Method **multiplication by a column**, allows you to simplify the multiplication of numbers. Multiplication by a column assumes **sequential multiplication** the first number, for all digits of the second day of the subsequent addition of the received works, taking into account **indentation**depending on the position of the digits of the second number.

Let's consider how to multiply a column by the example of finding the product of two numbers **625** × **25**.

- 1 Write the numbers one below the other and draw a line.
- 2 number
**25**, consists of**2**digits**2**and**5**, we will multiply the first number**625**, on the digits of the second number in reverse order. We start the calculation by finding the product**625**×**5**, write the result below the line, start recording on the right side, we get:. - 3 Now we multiply
**625**on**2**, and write the result on the next line, shifting the record one cell to the left of the previous product, we get.

With more numbers in the second number, we get that our works line up on the right in the form of a "ladder".

4 As a result of multiplication, we obtain **2** works **3125** and **1250**, we will sequentially from right to left add their numbers to each other, in the order they go, and write down the result of their addition below. If the sum of the digits in addition exceeds **9**, then divide the amount by **10**, the remainder of the division is written under the current numbers, and the whole part of the division is moved to the left.

As a result, we get.

### Instructions for using a calculator for multiplication by a column

To calculate, just enter the numbers (integers or decimal fractions) and press the "=" button.

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### Natural numbers Edit

= ⋅ = ,

To multiply natural numbers in a positional notation, a bitwise multiplication algorithm is used. If two positive integers a < displaystyle a> and b < displaystyle b> are given such that:

- t n - 1, 0 = m o d (a n - 1 ⋅ b 0 + r n - 1, P), r n = d i v (a n - 1 ⋅ b 0 + r n - 1, P), t 0 ⋅ P k, < displaystyle t_<>

P ^

Arithmetic operations on numbers in any positional number system are carried out according to the same rules as in the decimal system, since they are all based on the rules for performing actions on the corresponding polynomials. In this case, you need to use the multiplication table corresponding to the given base P < displaystyle P> of the number system.

An example of the multiplication of natural numbers in binary, decimal and hexadecimal number systems, for convenience, the numbers are written under each other according to the digits, the hyphenation is written above:

### Rational numbers Edit

The set of rational numbers is denoted by Q < displaystyle mathbb > (from the English quotient "private") and can be written in this form:

To multiply rational numbers in the form of ordinary (or simple) fractions of the form: ± m n < displaystyle pm < frac

The arithmetic operation "multiplication" over rational numbers refers to closed operations.

### Real numbers Edit

Arithmetic operations on real numbers represented by infinite decimal fractions are defined as *continuous continuation* corresponding operations on rational numbers.

If you are given two real numbers, represented by infinite decimal fractions:

∀ a ′, a ″, b ′, b ″ ∈ Q, (a ′ ⩽ α ⩽ a ″) ∧ (b ′ ⩽ β ⩽ b ″) ⇒ (a ⋅ b ′ ⩽ α × β ⩽ a ″ ⋅ b ″) ⇒ (a ′ ⋅ b ′ ⩽ γ ⩽ a ″ ⋅ b ″). < displaystyle forall a ', a' ', b', b '' in mathbb ,

(a ' leqslant alpha leqslant a' ') land (b' leqslant beta leqslant b '') Rightarrow (a ' cdot b' leqslant alpha times beta leqslant a '' cdot b``) Rightarrow (a ' cdot b' leqslant gamma leqslant a '' cdot b '').>

### Complex numbers Edit

The set of complex numbers with arithmetic operations is a field and is usually denoted by C < displaystyle mathbb

The product of two complex numbers in an algebraic form of writing is called a complex number equal to:

c + f i = (a + d i) ⋅ (b + e i) = (a ⋅ b - d ⋅ e) + (a ⋅ e + b ⋅ d) i,

In order to multiply two complex numbers in the trigonometric form of writing, you need to multiply their modules, and add the arguments:

where: r = | z | = | a + i b | = a 2 + b 2, φ = arg (z) = arctan (b a), < displaystyle r = | z | = | a + ib | = < sqrt + b ^ <2> >>,

### Multiplication of Arbitrary Numbers Edit

The unit of measurement of a physical quantity has a specific name (dimension), for example, for length - meter (m), for time - second (s), for mass - gram (g) and so on. The result of measuring one or another quantity is not just a number, but a number with a dimension, for example, 10 m, 145 s, 500 g. Dimension is an independent object that participates on an equal footing in the multiplication operation. When multiplying physical quantities, both the numerical values themselves and their dimensions are multiplied, giving rise to a new number with a new dimension.

In addition to dimensional physical quantities, there are dimensionless (quantitative) quantities that are formally numbers that are not associated with specific physical phenomena (measured by "pieces", "times" and the like). When you multiply a number with a dimension by a dimensionless quantity, the result preserves the original dimension. For example, if we take 5-meter rails in the amount of 3 pieces, then as a result of multiplication we get the total length of the rails 15 meters:

Multiplication of heterogeneous physical quantities should be considered as finding a new physical quantity that is fundamentally different from the values that we multiply. If it is physically possible to create such a work, for example, when finding work, speed or other quantities, then this quantity forms a set that is different from the initial ones. In this case, the composition of these quantities is assigned a new designation (new term), for example: density, acceleration, power, etc.

For example, if we multiply the speed of a uniformly and rectilinearly moving body equal to 4 m / s by a time equal to 2 s, we get a named number (physical quantity) called “length” or “distance” and is measured in meters:

4 m / s2 s = 8 (m / s) s = 8 m.

The product of the elements of the sequence can be compactly written using a special multiplication symbol, which goes back to the capital letter Π (pi) of the Greek alphabet, as shown in the example:

Such a record can be “expanded” into an expression in which the values of the multiplication index from the initial to the final value are sequentially substituted, for example:

More formally, a notation is defined as follows:

Where *m* and *n* there are integers or expressions that are calculated into integer values.

If the index values are given by some set, then a multiple product can be written using it, for example