maths > decimals

Decimal Multiplication

    what you'll learn...


Decimals are fractions with standard place values. In this page, the following for decimals are explained.

 •  multiplication in first principles -- repeatedly combining a quantity and measuring the combined and

 •  simplified procedure : Sign property of multiplication and Multiplication by Place value for Decimals.

decimal repeated addition

Multiplication can be understood as "multiplicand repeated multiplier number of times".

Decimals are fractions with standardized place-values.

Consider 0.2×0.3.

By first principles, the multiplicand 0.2=210 is repeated multiplier 0.3=310 number of times.

This is done in two steps,

first step: 210 repeated 110 (denominator of 310) times is 2100.

Second step: the result from the first step 2100 is repeated 3 (numerator of 310) times is 2100+2100+2100 =6100

The product is 6100, or equals 0.06 in decimals.

aligned or opposed to direction

Decimals are directed numbers, that is decimals are either positive or negative.

0.2 aligned in direction is "+0.2".

0.2 opposed in direction is "-0.2".

Directed numbers, positive and negative, are explained as "aligned in direction" and "opposed in direction" respectively.

Consider 0.2×(-0.3).

By first principles, the multiplier 0.2=210 aligned in direction is repeated multiplier 0.3=310 opposed in direction number of times.

This is done in two steps,

first step: 210 repeated 110 (denominator of 310) times is 2100.

Second step: the result from the first step 2100 is repeated 3 (numerator of 310) timesopposed in direction is 2100+2100+2100 =6100 opposed in direction.

The product is 6100 opposed in direction, or equals -0.06 in decimals.

Decimal multiplication by first principles : Decimal multiplication is repeating the multiplicand, multiplier number of times with sign of the numbers (direction) handled appropriately.

simplifications recap

multiplication by place value

In whole numbers, we have studied Multiplication by Place-value as illustrated in the figure. This procedure is used in decimals in a later step.

In Integers, we have studied Sign-property of Multiplication.
+ve × +ve = +ve
+ve × -ve = -ve
-ve × +ve = -ve
-ve × -ve = +ve
This is applicable to decimals.

In Fractions, we have studied Multiplication of Numerators and Denominators.
For example, to multiply 45×32, the numerators are multiplied and denominators are multiplied. The product is 4×35×2.

Similarly decimals are multiplied keeping in mind the place-value representation.
For example, to multiply 0.8×1.5, it is equivalently thought as 810×1510 and so the product is 8×1510×10. Note that 8×15 is multiplied as per "multiplication by place-value".

simplify decimal multiplication

Consider multiplication of 0.007×0.05
This is equivalently 71000×5100

Understanding the above, a simplified procedure to multiply the decimals is devised.

The decimal point of multiplicand and multiplier are removed and the numbers are multiplied as integers.
eg: 0.007 is modified to the integer form 7.
0.05 is modified to the integer form 5.

The number of decimal-places in the multiplicand and multiplier are counted.
eg: 0.007 has 3 decimal-places.
0.05 has 2 decimal-places.

Now the integer forms are multiplied.
eg: 7×5=35

The number of decimal places of multiplicand and multiplier are added.
eg: 3+2=5.

The product of integer forms is modified to have the number of decimal points give by the sum above.
eg: 35 is modified to 0.00035 which has 5 decimal points.


What is 12.55×0.002
The answer is "0.02510".

Total number of decimal places in the multiplicand and multiplier is 2+3=5
So the product decimal place moves 5 places



Decimal Multiplication -- Simplified Procedure : The signs (+ve / -ve) are handled as in Sign-property of Integer Multiplication
 •  +ve × +ve = +ve
 •  +ve × -ve = -ve
 •  -ve × +ve = -ve
 •  -ve × -ve = +ve

The decimal places are removed and the multiplication is carried out as per Whole number Multiplication by Place Value.
 •  The decimal place is re-introduced in to the product.
 •  The decimal-point is moved to the left -- a number of digits equal to the total number of decimals in multiplier and multiplicand


The outline of material to learn "decimals" is as follows.

Note: goto detailed outline of Decimals

•   Decimals - Introduction

    →   Decimals as Standard form of Fractions

    →   Expanded form of Decimals

•   Decimals - Conversion

    →   Conversion between decimals and fractions

    →   Repeating decimals

    →   Irrational Numbers

•   Decimals - Arithmetics

    →   Comparing decimals

    →   Addition & Subtraction

    →   Multiplication

    →   Division

•   Decimals - Expressions

    →   Expression Simplification

    →   PEMA / BOMA