5.1 GENERAL
The arithmetic operations, addition, subtraction, multiplication, and division, are performed in the octal system in a manner similar to the decimal operations, although octal counting is difficult. Table 2-3, for octal addition and subtraction, and table 2-4, for use in octal multiplication, should be used to check exam-ples. Octal arithmetic may also be performed in a roundabout fashion by converting to decimal, perform-ing the required operation, and then convertperform-ing back to octal. In the following paragraphs, only the table meth-ods will be shown; conversion methmeth-ods are given in Chapter 3.
5.2 OCTAL ADDITION
Octal addition is performed in much the same way as decimal addition. A sum and carry technique is used in which the sum and carry are determined by reference to an addition table. Addition and subtraction are given in table 2-3. The sum of two digits is found where the
column containing the addend digit and the row con-taining the augend digit intersect. For example, the sum of 7 and 6 is 15. The difference of two digits is found in the difference column. Find the minuend which is in the same column as the subtrahend digit. The row which contains this minuend also contains the difference. For instance, 12 - 6
==
4. Examples in the use of this table for addition are given below. Other than the difference in addition tables used, the addition processes used in octal and decimal are exactly the same. The carries are shown in parentheses.(1) (1) (112) carries
271.1 254.5 262.3
314.3 311.3 351.7
605.4 566.0 434.7
1271.1
TABLE 2-3. OCTAL ADDITION - SUBTRACTION
(SUBTRAHEND)
(DIFFERENCE) (SUBTRAHEND) ADDEND
AUGEND ADDEND 0 2 3 4 5 6 7 10
0 0 0 1 2 3 4 5 6 7 10
1 1 1 2 3 4 5 6 7 10 11
2 2 2 3 4 5 6 7 10 11 12
3 (Difference) 3 3 4 5 6 7 10 11 12 13
4 Augend 4 4 5 6 7 10 11 12 13 14
5 5 5 6 7 10 11 12 13 14 15
6 6 6 7 10 11 12 13 14 15 16
7 7 7 10 11 12 13 14 15 16 17
10 10 10 11 12 13 14 15 16 17 20
(Minuend) Sum
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Octal Multiplication
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PART 2 eH 5 5.3 OCTAL SUBTRACTION
Octal subtraction may be performed directly, as in decimal arithmetic, by a subtract and borrow routine.
In this case, the subtract portion of table 2-3 is used.
When the minuend is smaller than the subtrahend, a 1 must be borrowed from t1).e next column to the left.
This is indicated in the table by the two digit minuend numbers. In these numbers, the 1 stands for a carry whereas the other digit stands for the minuend. Exam-ples of the use of this table and the general methods of subtraction are given below. The borrows are shown in parentheses above the column they were borrowed from.
7254.3 6132.2 1122.1
(11) borrows 7356.3
7266.6 0067.5
5.4 OCTAL MULTIPLICATION
(111) borrows 5432.3
4567.0 0643.3
The operations used in octal multiplication are similar to the operations used in decimal arithmetic.
The multiplicand is multiplied by one digit of the multi-plier at a time to form a series of partial products that must be added to obtain the desired result. The
digit-by-digit multiplications are performed using the prod-ucts given in the octonary multiplication table, and the sums are obtained using the octal addition table. The position of the octonary point in the product, if either or both of the original numbers are fractional, is deter-mined exactly as in decimal multiplication; that is, if there are two digits to the right of the octonary point in the multiplier and four digits to the right of the octonary point in the multiplicand, the point is posi-tioned six places to the left of the least significant digit in the product.
Table 2-4 is a combination multiplication and division table. To use it for multiplication, read the numbers corresponding to the labels not in parenthe-ses. To use it as a division table, read the numbers corresponding to labels in the parentheses.
In multiplication, the product is found at the inter-section of the column containing the multiplicand digit and the row containing the multiplier digit. For in-stance, 6 x 7
==
52. In division, the quotient digit is found by searching the column which contains the divi-sion digit for the corresponding dividend digit (or digits). Then the row which contains this dividend digit intersects the quotient column where the proper quotient digit is located. For instance, 43 -;- 7==
5. Asan example, using the table, multiply 462(8) by 35(8)' TABLE 2-4. OCTAL MULTIPLICATION - DIVISION
MULTIPLICAND (DIVISION)
0 2 3 4 5 6 7 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0
0 1 2 3 4 5 6 7 10 11 12
2 0 2 4 6 10 12 14 16 20 22 24
Multiplier 3 0 3 6 11 14 17 22 25 30 33 36
(Quotient) 4 0 4 10 14 20 24 30 34 40 44 50
5 0 6 12 17 24 31 36 43 50 55 62
6 0 6 14 22 30 36 44 52 50 66 74
7 0 7 16 25 34 43 52 61 70 77 106
10 0 10 20 30 40 50 60 70 100 110 120
11 0 11 22 33 44 55 66 77 110 121 132
12 0 12 24 36 50 62 74 106 120 132 144
Product (Dividend)
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Octal Division 5.4-5.5
462 35 2772 1626 21252
The multiplicand is multiplied by each digit of the multiplicand, and the carry is added to the product of each individual multiplication. From the table, 5 x 2 = 12, write 2 and carry 1; from the table 5 x 6
=
36+
1 (carry)=
37, write 7 and carry 3; from the table 5 x 4=
24+
3=
27, write both digits. The same procedure is followed in the multiplication by 3. After the partial products are found, they are added accord-ing to the octal addition table.5.5 OCTAL DIVISION
Octal division is performed like decimal division except that the octal division and subtraction tables are used instead of decimal. As an example of octal divi-sion, divide 21252(8) by 35(8)'
4628 358/212528
164 265 256 72 72
o
30610
2910/887410
87 174 174
o
The most significant number in the quotient is gener-ated by examining the division of 212 by 35 and decid-ing, on a trial basis, the largest number that 35 can be multiplied by, resulting in a product less than 212. The number selected is 4; this is because 5 x 358 = 2218 , The multiplication of 4 x 35 is performed using the octonary multiplication table. The subtraction of the product from 212 is performed by direct octal subtraction. The process is continued as in decimal long division until the required number of octal digits have been gener-ated. Note that the divisors, dividends, and quotients agree in magnitude, but that the intermediate steps are different.
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