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Decimals

Scientific Notation

Scientists often deal with very large or very small numbers. For example,

There are approximately 70000000000000000000000 stars in the universe.

Avagadro's constant, the number of atoms in N grams of an element with atomic weight N, is 602214199000000000000000.

The atomic volume of carbon is approximately 0.00000000000000000000000997 cm3.

To make writing and dealing with these very large and very small numbers easier, scientists developed scientific notation.

Consider the following numbers written in decimal form and in exponential notation with base ten. Those numbers that are less than one are also written as fraction.

Fraction

Decimal

Exonential Notation

one thousandth

0.001

10-3

one thousandth

0.01

10-2

one thousandth

0.1

10--1

 

1

100

 

10

101

 

100

102

 

1000

103

So we could also write

30000 = 3*10000 = 3*10^4. 200 = 2*100 = 2*10^2. 0.005 = 5*0.001 = 5*10^(-3)

 

On your calculator, a very small or very large number is displayed in scientific notation as:

m*10^e

where 1 ≤ |m| < 10. That is, m is a number with one non-zero digit to the left of the decimal point and it can be positive or negative. The number m is called the mantissa and the number p is called the exponent.

Every non-zero number can be written in scientific notation. For example,

100 = 1*10^2. 4.321 = 4321*10^3. 1.23 = 1.23*10^0. 0.25 = 2.5*10^(-1). 0.0007925=7.925*10^(-4).

Note that the exponent of 10 is the number of places the decimal point is shifted from the number in decimal form. A positive exponent indicates that the decimal point is shifted to the right. The decimal number is larger than the mantissa.

A negative exponent indicates that the decimal point is shifted to the left.The decimal number is smaller than the mantissa. The mantissa indicates the significant figures in the number.

Examples

Convert 302400 to scientific notation.

Removing the place holding zeros, the mantissa is 3.024 . The decimal place has moved 5 positions from the end of the decimal number to its position in the mantissa. The decimal number exceeds the mantissa so the exponent will be positive.

302400 = 3.024 x 105

Convert 0.05607 to scientific notation.

Removing the place holding zeros, the mantissa is 5.607 . The decimal place has moved 2 positions from the decimal number to its position in the mantissa. The decimal number is less than the mantissa so the exponent will be negative.

0.05607 = 5.607 x 10-2

starsScientists talk of approximately 7 x 1022 stars in the universe, Avagadro's constant = 6.02214199 x 1023 and the atomic volume of carbon is approximately 9.97 x 10−24 cm3.

 

Exercise

Convert these decimal numbers to scientific notation.

Decimal
Scientific Notation
                     
×10
 

Conversion from scientific form to decimals involves writing the digits of the mantissa and moving the decimal point the required number of positions adding zeros if needed as place holders.

Examples

9.123 x 102 = 912.3

7 x 104 = 70000

1.1 x 10-2 = 0.011

Exercise

Scientific Notation

Decimal

 

 

× 10

 

 

<< Rounding | Decimals Index | Significant Figures >>

 

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