔ 4 ›ӷÈ:
∆Ô ËÏ›ÎÔ Ù˘ ‰È·›ÚÂÛ˘ ·:‚ ‹
‚ -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:
Â͛ۈÛ˘ ‚ x=· . -7 7 7
(–7) : (–2) = –2 = 2 ‰ÈfiÙÈ (-2) 2 = (-7)
· 1 -3 1
∏ ‰È·›ÚÂÛË
‚
ÌÔÚ› Î·È Ó· ÁÚ·Ê› ·
‚
, ÂÔ̤ӈ˜ (–3) : (–4) = -4 = –3 (– 4 )
ÁÈ· Ó· ‰È·ÈÚ¤ÛÔ˘Ì ‰‡Ô ÚËÙÔ‡˜ ·ÚÈıÌÔ‡˜, ·ÚΛ Ó· 6 1
6 : (–7) = -7 = 6 (– 7 )
ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ÙÔ ‰È·ÈÚÂÙ¤Ô Ì ÙÔÓ ·ÓÙ›ÛÙÚÔÊÔ
ÙÔ˘ ‰È·ÈÚ¤ÙË. -5 1
(–5) : (+2) = 2 = –5 ( 2 )
· 1
=·
‚ ‚
¢È·›ÚÂÛË Ì ‰È·ÈÚ¤ÙË ÙÔ Ìˉ¤Ó ‰ÂÓ ÔÚ›˙ÂÙ·È.
¶∞ƒ∞¢∂π°ª∞∆∞ - ∂º∞ƒª√°∂™
¡· ˘ÔÏÔÁÈÛÙÔ‡Ó Ù· Ëϛη:
1. 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
- 14. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·134
- 134 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
2. ¡· Ï˘ıÔ‡Ó ÔÈ ÂÍÈÛÒÛÂȘ: (·) –6x = –24, (‚) –3x = +15, (Á) x : (–2) = –3
§‡ÛË
(·) –6x = –24 (‚) –3x = +15 (Á) x : (–2) = –3
x = (–24) : (–6) x = (+15) : (–3) x = (–3) (–2)
x = +(24 : 6) x = –(15 : 3) x = +(3 2)
x = +4 x = –5 x = +6
2
3. N· ‚ÚÂı› Ë ÙÈÌ‹ Ù˘ ·Ú¿ÛÙ·Û˘: [
3
(–3)–(–2)(–9)] : [0,4(–10)–(–0,2)(–5)]+7.
§‡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 = (οÓÔ˘Ì ÙȘ Ú¿ÍÂȘ ̤۷ ÛÙȘ ·ÚÂÓı¤ÛÂȘ)
= (–2–18) : (–4–1) + 7 =
= (–20) : (–5) + 7 = (οÓÔ˘Ì ÙÔ˘˜ ÔÏÏ·Ï·ÛÈ·ÛÌÔ‡˜ Î·È ÙȘ ‰È·ÈÚ¤ÛÂȘ)
= +(20 : 5) + 7 =
= +4 +7 = (οÓÔ˘Ì ÙȘ ÚÔÛı¤ÛÂȘ Î·È ÙȘ ·Ê·ÈÚ¤ÛÂȘ)
= +11
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
1. ™˘ÌÏ‹ÚˆÛ ٷ ·Ú·Î¿Ùˆ ÎÂÓ¿:
(·) ∆Ô ÚfiÛËÌÔ ÙÔ˘ ËÏ›ÎÔ˘ ‰‡Ô ÔÌfiÛËÌˆÓ ÚËÙÒÓ Â›Ó·È ¿ÓÙ· .................................. .
(‚) ∆Ô 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 .............................
ÌÂ 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)
x y x+y x–y xy x:y
-7 5
3 -6
3. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:
1,7 2,3
– K –1
10 –0,75 –120 1 2
4. YÔÏfiÁÈÛ ٷ Ëϛη: (·)
0,25
, (‚)
–0,5
, (Á)
(–12) + (–8)
, (‰) (–3 ):(–2 ).
5 3
2 4
5. §‡Û ÙȘ ÂÍÈÛÒÛÂȘ: (·) –3x = 74, (‚) –0,14x = –49, (Á) x(–2) = 12, (‰)
3
x=– .
6
–1 2 12 (–2)(–5)(–1) –7 5 3
6. K¿Ó ÙȘ Ú¿ÍÂȘ: (·)
3
+
–6
–
–15
, (‚) –
–10
, (Á) ( –
3 –3
):(– ).
2
–7 9
7. ÀÔÏfiÁÈÛ ÙËÓ ÙÈÌ‹ Ù˘ ·Ú¿ÛÙ·Û˘: [(–8)(
64
)–(–15) : (–8)](–8)+(–27) : (– ).
8
- 15. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·135
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 135 -
∞.7.7. ¢ Â Î · ‰ È Î ‹ Ì Ô Ú Ê ‹ Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó
¢ƒ∞™∆∏ƒπ√∆∏∆∞ 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 ÌÔÚÔ‡Ó Ó· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ÔÈ ‰‡Ô
ÛÔÎÔÏ¿Ù˜ ÛÙ· ÙÚ›· ·È‰È¿;
™ÎÂÊÙfiÌ·ÛÙ 5,0000... 3
∞Ó ‰ÂÓ ˘‹Ú¯Â Ô Ê›ÏÔ˜ ÙˆÓ ·È‰ÈÒÓ, ı· ¤ÙÚˆÁ ηı¤Ó· 20 1,666...
·fi Ù· ‰‡Ô ·È‰È¿ 5 : 2 =2,5 ÛÔÎÔÏ¿Ù˜. 20
∆ÒÚ·, fï˜, Ú¤ÂÈ Ó· ‰Ô‡Ì ÔÈÔ Â›Ó·È ÙÔ ·ÎÚÈ‚¤˜
20
·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 ˘fiÏÔÈÔ 2.
∞Ó Û˘Ó¯›ÛÔ˘Ì ÙË ‰È·›ÚÂÛË, ı· ¿ÚÔ˘Ì ËÏ›ÎÔ ÙÔ ‰Âη‰ÈÎfi ·ÚÈıÌfi: 1,666...
∂Âȉ‹, fï˜, ÙÔ ˘fiÏÔÈÔ Ù˘ ‰È·›ÚÂÛ˘ Â›Ó·È ÙÔ ›‰ÈÔ ¿ÓÙ·, Ù· ‰Âη‰Èο „ËÊ›·
·ӷϷ̂¿ÓÔÓÙ·È Î·È Â›Ó·È fiÏ· ›Û· Ì 6.
ÕÚ·, ‰ÂÓ ÌÔÚÔ‡Ó Ó· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ‰‡Ô ÛÔÎÔÏ¿Ù˜ Û ÙÚ›· ·È‰È¿.
¢ƒ∞™∆∏ƒπ√∆∏∆∞ 2 Ë
∂Ù¿ Ô‰ÔÛÊ·ÈÚÈΤ˜ ÔÌ¿‰Â˜ Ú¤ÂÈ Ó· ÌÔÈÚ·ÛÙÔ‡Ó ÂÍ›ÛÔ˘ ÌÈ· ÂȯÔÚ‹ÁËÛË 1.000.000 Q.
➣ µÚ˜, Ì fiÛÔ ÌÂÁ·Ï‡ÙÂÚË ·ÎÚ›‚ÂÈ· ÌÔÚ›˜, 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 . . . 7
‚Ï¤Ô˘Ì fiÙÈ Ë ‰È·›ÚÂÛË 1.000.000 : 7 3 0 142857,142857 142857
‰ÂÓ Â›Ó·È Ù¤ÏÂÈ·. ¢›ÓÂÈ ËÏ›ÎÔ 142.857 20
Î·È ˘fiÏÔÈÔ 1. ∞Ó Û˘Ó¯›ÛÔ˘Ì ÙË 60
‰È·›ÚÂÛË ı· ‚Úԇ̠ÙÔ ‰Âη‰ÈÎfi 40
·ÚÈıÌfi 142.857, 142857 142857... ÌÂ 50
¿ÂÈÚ· ‰Âη‰Èο „ËÊ›·, Ù¤ÙÔÈ· ÒÛÙÂ, 10
Ó· ·ӷϷ̂¿ÓÔÓÙ·È Û˘Ó¯Ҙ Ù· ›‰È· 30
¤ÍÈ „ËÊ›· 142857. 20
..
..
ª·ı·›ÓÔ˘ÌÂ
∆Ô˘˜ ·ÚÈıÌÔ‡˜ Ô˘ ‚ڋηÌ ·Ú·¿Óˆ ÙÔ˘˜ ÔÓÔÌ¿˙Ô˘Ì ÂÚÈÔ‰ÈÎÔ‡˜ ‰Âη‰ÈÎÔ‡˜ ·ÚÈıÌÔ‡˜.
∆Ô Ï‹ıÔ˜ ÙˆÓ Â·Ó·Ï·Ì‚·ÓÔÌ¤ÓˆÓ ‰Âη‰ÈÎÒÓ „ËÊ›ˆÓ οı ÂÚÈÔ‰ÈÎÔ‡ ·ÚÈıÌÔ‡
ÔÓÔÌ¿˙ÂÙ·È ÂÚ›Ô‰Ô˜.
°ÂÓÈÎfiÙÂÚ·, ÏÔÈfiÓ, ÌÔÚԇ̠ӷ ԇ̠fiÙÈ:
∫¿ı ÚËÙfi˜ ·ÚÈıÌfi˜ ÌÔÚ› Ó· ¤¯ÂÈ ÙË ÌÔÚÊ‹ ‰Âη‰ÈÎÔ‡ ‹ ÂÚÈÔ‰ÈÎÔ‡ ‰Âη‰ÈÎÔ‡
·ÚÈıÌÔ‡ Î·È Û˘Ì‚ÔÏ›˙ÂÙ·È fiˆ˜ Ê·›ÓÂÙ·È ÛÙ· ·Ú·‰Â›ÁÌ·Ù·.
5 1.000.000
.¯. = 1,6 Î·È = 142857,142857.
3 7
- 16. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:27 ™ÂÏ›‰·136
- 136 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
¶ÚÔËÁÔ˘Ì¤Óˆ˜, ›‰·Ì Ì ÔÈÔÓ ÙÚfiÔ ¤Ó·˜ ÚËÙfi˜ ·ÚÈıÌfi˜ ÌÔÚ› Ó· ÁÚ·Ê› Ì ÙË ÌÔÚÊ‹
ÂÚÈÔ‰ÈÎÔ‡ ‰Âη‰ÈÎÔ‡ ·ÚÈıÌÔ‡.
°ÂÓÓȤٷÈ, fï˜, ÙÔ ÂÚÒÙËÌ· ·Ó ÌÔÚԇ̠ӷ οÓÔ˘ÌÂ Î·È ÙÔ ·ÓÙ›ÛÙÚÔÊÔ. ¢ËÏ·‰‹, ·Ó
ÌÔÚԇ̠¤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˘Ì x =
0,2 Î·È ¤¯Ô˘Ì ‰È·‰Ô¯Èο: (‚) ∞Ó x =
1 ,64 ¤¯Ô˘ÌÂ
x =
0,222... x =
1,646464...
10x =
2,222... 100x =
164,646464...
10x =
2+0,222... 100x =
164+0,646464...
10x =
2+x 100x =
164+x–1
(10–1)x =
2 (100–1)x =
163
9x =
2 99x =
163
2 2 163 163
x = 9 ¢ËÏ·‰‹: 0,2= 9 x = 99 ¢ËÏ·‰‹: 1 ,64= 99
™˘ÌÂÚ·›ÓÔ˘Ì fiÙÈ:
∫¿ı ÂÚÈÔ‰ÈÎfi˜ ‰Âη‰ÈÎfi˜ ·ÚÈıÌfi˜ ÌÔÚ› Ó· ¤¯ÂÈ ÙË ÌÔÚÊ‹ ÎÏ·ÛÌ·ÙÈÎÔ‡ ÚËÙÔ‡.
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
15 5 13 20 32
1. µÚ˜ ÙË ‰Âη‰È΋ ÌÔÚÊ‹ ÙˆÓ ÚËÙÒÓ: (·) –
10
, (‚)
8
, (Á)
14
, (‰)
11
, (Â)
31
.
2. µÚ˜ ÙËÓ ÎÏ·ÛÌ·ÙÈ΋ ÌÔÚÊ‹ ÙˆÓ ·ÚÈıÌÒÓ:
(·) 57,92, (‚) 2,8, (Á) 3,83, (‰) 7,4561, (Â) 15,399.
3. BÚ˜ ÌÈ· ¿ÏÏË ‰Âη‰È΋ ÌÔÚÊ‹ ÙˆÓ ·ÚÈıÌÒÓ: (·) 2,9, (‚) 7,69, (Á) 7,3259.
¢ƒ∞™∆∏ƒπ√∆∏∆∞ °π∞ ∆√ ™¶π∆π
√ ·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· ÌÔÚ¤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 ÂȂ‚·ÈÒÛÂȘ ‹ Ó· ·ÔÚÚ›„ÂȘ ÙÔ ÏfiÁÔ ÁÈ· ÙÔÓ ÔÔ›Ô
Ô ∑‹ÓˆÓ·˜ ÈÛ¯˘Ú›˙ÂÙ·È Î¿ÙÈ Ù¤ÙÔÈÔ.
- 17. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:28 ™ÂÏ›‰·137
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 137 -
∞.7.8. ¢ ˘ Ó ¿ Ì Â È ˜ Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó Ì Â Â Î ı ¤ Ù Ë Ê ˘ Û È Î fi
¢ƒ∞™∆∏ƒπ√∆∏∆∞
ŒÓ·˜ ˘ÔÏÔÁÈÛÙ‹˜ ÌÔχÓıËΠ·fi οÔÈÔ Èfi, Ô ÔÔ›Ô˜ ›¯Â
ÙËÓ È‰ÈfiÙËÙ· Ó· ηٷÛÙÚ¤ÊÂÈ Ù· ËÏÂÎÙÚÔÓÈο ·Ú¯Â›· Ì ÙÔÓ
ÂÍ‹˜ ÙÚfiÔ:
∫¿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˘ÌÂ
™˘Ì‚ÔÏÈÛÌÔ›
Ó ·Ú¿ÁÔÓÙ˜
}
∆Ô ÁÈÓfiÌÂÓÔ · · · ... · (›ÙÂ Ô · Â›Ó·È ıÂÙÈÎfi˜ ÂÎı¤Ù˘
›Ù ·ÚÓËÙÈÎfi˜ ÚËÙfi˜), Û˘Ì‚ÔÏ›˙ÂÙ·È Ì ÙÔ ·Ó Î·È Ï¤ÁÂÙ·È
·Ó = · · · ... ·
‰‡Ó·ÌË Ì ‚¿ÛË ÙÔ · Î·È ÂÎı¤ÙË ÙÔ Ê˘ÛÈÎfi Ó>1.
}
‚¿ÛË Ó ·Ú¿ÁÔÓÙ˜
°È· Ó = 1, ÁÚ¿ÊÔ˘Ì ·1 =·
∏ ‰‡Ó·ÌË ·Ó ‰È·‚¿˙ÂÙ·È Î·È ÓÈÔÛÙ‹ ‰‡Ó·ÌË ÙÔ˘ ·.
∏ ‰‡Ó·ÌË ·2 ϤÁÂÙ·È Î·È ÙÂÙÚ¿ÁˆÓÔ ÙÔ˘ · ‹ · ÛÙÔ ÙÂÙÚ¿ÁˆÓÔ.
∏ ‰‡Ó·ÌË ·3 ϤÁÂÙ·È Î‡‚Ô˜ ÙÔ˘ · ‹ · ÛÙÔÓ Î‡‚Ô.
¶ÚfiÛËÌÔ ‰‡Ó·Ì˘
¶·Ú·ÙËÚԇ̠fiÙÈ: °ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:
5 ¢‡Ó·ÌË Ì ‚¿ÛË ıÂÙÈÎfi ·ÚÈıÌfi Â›Ó·È ıÂÙÈÎfi˜
(+2) = (+2)(+2)(+2)(+2)(+2) = +32 > 0 ·ÚÈıÌfi˜.
∞Ó · > 0, ÙfiÙÂ ·Ó > 0
¿ÚÙÈÔ Ï‹ıÔ˜
¢‡Ó·ÌË Ì ‚¿ÛË ·ÚÓËÙÈÎfi ·ÚÈıÌfi Î·È ÂÎı¤ÙË
}
¿ÚÙÈÔ Â›Ó·È ıÂÙÈÎfi˜ ·ÚÈıÌfi˜.
4
(–2) = (–2)(–2)(–2)(–2) = +16 > 0 ∞Ó · < 0 Î·È Ó ¿ÚÙÈÔ˜, ÙfiÙ ·Ó > 0
¢‡Ó·ÌË Ì ‚¿ÛË ·ÚÓËÙÈÎfi ·ÚÈıÌfi Î·È ÂÎı¤ÙË
ÂÚÈÙÙfi Ï‹ıÔ˜ ÂÚÈÙÙfi Â›Ó·È ·ÚÓËÙÈÎfi˜ ·ÚÈıÌfi˜.
}
5
(–2) = (–2)(–2)(–2)(–2)(–2) = –32 < 0 ∞Ó · < 0 Î·È Ó ÂÚÈÙÙfi˜, ÙfiÙ ·Ó < 0
- 18. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:28 ™ÂÏ›‰·138
- 138 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
π‰ÈfiÙËÙ˜ ‰˘Ó¿ÌÂˆÓ ÚËÙÒÓ Ì ÂÎı¤ÙË Ê˘ÛÈÎfi
¶·Ú·ÙËÚԇ̠fiÙÈ: °ÂÓÈο ÈÛ¯‡ÂÈ fiÙÈ:
°È· Ó· ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì ‰˘Ó¿ÌÂȘ
(–3) 3 (–3) 5 =
3 ·Ú¿ÁÔÓÙ˜ 5 ·Ú¿ÁÔÓÙ˜ Ì ÙËÓ ›‰È· ‚¿ÛË, ·Ê‹ÓÔ˘Ì ÙËÓ ›‰È·
‚¿ÛË Î·È ‚¿˙Ô˘Ì ÂÎı¤ÙË ÙÔ
= (–3)(–3)(–3)(–3)(–3)(–3)(–3)(–3) =
¿ıÚÔÈÛÌ· ÙˆÓ ÂÎıÂÙÒÓ.
8 ·Ú¿ÁÔÓÙ˜
= (–3) = (–3) 3+5
8 ·Ì ·Ó = ·Ì+Ó
°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˘
78 7 7 7 7 7 7 7 7
78 : 73 = = 7 7 7 = ÂÎı¤ÙË ÙÔ˘ ‰È·ÈÚ¤ÙË ·fi ÙÔÓ ÂÎı¤ÙË
73
ÙÔ˘ ‰È·ÈÚÂÙ¤Ô˘.
= 7 7 7 7 7 = 7 5 = 7 8–3
·Ì : ·Ó = ·Ì–Ó
6 °È· Ó· ˘„ÒÛÔ˘Ì ¤Ó· ÁÈÓfiÌÂÓÔ ÛÂ
(2 7) = (2 7)(2 7)(2 7)(2 7)(2 7)(2 7)
ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì οı ·Ú¿ÁÔÓÙ·
=(2 2 2 2 2 2) (7 7 7 7 7 7)= ÙÔ˘ ÁÈÓÔ̤ÓÔ˘ ÛÙÔÓ ÂÎı¤ÙË ·˘Ùfi.
= 26 76
(· ‚)Ó = ·Ó ‚Ó
°È· Ó· ˘„ÒÛÔ˘Ì ¤Ó· ËÏ›ÎÔ Û ¤Ó·Ó
5
(2) 2 2 2 2 2 ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ηı¤Ó· ·fi ÙÔ˘˜
9 = 9 9 9 9 9 = fiÚÔ˘˜ ÙÔ˘ ËÏ›ÎÔ˘ ÛÙÔÓ ÂÎı¤ÙË ·˘Ùfi.
2 2 2 2 2 25 Ó
= 9 9 9 9 9 =
95 ( · )Ó= ·
‚ ‚ Ó
°È· Ó· ˘„ÒÛÔ˘Ì ̛· ‰‡Ó·ÌË Û ¤Ó·Ó
(8 3 ) 7 = 83 83 83 83 83 83 83 = ÂÎı¤ÙË, ˘„ÒÓÔ˘Ì ÙË ‚¿ÛË Ù˘
= 8 3+3+3+3+3+3+3 = ‰‡Ó·Ì˘ ÛÙÔ ÁÈÓfiÌÂÓÔ ÙˆÓ ÂÎıÂÙÒÓ.
= 87 3 = 8 21 (·Ì)Ó = ·ÌÓ
- 19. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 10-11-06 15:01 ™ÂÏ›‰·139
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 139 -
¶∞ƒ∞¢∂π°ª∞∆∞ - ∂º∞ƒª√°∂™
1. N· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: (·) –33, (‚) (–3)3, (Á) –34 ,(‰) (–3)4.
∆È ·Ú·ÙËÚ›ÙÂ;
§‡ÛË
(·) ∏ ·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.
(Á) ∏ ·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
2. 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
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
1. ™˘ÌÏ‹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.
2. µÚ˜ Ì ÔÈÔ ÛÙÔÈ¯Â›Ô Ù˘ 2˘ Î·È Ù˘ 3˘ ÁÚ·ÌÌ‹˜ ·ÓÙ›ÛÙÔȯ· Â›Ó·È ›ÛÔ Î¿ı ÛÙÔȯ›Ô
Ù˘ 1˘ ÁÚ·ÌÌ‹˜ ÙÔ˘ ·Ú·Î¿Ùˆ ›Ó·Î·.
32 3 2
3 + 52 (3 + 5)2 3 52 (3 5)2 3 – 52 (3 – 5)2 5
5
¢È·ÊÔÚ¿ ÕıÚÔÈÛÌ· °ÈÓfiÌÂÓÔ ¶ËÏ›ÎÔ ∆ÂÙÚ¿ÁˆÓÔ ∆ÂÙÚ¿ÁˆÓÔ ∆ÂÙÚ¿ÁˆÓÔ ∆ÂÙÚ¿ÁˆÓÔ
ÙˆÓ 3 Î·È 5 2 ÙˆÓ 3 Î·È 5 2 ÙˆÓ 3 Î·È 5 2 ÙˆÓ 3 2 Î·È 5 Ù˘ ‰È·ÊÔÚ¿˜ ÙÔ˘ ËÏ›ÎÔ˘ ÙÔ˘ ·ıÚÔ›ÛÌ·ÙÔ˜ ÙÔ˘ ÁÈÓÔ̤ÓÔ˘
3 ÏËÓ 5 3 ‰È· 5 3 Î·È 5 3 › 5
75 4 28 64 0,36 225 1,8 –22
3. YÔÏfiÁÈÛ ÙȘ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ: ∞ = (–1)1 +(–1)2 +(–1)3 + (–1)4 + (–1)5,
(– 6)5 84 103
µ = 32 54 – 25 45 + 87,5 43, °=– – +
35 (–4)4 (–5)3
- 20. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:28 ™ÂÏ›‰·140
- 140 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
∞.7.9. ¢ ˘ Ó ¿ Ì Â È ˜ Ú Ë Ù Ò Ó · Ú È ı Ì Ò Ó Ì Â Â Î ı ¤ Ù Ë · Î ¤ Ú · È Ô
£˘ÌfiÌ·ÛÙ - ª·ı·›ÓÔ˘ÌÂ
™‡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 ÂÔ̤ӈ˜, 50 = 1. °ÂÓÈο ÈÛ¯‡ÂÈ:
57 5
∏ ‰‡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:
57 57 1 1
7 = 57–8 = 5–1, ÁÓˆÚ›˙Ô˘Ì fiÙÈ Â›Ó·È Î·È 8 = 5 5 5 5 5 5 5 = ,¿Ú· 5–1 =
5 5 5 5 5 5 5 5 5 5 5 5
56 56 1 1
8 = 56–8 = 5–2, ÁÓˆÚ›˙Ô˘Ì fiÙÈ Â›Ó·È Î·È 8 = 5 5 5 5 5 5 = 2 ,¿Ú· 5–2 = 2
5 5 5 5 5 5 5 5 5 5 5 5
Î.Ô.Î. ™˘ÌÂÚ·›ÓÔ˘ÌÂ, ÏÔÈfiÓ fiÙÈ:
∏ ‰‡Ó·ÌË Î¿ı ·ÚÈıÌÔ‡, ‰È¿ÊÔÚÔ˘ ÙÔ˘ ÌˉÂÓfi˜,
1 1 Ó
Ì ÂÎı¤ÙË ·ÚÓËÙÈÎfi Â›Ó·È ›ÛË Ì ÎÏ¿ÛÌ· Ô˘ ¤¯ÂÈ ·–Ó = =( )
·Ó ·
·ÚÈıÌËÙ‹ ÙË ÌÔÓ¿‰· Î·È ·ÚÔÓÔÌ·ÛÙ‹ ÙË ‰‡Ó·ÌË
ÙÔ˘ ·ÚÈıÌÔ‡ ·˘ÙÔ‡ ÌÂ ·ÓÙ›ıÂÙÔ ÂÎı¤ÙË .
· ‚
∂Âȉ‹ Ù· Î·È Â›Ó·È ·ÓÙ›ÛÙÚÔÊÔÈ ·ÚÈıÌÔ›,
‚ ·
1
fiˆ˜ Î·È Ù· · Î·È ÛÙËÓ ÚÔËÁÔ‡ÌÂÓË Û¯¤ÛË,
·
ÂÍ¿ÁÔ˘Ì ÙÔ Û˘Ì¤Ú·ÛÌ· fiÙÈ ÈÛ¯‡ÂÈ:
( · )–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.
- 21. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:28 ™ÂÏ›‰·141
ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ› - 141 -
¶∞ƒ∞¢∂π°ª∞∆∞ - ∂º∞ƒª√°E™
1. ¡· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ‰˘Ó¿ÌÂȘ: (·) (–2)–5, (‚) –3–3, (Á) (–234567)0.
§‡ÛË
1 1 1 1 1
(·) (–2) –5 = –3 0
5 = –32 =– 32 , (‚) –3 =– 33 =– 27 , (Á) (–234567) =1
(–2)
2. ¡· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ:
12-3
(·) [(–3)3]2, (‚) 33 : 3–2, (Á) (–2)4 (–2)6, (‰) .
3-3
§‡ÛË
(·) [ ( – 3 ) 3] 2 = ( – 3 ) 3 2 = (–3)6 = 729
(‚) 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 –3 3
12–3 12 4 1 1 1
(‰) = ( ) = ( ) = ( ) = =
3–3 3 1 4 43 64
3. ¡· ˘ÔÏÔÁÈÛÙÔ‡Ó ÔÈ ‰˘Ó¿ÌÂȘ: 10–1, 10–2, 10–3, 10–4, 10–5, 10–6, 10–7.
§‡ÛË
1 1
10 –1 = = =0,1
101 10
1 1
10 –2 = = =0,01
102 100
1 1
10 –3 = = =0,001
103 1000
1 1
10 –4 = = =0,0001
104 10000
1 1
10 –5 = 5 = 100000 =0,00001
10
1 1
10 –6 = 6 = 1000000 =0,000001
10
1 1
10 –7 = 7 = 10000000 =0,0000001
10
- 22. ∂º∞ §√-7(113-146)-(20,5 à 28)
∫ π∞ 1-11-06 22:28 ™ÂÏ›‰·142
- 142 - ª¤ÚÔ˜ ∞ - KÂÊ¿Ï·ÈÔ 7Ô - £ÂÙÈÎÔ› Î·È ∞ÚÓËÙÈÎÔ› ∞ÚÈıÌÔ›
∞™∫∏™∂π™ ∫∞𠶃√µ§∏ª∞∆∞
1. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:
· ‚ Á (·+‚)2 (·‚)2
2
(‚ )
·
(–·)–2 (Á‚)–1
1 –2 –4
–1 –1 G
10 –10 0,01
2. ÀÔÏfiÁÈÛ ÙȘ ÙÈ̤˜ ÙˆÓ ·Ú·ÛÙ¿ÛˆÓ:
∞ = (–1)–3+(–1)–2+(–1)–1+(–1)0+(–1)1+(–1)2
(–6)–5 16–4 5–3
µ = [(–2)2]5[(–3)2]–2+[(–23,5)2(23,5)–2]5, °= –5
+ –4
–
12 (–32) (–10)–3
1 1
3. µÚ˜ ÔÈÔ˜ ·fi ÙÔ˘˜ ·ÚÈıÌÔ‡˜:
10
,103 5 2, 3 ,103+102, ‰ÂÓ Â›Ó·È ‰‡Ó·ÌË ÙÔ˘ 10.
10
4. ™˘ÌÏ‹ÚˆÛ ÙÔÓ ›Ó·Î·:
x 0,001 0,01 0,1 –10 –100 2 104 5 10–3 1 G –4
x–3
x3
x–1
5. ™˘ÌÏ‹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
