Multiplication
Along with addition, subtraction, and division, multiplication is one of the four fundamental arithmetic operations
Mathematics has enormously broadened its horizons in the last centuries, and multiplication has followed its path, finding application in gradually different fields. Multiplication is possible between integers, fractions, real numbers, complex numbers, polynomials, sets, vectors, matrices, and more. But the most elementary and ancient form of multiplication is and remains that between natural numbers.
Multiplication as a recursive addition
This fundamental type of multiplication is based upon the idea of a recursive addition, i.e., an addition repeated as many times as required by the factor by which a number is multiplied. For instance, what does it mean to multiply 4×6? It means repeating the addition 4+4 six times in a row. Thus, multiplication is essentially a shortcut to make the addition operation more concise:
The more numerous the addends are, the greater the advantage of this compact form of writing:
However, transforming a recursive addition into a multiplication is possible only if the addends to be replaced are all the same. The addition
can be transformed into multiplication
But the addition
cannot become a multiplication, at least not integrally (we can transform it eventually into 3×4+1).
Intuitive representation of multiplication
From an intuitive perspective, the truth of multiplication, i.e., its equivalence with a repeated addition, can be easily verified by counting the elements in finite sets of objects, as in the following example.
We have six groups of 5 cherries each. If we count all the cherries one by one, starting with group 1 and ending with group 6, we arrive at a total of 30. But we get 30 even if we add the contents of each group: 5+5+5+5+5+5 = 30. Finally, we get 30 if we multiply the number of cherries in each group by the total number of groups: 5×6 = 30. Once again, we note that each group contains the same number of elements.
We can get another intuitive representation of the equivalence between multiplication and repeated addition using a number line, as shown in the following image.
Long multiplication
Throughout centuries and millennia, mathematicians developed several algorithms to perform multiplications. The algorithm that prevailed, becoming the standard model taught in schools, is the so-called long multiplication, of which the following image provides an example, complete with the proper terminology.
The numbers to be multiplied, generically called factors, in long multiplication are called the multiplicand and the multiplier. They must be vertically aligned so that ones correspond to ones, tens to tens, hundreds to hundreds, etc.
Each digit of the multiplier, starting from the one furthest to the right, is multiplied by all the digits of the multiplicand, always starting from the right. If the product of each multiplication is greater than 9, a carry is generated corresponding to the figure representing tens.
The partial products of the multiplications must take into account the carry. For example, in the multiplication shown above, the 8 at the end of the multiplier, multiplied by the 3 at the end of the multiplicand, gives 24. The 4 is written in the column under the multiplier. The 2 representing tens is the carry. It must be added to the product of the next multiplication: 8×2 = 16, 16+2 (the carry) =18. Write the 8 to the left of the 4 and shift the 1 to the next left column.
The partial products of the multiplication must be written under the multiplier, taking care to offset all but the first by one column to the left with respect to the previous one. This way, the correspondence between ones, tens, hundreds, etc., is preserved.
We get the end product of the multiplication by adding the partial products column by column, always starting from the right. Again, carry must be taken into account, according to the addition model.
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Note that the decimal digits, which were two in the multiplier and the multiplicand (that is, tenths and hundredths), become four in the final product, which increases the precision with the addition of thousandths and ten thousandths.
Grid method multiplication
A simplified form of multiplication is the grid (or box) method. First, a rectangular grid is drawn that has as many columns as the multiplicand digits and as many rows as the multiplier digits. Next, a part of the multiplicand is associated with each column — ones to the rightmost column, tens to the next column to the left, then hundreds, etc., depending on the length of the number. The same is done with the rows, dividing the multiplier into ones, tens, hundreds, etc. Each cell of the grid is then filled with the product of the multiplication between the parts of the multiplicand and the multiplier to which each cell corresponds. Below is an example of this system applied to multiplication 9,746 × 825.
The next step is to add up all the partial products to get the result of the multiplication:
7,200,000 +
560,000 +
32,000 +
4,800 +
180,000 +
14,000 +
800 +
120 +
45,000 +
3,500 +
200 +
30 =
------------
8,040,450 The advantage of this method is to simplify the partial products, reducing them to the result of only one multiplication. However, it is necessary to pay attention to the number of zeros to be added to the various partial products. Their number is equivalent to the sum of the zeros of the two factors. For example, 700 and 20 have three zeros in all, so their product, 14,000, has three zeros.
Multiplication tables
As seen above, even the longest multiplication can be reduced to multiple multiplications between two single-digit numbers, followed by the necessary additions. From a mnemonic point of view, this requires that at least the products of multiplications between factors between 1 and 10 are well known. There are tables built to facilitate this learning. They report in the form of a grid the products corresponding to each pair of factors, as in the following example.
Multiplication symbols
In arithmetic, the multiplication operator is the cross sign ×. Since this symbol generates confusion with the variable 𝑥, the use of the symbol ⋅ has prevailed in algebra (as in 3𝑥 ⋅ 23𝑦.) However, multiplication is often implicitly marked by the simple juxtaposition of the terms to be multiplied. This typically happens when there are parentheses:
Juxtaposition is also the usual way of multiplying coefficients and variables in the terms of a polynomial. Therefore, in the previous example:
Finally, in computer usage, the multiplication operator is the asterisk character, as in
Rule of signs
In multiplications between integers, the following rule of signs applies:
In short, the rule states that when the sign is the same, the product of multiplication is positive; otherwise, the product is negative. Here are some examples:
Properties of multiplication
Multiplication has the following properties:
- Commutative. The order of the factors in a multiplication does not matter: 𝑥 ⋅ 𝑦 =𝑦 ⋅ 𝑥. E.g., 2 ⋅ 7 = 7 ⋅ 2 =14. This property applies to multiplications between natural, whole, rational, real, complex numbers. However, it does not apply in some other mathematical fields, such as in multiplication between matrices, which is not commutative.
- Associative. The order in which the multiplications are performed does not matter. Given three numbers x, y and z, (𝑥 ∙ y) ⋅ z =𝑥 ⋅ (y ⋅ z). E.g., (4 ⋅ 3) ⋅ 6 = 4 ⋅ (3 ⋅ 6) → 12 ⋅ 6 = 4 ⋅ 18 = 72.
- Distributive over addition. The product of a number times a sum of numbers is equal to the sum of the products of that number multiplied by each of the addends. Given three numbers x, y and z, x ⋅ (y + z) = x ⋅ y + x ⋅ z. E.g., 4 ⋅ (3 + 6) = 4 ⋅ 3 + 4 ⋅ 6 → 4 ⋅ 9 = 12 + 24 = 36.
- Identity element. Any number multiplied by 1 is equal to itself: x ⋅ 1 = 1 ⋅ x = x.
- Zero property. Any number multiplied by zero is zero: x ⋅ 0 = 0 ⋅ x = x. Also, note that it is necessary and sufficient that at least one factor in a series of multiplications be zero for the whole product to be zero. E.g., 9 ⋅ 6 ⋅ 11 ⋅ 7 ⋅ 4 ⋅ 0 = 0.
- Negation. Any number multiplied by −1 is equal to its additive inverse: x⋅(−1)= −x. E.g., 3 ⋅ (−1) = −3; −7 ⋅ (−1) = 7. Note that the additive inverse of a natural number is a negative number, which does not belong to the set of natural numbers.
- Inverse element. Any number 𝑥, except zero, has a multiplicative inverse 1/x such that x ⋅ 1/x = 1. E.g., −9 ⋅ (−1/9) = 1.
