In the International System prefixes to indicate decimal fractions or multiples of the units are used.

For example, the “mili” prefix, which represents one thousandth of a unit [written in exponential notation, 10-3]: a milligram (mg) is 0.001 g [10-3 g], one millimeter (mm) is 10-3 m.

The following table lists the most common prefixes.

0008 Exponential notation. TableExponential notation.

The exponential notation or “scientific”, used to easily express very large or very small numbers, is a quick way to represent numbers using base ten powers.

Any number “N” is written as a product:

N = a x 10n

where “a” is a real number greater than or equal to 1 and less than 10, and called coefficient;

“n” is an integer, which is called the exponent or order of magnitude (written as a superscript to the right of ten).

A couple of examples will help us to understand:

2,500,000 equals 2.5 x 106

[which reads: “two point five to the (power) six”]

0.000705 equals 7.05 x 10-4

[that reads: “seven point zero five by ten to the (power) minus four”]

A positive exponent, as in the first case, tells us how many times you have to multiply a number by 10 to get the “long” form of the number:

1,2 x 106 = 1,2 x 10 x 10 x 10 x 10 x 10 x 10 (six times ten) = 1.200.000

It can also be thought that the positive exponent is equal to the number of times that the decimal point is shifted to the left to get a number greater than 1 and less than 10.

If, for example, we start with the number 5270.00 and we “move” the point three “positions” to the left, we get 5.27 x 103 (and as an exponent for the 10 appears the number of positions).

Similarly, a negative exponent indicates how many times we have to divide a number by 10 to get the “long” form of that number:

0008 Notación exponencial. Ecuación 0As before, the negative exponent is equal to the number of times that we have to move the decimal point to the right to get a number greater than 1 and less than 10.

If we take the number 0.0067 and we “move” the point three “places” (positions) to the right, we will have 6.7 x 10-3.

Therefore, in the exponential notation system, with each shift to the right of the decimal point, the exponent decreases a unit:

6,7 x 10-3 = 67 x 10-4

Likewise, each shift to the left of the decimal point increases the exponent by one unit:

6,7 x 10-3 = 0,67 x 10-2

Mathematical operations with scientific notation.

Clearly, the purpose of using this notation is to simplify the math related with scientific studies. We will describe the basic operations with this notation.

Addition and subtraction.

Whenever the powers of 10 are of the same order, ie with the same exponent [remember that it is the number placed to the right of the 10 at the top], the coefficients must be added, leaving the power of 10 with the same degree. In case that the powers do not have the same exponent, the coefficient must be transformed: multiplying or dividing it by 10 (moving the decimal point right or left) as many times as needed to get the same exponent.

Examples:

  • same order (same exponent):

3×105 + 6×105 = 9×105

6×106 – 2,2×106 = 4,8×106

  • conversion (different order):

4×104 + 3×105 – 2×103 = [taking the exponent 5 as reference]

= 0,4×105 + 3×105 – 0,02×105 = (0,40 + 3,00 – 0,02)x105 = 3,38×105

Multiplication.

To multiply numbers written in scientific notation multiply the coefficients and add the exponents.

Example:

(3×109)×(2×106) = 6×1015

[first coefficient “3” by second coefficient “2”, 3×2 = 6

and first exponent “9” plus second exponent “6”, 9 + 6 = 15]

Division.

To divide numbers written in scientific notation divide the coefficients and subtract the exponents.

Example:

0008 Notación exponencial. Ecuación 1[first coefficient “24” divided by second coefficient “3”, 24/3 = 8

and first exponent “-8” minus second exponent “-2”, (-8)-(-2)= -6]

Powers.

Rise the coefficient to the power and multiply the exponents.

Example:

(4×104)2 = 1,6 ×109

[coefficient “4” squared, 42 = 4×4 = 16

exponent (of the 10)“4” mutiplied by exponent (global) “2”, 4×2 = 8

16×108 = 1,6×109]

Roots.

The root of the exponent must be calculated the coefficient is divided by the index of the root.

Examples:

0008 Notación exponencial. Ecuación 2[square root of the coefficient “9” = 3

exponent “-12” divided by the root index “2”, -12/2 = -6]

3×10-6

0008 Notación exponencial. Ecuación 3[cubic root of the coefficient “8” = 2

exponent “15” divided by the root index “3”, 15/3 = 5]

2×105