Scientific Notation

I am giving you examples of how to convert to scientific notation.

3450 = 3.45x103

345 = 3.45x102

34.5 = 3.45x101

3.45 = 3.45x100

0.345 = 3.45x10-1

0.0345 = 3.45x10-2

0.00345 = 3.45x10-3

0.000345 = 3.45x10-4

0.0000345 = 3.45x10-5

and so on.

I hope you see a pattern that 3.45 is always expressed. The rest is to figure out what is the exponent of 10. Memorizing one will help you expressing other numbers. Seeing the pattern is critical.

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Numbers in scientific notation have two parts: a mantissa and an exponent:

mantissa x 10exponent

• The mantissa is a number that must be greater than or equal to 1 and less than 10.
• The exponent is any integer (positive or negative or zero).

The exponent tells how many digits the decimal point needs to move from after the first (non-zero) digit of the mantissa to get back to where it was in the original number - if it is negative, it needs to move to the left, otherwise (if positive) it moves to the right. (If zero the decimal point does not need to move!)

After any operation, the result should be corrected if necessary to have the mantissa in the correct range (above) by moving the decimal point and correcting the exponent by adding/subtracting the number of digits required to move the digit back to its original position (add if to right, subtract if left) to/from the current exponent.

If the mantissa is 1, it is sometimes omitted completely, eg 1 million = 1 x 106 which is sometimes just written as 106; similarly, if the exponent is 1 it may be omitted, eg 1.2 x 101 may be written as 1.2 x 10

Doing maths with numbers in scientific notation:

When adding or subtracting numbers in scientific form, change the mantissa with the smaller exponent into a number with the same exponent as the larger number by moving the decimal point to the left the difference between the exponents. Now add or subtract as normal the mantissas (with the decimal points aligned) and correct the result if necessary (as above).

Examples:

• 1.23 x 103 + 4.5 x 102 = 1.23 x 103 + 0.45 x 103 = (1.23 + 0.45) x 103 = 1.68 x 103
• 1.23 x 103 - 4.5 x 102 = 1.23 x 103 - 0.45 x 103 = (1.23 - 0.45) x 103 = 0.78 x 103 = 7.8 x 103 - 1 = 7.8 x 102

When multiplying or dividing numbers in scientific form, multiply/divide the mantissas as normal numbers and add/subtract the exponents, correcting the result if necessary (as above).

Examples:

• (3.69 x 103) x (4.5 x 102) = (3.69 x 4.5) x 10(3 + 2) = 16.605 x 105 = 1.6605 x 105 + 1 = 1.6605 x 106
• (3.69 x 103) ÷ 4.5 x 102 = (3.69 ÷ 4.5) x 10(3 - 2) = 0.82 x 101 = 8.2 x 101 - 1 = 8.2 x 100

When raising a number in scientific notation to a power, raise the mantissa to the power and multiply the exponent by the power, correcting the result if necessary (as above).

Examples:

• (1.23 x 103)2 = 1.232 x 103 x 2 = 1.5129 x 106
• (1.6 x 103)-1 = 1.6-1 x 103 x -1 = 0.0625 x 10-3 = 6.25 x 10-3 - 2 = 6.25 x 10-5

As scientific numbers are often rounded to a number of significant figures, the [final] result of a calculation should be rounded to a similar/the same number of significant figures.

Not sure what you mean by writing it "in measurement." Wherever or however it's employed, numbers or values expressed in scientific notation have the following format: a x 10b, where bis an integer (positive or negative) chosen such that a is a real number between at least one but less than ten. In other words 1 <= a < 10. Here are some examples of numbers and their representations in scientific notation (also known as standard form): 1543 = 1.543 x 103 0.00345 = 3.45 x 10-3 1,000,000 = 1 x 106

1,000,000.0 = 1.0000000 x 106

Note the difference between the last two numbers, which are not quite the same. Both numbers represent one million, but the first value has only one significant digit, whereas the second number indicates a much greater degree of precision. In other words, the first number may represent a measurement that was rounded to the nearest million, whereas the second number represents a measurement that was rounded to the nearest tenth. (That is, the uncertainty is in the tenth's place.) Do you understand the differences among the following values? 3.0 x 106

3.00 x 106

3.000 x 106

Also note that a shorthand version of scientific notation exists. Here are two examples: 1.23 x 103 = 1.23E3

2.47 x 10-6 = 2.47E-6 In the shorthand version, the number after the E represents the power of ten.

14575 + 23x101 =

14805 =

1.4805 x 104