Decimals

Decimals are numbers written using 10 as the base.  Each position of the decimal number represents a power of 10 and the decimal point, the period '.',  is used to indicate how the powers of ten advance as you move away from the decimal. 

So, a decimal number is written like this: +/-  D---.d---

I used '-'s to indicate zero or more Ds or ds, the '+' is normally omitted if the decimal is a positive number.  

examples:     1.23,     1034.9123,    -0.15,   -156.2

We speak of positions or place holders for each number to the left or to the right of the decimal point.    I will use letters to name the position to which I refer in what follows.

Consider the decimal     EDCBA.abcde

The decimal point is located between the positions marked with the letters 'A' and 'a'.

The numbers 0,1,2,3,4,5,6,7,8,9 are used to "fill" these positions and name the number of 1s, 10s, 100s, ..., 1/10s, 1/100s, 1/1000s, etc.

As we move left from the decimal point, A represents the number of 1s, B the number of 10s, C the number of 100s, D the number of 1,000s, and E the number of 10,000s. 

As we move right from the decimal point, a represents the number of 1/10s, b the number of 1/100s, c the number of 1/1000s, d the number of 1/10,000s, and e the number of 1/100,000s.   

Example:   325.16   means:  3 * 100   + 2 * 10 +   5 * 1 +   1 * 1/10 +   6 * 1/100 

Example:    0.125    means:  0 * 1   +  1* 1/10  +  2 * 1/100  + 5 * 1/1000

Example:  -2.93      means:  -2 * 1 + -9 * 1/10  + -3 * 1/100     

NOTE how the negative sign moves through the decimal!

When adding/subtracting decimals, write the decimals one per line, lining up the decimal points and write zeros for missing place values.  Then perform the operations in the normal way.

Example:   3.56  + 206.1

rewrite    as     003.56

                +    206.10

                      209.66

Example:   0.0101 + 100.0

                     000.0101

                +   100.0000

                     100.0101

Example:   27.52  -  3.0025

                      27.5200

                 -    03.0025

                       24.5175

Example:   3.0025 - 27.52

                    03.0025

                -   27.5200

In this example recall how to add mixed-signed numbers when the negative number is larger in magnitude than the positive number:  Subtract the smaller magnitude from the larger magnitude then add the negative sign to the result.

So we have:

                       24.5175      now add the negative sign and get -24.5175 as the answer!