Carry and Borrow Principles

Soon after you learned how to count, you were taught how to add and subtract. At that time, you learned some concepts that you use almost everyday. Those concepts will be reviewed using the decimal system. They will also be applied to the other number systems you will study.

ADDITION. - Addition is a form of counting in which one quantity is added to another. The following definitions identify the basic terms of addition:

CARRY - A carry is produced when the sum of two or more digits in a vertical column equals or exceeds the base of the number system in use

How do we handle the carry; that is, the two-digit number generated when a carry is produced? The lower order digit becomes the sum of the column being added; the higher order digit (the carry) is added to the next higher order column. For example, let's add 15 and 7 in the decimal system:

Starting with the first column, we find the sum of 5 and 7 is 12. The 2 becomes the sum of the lower order column and the 1 (the carry) is added to the upper order column. The sum of the upper order column is 2. The sum of 15 and 7 is, therefore, 22.

The rules for addition are basically the same regardless of the number system being used. Each number system, because it has a different number of digits, will have a unique digit addition table. These addition tables will be described during the discussion of the adding process for each number system.

A decimal addition table is shown in table 1-1. The numbers in row X and column Y may represent either the addend or the augend. If the numbers in X represent the augend, then the numbers in Y must represent the addend and vice versa. The sum of X + Y is located at the point in array Z where the selected Xrow and Y column intersect.

Table 1-1. - Decimal Addition Table

To add 5 and 7 using the table, first locate one number in the X row and the other in the Y column. The point in field Z where the row and column intersect is the sum. In this case the sum is 12.

SUBTRACTION. - The following definitions identify the basic terms you will need to know to understand subtraction operations:

SUBTRACT - To take away, as a part from the whole or one number from another

BORROW - To transfer a digit (equal to the base number) from the next higher order column for the purpose of subtraction.

Use the rules of subtraction and subtract 8 from 25. The form of this problem is probably familiar to you:

It requires the use of the borrow; that is, you cannot subtract 8 from 5 and have a positive difference. You must borrow a 1, which is really one group of 10. Then, one group of 10 plus five groups of 1 equals 15, and 15 minus 8 leaves a difference of 7. The 2 was reduced by 1 by the borrow; and since nothing is to be subtracted from it, it is brought down to the difference.

Since the process of subtraction is the opposite of addition, the addition table in table 1-1 may be used to illustrate subtraction facts for any number system we may discuss.

In addition,

In subtraction, the reverse is true; that is,

Thus, in subtraction the minuend is always found in array Z and the subtrahend in either row X or column Y. If the subtrahend is in row X, then the remainder will be in column Y. Conversely, if the subtrahend is in column Y, then the difference will be in row X. For example, to subtract 8 from 15, find 8 in either the X row or Y column. Find where this row or column intersects with a value of 15 for Z; then move to the remaining row or column to find the difference.

THE BINARY NUMBER SYSTEM

The simplest possible number system is the BINARY, or base 2, system. You will be able to use the information just covered about the decimal system to easily relate the same terms to the binary system.

Unit and Number

The base, or radix - you should remember from our decimal section - is the number of symbols used in the number system. Since this is the base 2 system, only two symbols, 0 and 1, are used. The base is indicated by a subscript, as shown in the following example:

When you are working with the decimal system, you normally don't use the subscript. Now that you will be working with number systems other than the decimal system, it is important that you use the subscript so that you are sure of the system being referred to. Consider the following two numbers:

With no subscript you would assume both values were the same. If you add subscripts to indicate their base system, as shown below, then their values are quite different:

The base ten number 11₁₀ is eleven, but the base two number 11₂ is only equal to three in base ten. There will be occasions when more than one number system will be discussed at the same time, so you MUST use the proper subscript.

Positional Notation

As in the decimal number system, the principle of positional notation applies to the binary number system. You should recall that the decimal system uses powers of 10 to determine the value of a position. The binary system uses powers of 2 to determine the value of a position. A bar graph showing the positions and the powers of the base is shown below:

All numbers or values to the left of the radix point are whole numbers, and all numbers to the right of the radix point are fractional numbers.

Let's look at the binary number 101.1 on a bar graph:

Working from the radix point to the right and left, you can determine the decimal equivalent:

Table 1-2 provides a comparison of decimal and binary numbers. Notice that each time the total number of binary symbol positions increases, the binary number indicates the next higher power of 2. By this example, you can also see that more symbol positions are needed in the binary system to represent the equivalent value in the decimal system.

Table 1-2. - Decimal and Binary Comparison

MSD and LSD

When you're determining the MSD and LSD for binary numbers, use the same guidelines you used with the decimal system. As you read from left to right, the first nonzero digit you encounter is the MSD, and the last nonzero digit is the LSD.

If the number is a whole number, then the first digit to the left of the radix point is the LSD.

Here, as in the decimal system, the MSD is the digit that will have the most effect on the number; the LSD is the digit that will have the least effect on the number.

The two numerals of the binary system (1 and 0) can easily be represented by many electrical or electronic devices. For example, 1₂ may be indicated when a device is active (on), and 0₂may be indicated when a device is nonactive (off).

Look at the preceding figure. It illustrates a very simple binary counting device.Notice that 1₂ is indicated by a lighted lamp and 0₂ is indicated byan unlighted lamp. The reverse will work equally well. The unlighted state of the lamp canbe used to represent a binary 1 condition, and the lighted state can represent the binary0 condition. Both methods are used in digital computer applications. Many other devicesare used to represent binary conditions. They include switches, relays, diodes,transistors, and integrated circuits (ICs).