Multiplication

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Multiplication with an integer multiplier $\scriptstyle n \,$ is repetitive addition (a 2^nd iteration "hyper-addition"): a given number $\scriptstyle m \,$ is repeatedly added a number of times $\scriptstyle n \,$ ; this can be notated $\scriptstyle m \times n\,$ or $\scriptstyle m \cdot n \,$ (or even $\scriptstyle mn \,$ ) and read " $\scriptstyle m \,$ times $\scriptstyle n \,$ ." For example, $\scriptstyle 7 \times 4 \,=\, 7 + 7 + 7 + 7 \,=\, 28 \,$ . In most computer programming languages, and in TeX source, the asterisk character * is used as the multiplication operator: m*n. Like addition (but unlike exponentiation, tetration, pentation, ...), multiplication is commutative^[1]. Thus, $\scriptstyle 4 \times 7 \,=\, 4 + 4 + 4 + 4 + 4 + 4 + 4 \,=\, 28 \,$ .

Multiplication table

Cf. Multiplication table.

Iterated multiplication

Iterated multiplication can be abbreviated by the use of the product operator (denoted with the capital letter pi of the Greek alphabet), i.e.

$\prod_{i = 1}^n a_i \equiv a_1 \cdot a_2 \cdot a_3 \cdot \ldots \cdot a_n. \,$

Multiplicative identity

The multiplicative identity is 1: any number multiplied by 1 remains the same. For example, –43 × 1 = –43, 0 × 1 = 0, 3.7 × 1 = 3.7, etc.

Multiplicative inverse

The multiplicative inverse (denoted $\scriptstyle /n \,$ ) of $\scriptstyle n \,$ is defined by

$(/n) \cdot n = (1/n) \cdot n = 1 \,$

Division is multiplication with multiplicative inverse of second term (the divisor,) which by definition makes it non-commutative.

Generating function

The generating function of multiples sequences

$M_m(n) \equiv mn,\ n \ge 0\,$

is given by

$G_{\{M_m(n)\}}(x) = G_{\{mn\}}(x) = m\ G_{\{n\}}(x) = \frac{mxA_1(x)}{(1-x)^2} = \frac{mx}{(1-x)^2},\,$

where $\scriptstyle A_1(x)\,=\,1\,$ is an Eulerian polynomial.

Table of sequences of nonnegative multiples of nonnegative integers

A sequence of integers $\scriptstyle \{m \times i\}_{i=0}^{\infty}$ is called "the multiples of $\scriptstyle m\,$ ." Some sequences of multiples in the OEIS are:

**Nonnegative multiples of nonnegative integers**
$m\,$	OEIS number	$mn,\ n \ge 0\,$ sequences
$0\,$	A000004( $\scriptstyle n \,$ )	{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}
$1\,$	A001477( $\scriptstyle n \,$ )	{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, ...}
$2\,$	A005843( $\scriptstyle n \,$ )	{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, ...}
$3\,$	A008585( $\scriptstyle n \,$ )	{0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, ...}
$4\,$	A008586( $\scriptstyle n \,$ )	{0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, ...}
$5\,$	A008587( $\scriptstyle n \,$ )	{0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, ...}
$6\,$	A008588( $\scriptstyle n \,$ )	{0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, ...}
$7\,$	A008589( $\scriptstyle n \,$ )	{0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, ...}
$8\,$	A008590( $\scriptstyle n \,$ )	{0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, ...}
$9\,$	A008591( $\scriptstyle n \,$ )	{0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180, 189, 198, 207, 216, 225, 234, 243, 252, 261, 270, 279, 288, 297, 306, 315, ...}
$10\,$	A008592( $\scriptstyle n \,$ )	{0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 340, ...}
$11\,$	A008593( $\scriptstyle n \,$ )	{0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297, 308, 319, 330, 341, 352, 363, 374, ...}
$12\,$	A008594( $\scriptstyle n \,$ )	{0, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, ...}
$13\,$	A008595( $\scriptstyle n \,$ )	{0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, ...}
$14\,$	A008596( $\scriptstyle n \,$ )	{0, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 378, 392, 406, 420, 434, 448, 462, 476, ...}
$15\,$	A008597( $\scriptstyle n \,$ )	{0, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360, 375, 390, 405, 420, 435, 450, 465, 480, 495, ...}
$16\,$	A008598( $\scriptstyle n \,$ )	{0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 528, ...}
$17\,$	A008599( $\scriptstyle n \,$ )	{0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459, 476, 493, 510, 527, 544, 561, ...}
$18\,$	A008600( $\scriptstyle n \,$ )	{0, 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270, 288, 306, 324, 342, 360, 378, 396, 414, 432, 450, 468, 486, 504, 522, 540, 558, 576, 594, ...}
$19\,$	A008601( $\scriptstyle n \,$ )	{0, 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342, 361, 380, 399, 418, 437, 456, 475, 494, 513, 532, 551, 570, 589, 608, 627, ...}
$20\,$	A008602( $\scriptstyle n \,$ )	{0, 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260, 280, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 500, 520, 540, 560, 580, 600, 620, 640, 660, ...}
$21\,$	A008603( $\scriptstyle n \,$ )	{0, 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, 231, 252, 273, 294, 315, 336, 357, 378, 399, 420, 441, 462, 483, 504, 525, 546, 567, 588, 609, 630, 651, 672, 693, ...}
$22\,$	A008604( $\scriptstyle n \,$ )	{0, 22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 242, 264, 286, 308, 330, 352, 374, 396, 418, 440, 462, 484, 506, 528, 550, 572, 594, 616, 638, 660, 682, 704, 726, ...}
$23\,$	A008605( $\scriptstyle n \,$ )	{0, 23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 529, 552, 575, 598, 621, 644, 667, 690, 713, 736, 759, ...}
$24\,$	A008606( $\scriptstyle n \,$ )	{0, 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360, 384, 408, 432, 456, 480, 504, 528, 552, 576, 600, 624, 648, 672, 696, 720, 744, 768, 792, ...}
$25\,$	A008607( $\scriptstyle n \,$ )	{0, 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475, 500, 525, 550, 575, 600, 625, 650, 675, 700, 725, 750, 775, 800, 825, ...}