Multiplication table
A multiplication table is used to define a 'multiplication' operation for an algebraic system. Multiplication tables as they are used to teach schoolchildren multiplication are a grid where rows and columns are headed by the numbers to multiply, and the entry in each cell is the product of the column and row headings:| 9×9=81 | 9×8=72 | 9×7=63 | 9×6=54 | 9×5=45 | 9×4=36 | 9×3=27 | 9×2=18 |
| 8×8=64 | 8×7=56 | 8×6=48 | 8×5=40 | 8×4=32 | 8×3=24 | 8×2=16 | |
| 7×7=49 | 7×6=42 | 7×5=35 | 7×4=28 | 7×3=21 | 7×2=14 | ||
| 6×6=36 | 6×5=30 | 6×4=24 | 6×3=18 | 6×2=12 | |||
| 5×5=25 | 5×4=20 | 5×3=15 | 5×2=10 | ||||
| 4×4=16 | 4×3=12 | 4×2=8 | |||||
| 3×3=9 | 3×2=6 | ||||||
| 2×2=4 | |||||||
This table doesn't give the ones and zeros. That is because:
- Anything times zero is zero.
- Anything times one is itself. For example, 5×1=5.
Adding a number to itself is the same as multiplying it by two. For example, 7+7=14, which is the same as 7×2.
Multiplication tables can define 'multiplication' operations for groups, fields, rings, and other algebraic systems.
The following table is an example of a multiplication table for the unit octonions (see octonion, from which this example is taken).
| · | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| 1 | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| e1 | e1 | -1 | e4 | e7 | -e2 | e6 | -e5 | -e3 |
| e2 | e2 | -e4 | -1 | e5 | e1 | -e3 | e7 | -e6 |
| e3 | e3 | -e7 | -e5 | -1 | e6 | e2 | -e4 | e1 |
| e4 | e4 | e2 | -e1 | -e6 | -1 | e7 | e3 | -e5 |
| e5 | e5 | -e6 | e3 | -e2 | -e7 | -1 | e1 | e4 |
| e6 | e6 | e5 | -e7 | e4 | -e3 | -e1 | -1 | e2 |
| e7 | e7 | e3 | e6 | -e1 | e5 | -e4 | -e2 | -1 |
Shall smile on your decay--oh, ask not me
I cannot die as ye must--over me
Shall be my paths henceforth, and so--farewell!'-- _240
25.
Sparkled beneath the shower of her bright tears,
Flung to the cavern-roof inconstant spheres
Of sobbing voices came upon her ears
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26.
Spelling out scrolls of dread antiquity, _250
Or broidering the pictured poesy
Which the sweet splendour of her smiles could dye
Added some grace to the wrought poesy.
27.
Of sandal wood, rare gums, and cinnamon;
Each flame.html">flame of it is as a precious stone _260
Belongs to each and all who gaze upon.
She held a woof that dimmed the burning brand.
28.
All night within the fountain--as in sleep.
Through the green splendour of the water deep
Like fire-flies--and withal did ever keep _270
With open eyes, closed feet, and folded palm.
29.
From the white pinnacles of that cold hill,
Where in a lawn of flowering asphodel
There yawned an inextinguishable well
And overflowing all the margin trim. _280
30.
Of wintry winds shook that innocuous liquor
O'er woods and lawns;--the serpent heard it flicker
And when the windless snow descended thicker
Melt on the surface of the level flame.
31.
For Venus, as the chariot of her star; _290
With all the ardours in that sphere which are,
And gave it to this daughter: from a car
Which ever upon mortal stream did float.
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