SCIENTIFIC NOTATION

1. What is "scientific notation"?

Scientists have a need to express very large (or very small numbers) in some convenient way. For example, astronomers measure the distance to stars in LIGHT YEARS, a unit of distance which conveniently expresses the huge distances that are involved. A light year is the distance which light travels in one year. Let us see how many metres there are in one light year. Light travels at a speed of 300 000 000 m/s (three hundred million metres per second).

This number is very inconvenient to read and to write, and it is very easy to make mistakes! The solution to this problem is to write such very large (or very small) numbers in SCIENTIFIC NOTATION. Expressions such as "billion", "trillions", "million billions", beloved of the media who frequently get these wrong, are to be avoided! ()

Scientific notation involves writing a number as the product of two numbers, as shown here on the left. The first one, the DIGIT VALUE, is always more than one and less than 10. The other, the EXPONENTIAL TERM, is expressed as a power of 10.

Computers normally have problems displaying the exponential term. So, one can also write a number in scientific notation in the following way:

9.4608E15 (or 9.4608e15) stands for 9.4608 x 1015 (numerical size of one light year)
3.7E-11 (or 3.7e-11) stands for 3.7 X 10-11 (numerical size of the hydrogen atom).

NOTE: In the numerical problems that are set in the Electronic Science Tutor®, use the above format for large or small numbers.

2. Converting numbers to scientific notation

To convert a large number to scientific notation, divide the number by 10 as many times as is needed to get a digit value less than 10 and greater than 1. Multiply this digit value by an exponential term where the exponent is the number of times that you have divided by 10.

In the example on the right, the number 1230000000, when divided by 10 nine times in succession gives a result of 1.23, so the number may be expressed in scientific notation as 1.23x109.

This gives rise to a simple rule:


For numbers greater than 1, if n is the number of digits TO THE LEFT of the decimal marker, then the value of the exponent is n - 1.

For example, the number 67342.085 has five digits to the left of the decimal point (n = 5). The exponent will have a value of n - 1 = 4, so the number can be written as 6.7342085x104.


To convert a small number to scientific notation, multiply the number by 10 as many times as is needed to get a digit value greater than 1 and less than 10. Multiply this digit value by an exponential term where the exponent is the negative number of times you have multiplied by 10.

In the example on the left, the number 0.0000000123 has to be multiplied 8 times by 10 in order to get a value of 1.23, and can therefore be expressed in scientific notation as 1.23 x 10-8.

Again, we have a simple rule:

For numbers smaller than 1, if n is the number of zeros IMMEDIATELY TO THE RIGHT of the decimal marker, then the value of the exponent will be -(n+1).

For example, the number 0.003257 has 2 zeros immediately to the right of the decimal point, so n = 2. The value of the exponent will therefore be -(n + 1) = -(2 + 1 ) = -3, and we can write the number as 3.257x10-3. Similarly, 0.1234 (n = 0) can be written as 1.234x10-1.

3. Additional questions