Chapter 1.2

Using the Numbers - Basic Algebra

In electronics, as with the rest of our lives, numbers matter. You might hear someone say, I need two 9-volt batteries, I want a 1500 watt portable heater, I have version 8.3 software, I need one-forth teaspoon of pepper, lets meet for lunch at 12:30, and so on. Numbers tell us something specific about our world. Without numbers it would be more challenging to communicate. In electronics this is no different.

In these few pages we sill start our discussions on numbers. I will address numbering systems by their BASE number and by their system number. With that idea in mind, we start off very slow with base-10 number system, and then show how to jump to other numbering systems. Base-10 numbers are also called decimal numbers. So what are some of the attributes that we know about the base-10 numbering system?

A look At Base-10 Or Decimal Number System

 Base-10 numbers use these ten symbols that we call numbers
   0  1  2  3  4  5  6  7  8  9 
   These ten symbols represent the base-10 number system.
   All numbers within that system use some combination of
   one or more of these symbols.

Lets start by thinking of the POWER of MONEY, which is based on the base-10 numbering system. Looking at the illustration to the right, the top row displays the BASE and something called an EXPONENT. This row had 10 as the base and the exponents 3, 2, 1, and 0. In simple terms, if the exponent is the number 0 you have what you have in the number set. In base-10 that is the range of numbers 0 through 9. Look at the far right-hand side of this illustration and note the 10 raised to the power of 0. This column with exponent 0 represents the numbers from 0 through 9 (cents in this example). The two columns with exponents 1 and 0 can represent any two-digit combinations within the number set. For my money chart that would be 0 to 99 cents. Column with exponents 2, 1 and 0 represent any three-digit number combination within the number set.
As we change from base-10 to different BASE-NUMBERS, this concept of an exponent to a base-number will become a key in understanding numbers that are not base-10. ONE MORE NOTE: We generally do not address what we do not have. If I said I have $50.00 that is equivalent to $0050.00 but the two leading zeros before the 5 are considered bad form. There is no need to tell the reader there are no hundreds and no thousands in the number. Even with the leading 0 being a valid number it adds no value to the statement $50.00. There may be times especially in binary tables where the 0 is handy as a place holder.

OK, time to dig in. Lets start by adding the number 123 to the number 345. In reality I am adding 100 plus 20 plus 3 plus 300 plus 40 plus 5. In breaking this process apart, we can look at the parts, in order.

  so in the "ones column" 3 plus 5 is 8. 8x1=    8
  In the "10 column" 2 plus 4 is 6.     6x10=   60
  In the "100 column" 1 plus 3 is 4.   4x100=  400
  The answer is                                468

We also know something about the word carry. To see how this works, replace the number 123 with the number 167 in this simple equation.

  so in the "ones column" 7 plus 5 is 12.     (carry the 1) 2x1=   2  
  The 10 is too large for the 1's column and 
      is carried into the 10's column.
  In the "10 column" 1 plus 6 plus 4 is 11.  (carry the 1) 1x10=  10
  Again the 100 is too large for 10's column
      and is carried into the 100's column.
  In the "100 column" 1 plus 1 plus 3 is 5.  (no carry)   5x100= 500

  The answer is                                                  512

Base-2 The Binary Number System

The idea of having a carry ALWAYS EXISTS. In the first example, all the carry values were 0, so they were easily ignored. in the second example we needed to remember the carry. Now as we move from base-10 numbers and into base-2 the binary numbering system, the carry is used a lot more often.

 In base-2 the carry is used in nearly EVERY math function.
 In base-10 the numbering system went from 0 to 9. 
 In base-2 numbers 0 and 1 are the only two symbols
 All numbers within that system use some combination of
   one or more of these numbers.

All is good so far. In this illustration we will step through the same process as we did with the base-10 system. Here we will add 2 bass-10 numbers, namely 17 and 23 (base-10). We will look at both the base-10 and base-2 math.

Why do we care, one might ask? We care because many logic operators use base-2 logic. Consider the experience of going to the eye doctor. The exam process is based on logic. The exam might go something like this. The doctor would say

    Looking at the chart, is number one or number two clearer?
  This is number ONE! This is number TWO! Again ONE! And TWO!

In electronics we want to know how to do operations based on binary logic. Electronic functions can answer a questions including: Is it dark outside, is the door open, did the car start, did somebody press a key on the keyboard, and so on.

Take a quick look at the chart illustration. Along the top we see the base number is 2 and the exponent numbers are 5, 4, 3, 2, 1, and 0. We can easily have exponents larger then 5 but for this example this is large enough. The illustration also has a section on the left titled B-10 Carry and base-10 numbers below that written in base-10. Both the base-10 and their base-2 equivelen numbers are given here to help you convert between base-2 and base-10. OK Here we go.

 Convert 17 base-10 to base-2 (using B-10 Carry).
 Is 32 <= to 17 = no (0), Place 0----- in 32's column and 
                                    carry 17 to right        0-----
 Is 16 <= to 17 = yes(1), Place  1---- in 16's column          
            subtract 16 from 17 and carry 1 to right         01----
 Is 8 <= to 1 = no (0),   Place   0--- in 8 column and       
                                    carry 1 to right         010---
 Is 4 <= to 1 = no (0),   Place    0-- in 4 column and
                                    carry 1 to right         0100-- 
 Is 2 <= to 1 = no (0),   Place     0- in 2 column and
                                    carry 1 to right         01000-
 Is 1 <= to 1 = yes(1),   Place      1 in 1 column, end      010001     

                  The decimal number 17 is the binary number 010001

 Convert 23 base-10 to base-2 (using B-10 Carry).         
 Is 32 <= to 23 = no (0), Place 0----- in 32 column and
                                     carry 23 to right       0-----
 Is 16 <= to 23 = yes(1), Place 1---- in 16 column
             subtract 16 from 23 and carry 7 to right        01----
 Is 8 <= to 7 = no (0),   Place 0--- in 8 column and
                                     carry 7 to right        010---
 Is 4 <= to 7 = yes(1),   Place 1-- in 4 column
               subtract 4 from 7 and carry 3 to right        0101--  
 Is 2 <= to 3 = yes(1),   Place 1- in 2 column and
                                     carry 1 to right        01011-  
 Is 1 <= to 1 = yes(1),   Place 010111 in 1 column,  end     010111
    
                  The decimal number 23 is the binary number 010111

We just converted the numbers from 17 and 23 in decimal to 010001 and 010111 respectively in binary.

  The next step is to add these up:    0  1  0  0  0  1
                                    +  0  1  0  1  1  1
  Remember 0's 1's and carry only.
  A number 1 in the largest number in a column or the set of numbers.
 
  Start by adding the right-hand column:
                                      0  1  0  0  0  1
                                      0  1  0  1  1 +1
                                                     0 carry 1
  Next column with carry                          1
                                      0  1  0  0  0  
                                      0  1  0  1 +1  
                                                  0  0 carry 1
  Third column with carry                      1
                                      0  1  0  0  
                                      0  1  0 +1  
                                               0  0  0 carry 1
  Forth column with carry                   1
                                      0  1  0  
                                      0  1  0  
                                            1  0  0  0 no carry
  Column five no carry                0  1  
                                      0  1  
                                         0  1  0  0  0 carry 1
  Column six with carry               1
                                      0 
                                      0  

  The binary addition total is        1  0  1  0  0  0

Recall earlier mention of leading zeros. Using leading zeros helps make the columns look cleaner.

  Now to convert from base-2 to base-10.
  Returning to the illustration there is a 1 in the 32 column 1x32 = 32
                                there is a 1 in the 8  column 1x8  =  8
     32 + 8 is 40

  Describing this conversion from base-2 to base-10 in words, one would say:

                 1 times 2 to the 5th power
            plus 1 times 2 to the 3rd power = decimal number 40. 

Hexadecimal Number - Base-16

In this illustration we will take a look at base-16 numbers. Because our decimal number system does not have 16 symbols we need to add 6 additional symbols to the 10 we generally use.
Base-16 numbers use these symbols as numbers
0 1 2 3 4 5 6 7 8 9 a b c d e f
In base-16 also called hex and hexadecimal, all numbers within that system use some combination of one or more of these numbers.

The table is in three sections. Starting at the top of the left section in its fat left column is the number zero. Looking two lines below that, in the third line, it reads it reads 2 0000 0010 2. The first number 2 is a base-10 number. The numbers 0000 0010 is a base-2 number for 2. The last 2 is a base-16 number for 2. Moving to the last line in that section, the line reads 15 0000 1111 f. In this line the 15 represents the base-10 number for 15. The number set 0000 1111 is the binary number for the same value of 15 and the f is the hexadecimal number for 15. Note that when writing base-2 numbers it is easier to read them if they are split is sets of 4-digits starting at the right side. For base-16 numbers, the f is the largest single digit number in that set. Now, moving to the top line in the center column, this number represents the values for the number 16. Review the table and notice how the base-2 and base-16 numbers change over their number sets.

 When working with mixed
 numbers the base needs
 to be written. For example
 20h(hex) times 101b(binary)
 equals 160d(decimal).
    
 If you add
 
      5d = 0000 0101b
      8d = 0000 1000b
      7d = 0000 0111b
    + 4d = 0000 0100b
    ___    _________
     24d = 0001 1000b
          
  


 It is easy to use a chart or a 
 calculator to start with.
 This chart has the first 48 numbers (0 is a common number in electronics)

 The same number  24 as represented in base-10,
              is 0001 1000          in base-2
             and        18          in hex(base-16).

One more bit of useful information is using a conversion table. It is common to use letters to represent number. At the center of the table is the 1's place. To convert form 100 pennies to 1 dollars divide by 100. To convert a kilometer to meters multiply by 1000. A MICRO-something is very small at one-millionth of the something. If your camera has a 5.1 megapixel rating you will have 5.1 million pixels (5100000 pixels) of picture resolution. A tera-byte of data is one-thousand-billion bytes of data. That is a lot.

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