The mantissa is any number greater than or equal to 1 and less than 10.   The characteristic is a whole number or integer signifying the power of 10.

Powers of Ten and What They Mean

Positive Powers of Ten	Negative Powers of Ten
100 = 1 101 = 10 102 = 100 103 = 1000 104 = 10000 105 = 100000 106 = 1000000	100  = 1/1  = 1 10-1 = 1/10 = 0.1 10-2 = 1/100 = 0.01 10-3 = 1/1000 = 0.001 10-4 = 1/10000 = 0.0001 10-5 = 1/100000 = 0.00001 10-6 = 1/1000000 = 0.000001

Positive powers are relatively easy to visualize because, as you can see, the characteristic or exponent tells you how many zeroes follow the one.  What this means when we add another zero is that we are multiplying the previous number by 10; eg. 100 = 10*10, 1000 = 100*10, 10000 = 1000*10, and so on.  The numbers just keep getting bigger and bigger.

The negative exponents are all reciprocals of their positive counterparts.    For instance, if 10³ = 1000, 10^-3 = 1/1000 = 1/10³.  Negative exponents cause the resulting number to become smaller as the negative exponents become more negative.

Which way to move the decimal can be troublesome, just remember the easy numbers,  10¹ = 10 and 10^-1 = 0.1.  Understanding these simple numbers and what you learned about the number line you should have no trouble remembering that when you move right, the numbers get larger and when you move left, the numbers get smaller.

Scientific notation is commonly shown as illustrated above but you may also have seen numbers on your calculator followed by a capital E and a couple of other numbers, such as 1.0E6.  The capital E followed by a number is just another way of showing 10 with the exponent shown after the E.

Engineers and mathematicians used scientific notation almost exclusively prior to the invention of calculators.  Many still use it today to determine the order of magnitude of an answer.  The order of magnitude is another way of saying, "Approximate size."

     Example:  100 × 10000 = 1.0 × 10²   ×   1.0 × 10⁴ = 1000000 = 1.0 × 10⁶ = 1.0 million.

Converting Form

Given a decimal number, locate the left-most non-zero number and place a decimal point immediately to the right of that digit because 1 ≤ characteristic < 10.  Count how many digits there are between the newly placed decimal and the original location as this is useful in determining your characteristic.  Examine the mantissa of the scientific notation, if the decimal number was larger than the mantissa, the power of ten (characteristic) will be positive by the number of digits between the decimals.  If the decimal number was smaller than the mantissa, the power of ten will be negative by the number of digits between the decimals.

Instead of trying to memorize which way to move the decimal, let's just look at a couple of numbers and their conversions.  Remember, the two numbers have to be equal, so you should be able to tell if you got it right simply by looking at it.

12300.2 = 1.23002 × 10⁴     If you examine the mantissa of the scientific notation, you can tell that the decimal number is much larger than the mantissa.  The only way to make it larger is to multiply it by a positive power of ten.  In this case, it happens to be ten to the 4th power.  Likewise, if you are given the scientific notation, you should be able to tell that the decimal number must be much larger because the mantissa was multiplied by 10000.

0.0025 = 2.5 × 10^-3     Examination of the mantissa of the scientific notation reveals that the decimal number is much smaller than the mantissa.  The only way to make it smaller is to multiply it by a negative power of ten, specifically -3.  Given the scientific notation, you should be able to tell that the decimal number must be much smaller because the mantissa was multiplied by 1/1000.