Spis treści
Mnożenie to operacja matematyczna, którą można przedstawić jako sumę identycznych terminów.
Ogólna zasada mnożenia
Na przykład, a b (czytane jako „a razy b”) oznacza, że sumujemy wyrazy a, których liczba jest równa b. Wynik mnożenia nazywamy iloczynem.
przykłady:
- 2 ⋅ 6 = 2 + 2 + 2 + 2 + 2 + 2 = 12
(sześć razy dwa)
- 5 4 = 5 + 5 + 5 + 5 = 20
(cztery razy pięć)
- 3 ⋅ 8 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 24
(osiem razy trzy)
Jak wiemy, z permutacji miejsc czynników iloczyn się nie zmienia. Dla powyższych przykładów okazuje się:
- 6 2 = 6 + 6 = 12
(dwa razy sześć)
- 4 ⋅ 5 = 4 + 4 + 4 + 4 + 4 = 20
(pięć razy cztery)
- 8 3 = 8 + 8 + 8 = 24
(trzy razy osiem)
Praktyczne korzyści
Dzięki mnożeniu można znacznie zmniejszyć łączną liczbę sztuk tego samego typu itp. Np. jeśli mamy 7 opakowań, z których każda zawiera 5 pisaków, to łączną liczbę pisaków obliczamy mnożąc te dwie liczby:
5 ⋅ 7 = 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35
(pięć długopisów siedem razy)
Pomnóż przez 0
Wynik jest zawsze zerowy.
- 0 0 = 0
- 1 0 = 0 ⋅ 1 = 0
- 2 0 = 0 ⋅ 2 = 0 + 0 = 0
- 3 0 = 0 ⋅ 3 = 0 + 0 + 0 = 0
- 4 ⋅ 0 = 0 ⋅ 4 = 0 + 0 + 0 + 0 = 0
- 5 ⋅ 0 = 0 ⋅ 5 = 0 + 0 + 0 + 0 + 0 = 0
- 6 ⋅ 0 = 0 ⋅ 6 = 0 + 0 + 0 + 0 + 0 + 0 = 0
- 7 ⋅ 0 = 0 ⋅ 7 = 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0
- 8 ⋅ 0 = 0 ⋅ 8 = 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0
- 9 ⋅ 0 = 0 ⋅ 9 = 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0
- 10 ⋅ 0 = 0 ⋅ 10 = 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 0
Pomnóż przez 1
Iloczyn jest równy innemu mnożnikowi innemu niż jeden.
- 1 1 = 1
- 2 1 = 2 ⋅ 1 = 2
- 3 1 = 3 ⋅ 1 = 3
- 4 1 = 4 ⋅ 1 = 4
- 5 1 = 5 ⋅ 1 = 5
- 6 1 = 6 ⋅ 1 = 6
- 7 1 = 7 ⋅ 1 = 7
- 8 1 = 8 ⋅ 1 = 8
- 9 1 = 9 ⋅ 1 = 9
- 10 1 = 10 ⋅ 1 = 10
Pomnóż przez 2
Dodaj do siebie pierwszy czynnik.
- 1 2 = 1 + 1 = 2
- 2 2 = 2 + 2 = 4
- 3 2 = 3 + 3 = 6
- 4 2 = 4 + 4 = 8
- 5 2 = 5 + 5 = 10
- 6 2 = 6 + 6 = 12
- 7 2 = 7 + 7 = 14
- 8 2 = 8 + 8 = 16
- 9 2 = 9 + 9 = 18
- 10 2 = 10 + 10 = 20
Pomnóż przez 3
Pierwszy czynnik mnożymy przez 2, a następnie dodajemy go do wyniku.
- 1 3 = (1 ⋅ 2) + 1 = 2 + 1 = 3
- 2 3 = (2 ⋅ 2) + 2 = 4 + 2 = 6
- 3 3 = (3 ⋅ 2) + 3 = 6 + 3 = 9
- 4 3 = (4 ⋅ 2) + 4 = 8 + 4 = 12
- 5 3 = (5 ⋅ 2) + 5 = 10 + 5 = 15
- 6 3 = (6 ⋅ 2) + 6 = 12 + 6 = 18
- 7 3 = (7 ⋅ 2) + 7 = 14 + 7 = 21
- 8 3 = (8 ⋅ 2) + 8 = 16 + 8 = 24
- 9 3 = (9 ⋅ 2) + 9 = 18 + 9 = 27
- 10 3 = (10 ⋅ 2) + 10 = 20 + 10 = 30
Pomnóż przez 4
Dodajemy tę samą kwotę do podwojonego pierwszego czynnika.
- 1 4 = (1 ⋅ 2) + (1 ⋅ 2) = 2 + 2 = 4
- 2 4 = (2 ⋅ 2) + (2 ⋅ 2) = 4 + 4 = 8
- 3 4 = (3 ⋅ 2) + (3 ⋅ 2) = 6 + 6 = 12
- 4 4 = (4 ⋅ 2) + (4 ⋅ 2) = 8 + 8 = 16
- 5 4 = (5 ⋅ 2) + (5 ⋅ 2) = 10 + 10 = 20
- 6 4 = (6 ⋅ 2) + (6 ⋅ 2) = 12 + 12 = 24
- 7 4 = (7 ⋅ 2) + (7 ⋅ 2) = 14 + 14 = 28
- 8 4 = (8 ⋅ 2) + (8 ⋅ 2) = 16 + 16 = 32
- 9 4 = (9 ⋅ 2) + (9 ⋅ 2) = 18 + 18 = 36
- 10 4 = (10 ⋅ 2) + (10 ⋅ 2) = 20 + 20 = 40
Pomnóż przez 5
Jeśli drugi mnożnik jest liczbą parzystą, wynik zakończy się zerem, jeśli nieparzysty, w liczbie 5.
- 1 5 = 5 ⋅ 1 = 5
- 2 5 = 5 ⋅ 2 = 5 + 5 = 10
- 3 5 = 5 ⋅ 3 = (5 ⋅ 2) + 5 = 15
- 4 5 = 5 ⋅ 4 = (5 ⋅ 2) + (5 ⋅ 2) = 20
- 5 ⋅ 5 = 5 + 5 + 5 + 5 + 5 = 25
- 6 5 = 5 ⋅ 6 = (5 ⋅ 5) + 5 = 30
- 7 ⋅ 5 = 5 ⋅ 7 = 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35
- 8 5 = 5 ⋅ 8 = (5 ⋅ 4) + (5 ⋅ 4) = 40
- 9 ⋅ 5 = 5 ⋅ 9 = (5 ⋅ 10) – 5 = 45
- 10 5 = 5 ⋅ 10 = 50
Pomnóż przez 6
Mnożymy pierwszy czynnik przez 5, a następnie dodajemy do niego wynik.
- 1 6 = (1 ⋅ 5) + 1 = 5 + 1 = 6
- 2 6 = (2 ⋅ 5) + 2 = 10 + 2 = 12
- 3 6 = (3 ⋅ 5) + 3 = 15 + 3 = 18
- 4 6 = (4 ⋅ 5) + 4 = 20 + 4 = 24
- 5 6 = (5 ⋅ 5) + 5 = 25 + 5 = 30
- 6 6 = (6 ⋅ 5) + 6 = 30 + 6 = 36
- 7 6 = (7 ⋅ 5) + 7 = 35 + 7 = 42
- 8 6 = (8 ⋅ 5) + 8 = 40 + 8 = 48
- 9 6 = (9 ⋅ 5) + 9 = 45 + 9 = 54
- 10 6 = (10 ⋅ 5) + 10 = 50 + 10 = 60
Pomnóż przez 7
Nie ma uproszczonego algorytmu mnożenia przez 7, więc stosujemy metody mające zastosowanie do innych czynników.
- 1 7 = 7 ⋅ 1 = 7
- 2 7 = 7 ⋅ 2 = 7 + 7 = 14
- 3 7 = 7 ⋅ 3 = (7 ⋅ 2) + 7 = 21
- 4 7 = 7 ⋅ 4 = (7 ⋅ 2) + (7 ⋅ 2) = 28
- 5 ⋅ 7 = 7 ⋅ 5 = 7 + 7 + 7 + 7 + 7 = 35
- 6 7 = 7 ⋅ 6 = (7 ⋅ 5) + 7 = 42
- 7 ⋅ 7 = 7 + 7 + 7 + 7 + 7 + 7 + 7 = 49
- 8 7 = 7 ⋅ 8 = (7 ⋅ 4) + (7 ⋅ 4) = 56
- 9 ⋅ 7 = 7 ⋅ 9 = (7 ⋅ 10) – 7 = 63
- 10 7 = 70
Pomnóż przez 8
Pierwszy czynnik mnożymy przez 4, a następnie dodajemy tę samą kwotę do wyniku.
- 1 8 = (1 4) + (1 ⋅ 4) = 8
- 2 8 = (2 4) + (2 ⋅ 4) = 16
- 3 8 = (3 4) + (3 ⋅ 4) = 24
- 4 8 = (4 4) + (4 ⋅ 4) = 32
- 5 8 = (5 4) + (5 ⋅ 4) = 40
- 6 8 = (6 4) + (6 ⋅ 4) = 48
- 7 8 = (7 4) + (7 ⋅ 4) = 56
- 8 8 = (8 4) + (8 ⋅ 4) = 64
- 9 8 = (9 4) + (9 ⋅ 4) = 72
- 10 8 = (10 4) + (10 ⋅ 4) = 80
Pomnóż przez 9
Pierwszy czynnik mnożymy przez 10, a następnie odejmujemy go od otrzymanego wyniku.
- 1 9 = (1 ⋅ 10) – 1 = 10 – 1 = 9
- 2 9 = (2 ⋅ 10) – 2 = 20 – 2 = 18
- 3 9 = (3 ⋅ 10) – 3 = 30 – 3 = 27
- 4 9 = (4 ⋅ 10) – 4 = 40 – 4 = 36
- 5 9 = (5 ⋅ 10) – 5 = 50 – 5 = 45
- 6 9 = (6 ⋅ 10) – 6 = 60 – 6 = 54
- 7 9 = (7 ⋅ 10) – 7 = 70 – 7 = 63
- 8 9 = (8 ⋅ 10) – 8 = 80 – 8 = 72
- 9 9 = (9 ⋅ 10) – 9 = 90 – 9 = 81
- 10 9 = (10 ⋅ 10) – 10 = 100 – 10 = 90
Pomnóż przez 10
Dodaj zero na końcu drugiego mnożnika.
- 1 10 = 10 ⋅ 1 = 10
- 2 10 = 10 ⋅ 2 = 20
- 3 10 = 10 ⋅ 3 = 30
- 4 10 = 10 ⋅ 4 = 40
- 5 10 = 10 ⋅ 5 = 50
- 6 10 = 10 ⋅ 6 = 60
- 7 10 = 10 ⋅ 7 = 70
- 8 10 = 10 ⋅ 8 = 80
- 9 10 = 10 ⋅ 9 = 90
- 10 10 = 10 ⋅ 10 = 100