Casting out nines

Nine and Eleven sample are methods to detect calculation errors in addition, subtraction or multiplication. The advantage of this sample is that it can be checked for credibility based lighter alternative calculation methods the accuracy of the result of a lengthy statement.

Colloquially, the term nines is also commonly used for a rough review of results.

Method

To determine the residual Neuner of a number, we first calculated the sum of the digits of that number, then the checksum of the checksum, and so on, until ultimately remains a one-digit number. The Nine rest of 5919, for example, 6 and is calculated as follows:

The Penalty residual is calculated similarly, except that the alternating sum of the digits is calculated. In this, the digits are alternately subtracted and added. The Penalty remainder of 5919, for example, 1 and is calculated as follows:

In the last step we check whether the remainder of the original result with the rest of the result of the deputy invoice matches using iterative or iterative alternating cross sums. This is not the case, it has revealed an error in the calculation. If the two results agree, however, the result is in 8 of 9 cases ( nine test ) and in 10 of 11 cases ( Penalty test) correctly. The tested result can only deviate just by multiples of 9 ( or 11) from the actual result. Performs one both tests are completed successfully, the result in 98 of 99 cases is correct. The tested result can differ only by multiples of 99 accurately the actual output.

The method can be equally applied to addition, subtraction and multiplication, in which any possible negative nine residues ( or penalty residues) must be converted to positive by adding 9 ( or 11).

Worked examples

Addition

Subtraction

( with transposed digits )

Note: Penalty sample is here an error, because the sum of the digits in the number 81 has been calculated incorrectly. The checksum for the penalty trial, it is important to make this starting from the one location. The correct checksum would be: 81 → 1-8 = -7 ≡ 4. The penalty trial order no longer leads to a contradiction.

Multiplication

Origin

The process is - probably by Arabic mediation - known since the 12th century in Europe.

In al - Khwarizmi " Algorismus " (9th century ), the nines, but without the use of checksums, discussed for the first time for the duplication and multiplication. The factors and the product is divided by 9 and the remainder is recorded. The remains so determined correspond to the nine residues of the factors or the product.

Restriction

With the nines, you can not, as is often assumed, to demonstrate the correctness, but only the faultiness of an invoice. While the failure of the nines clearly excludes that you have calculated correctly, leaves her success the correctness of the result only become more likely: the result could be wrong, for example, by interchanging the digits despite successful nines. To further test the Penalty sample can be connected, the additional success increases the probability of correctness at 98:99.

Note: It should be noted that in the nine sample itself, of course, can put miscalculation; especially if you use them infrequently ( in the computer age to accept ), you can already easily squander in determining the nine residues in the head.

Mathematical background and other bases

The special significance of Nine - Eleven and sample in the decimal system arises from the fact that the nine remaining simply be calculated as the sum of the digits penalty and the rest as an alternating sum of the digits. In a place value system to the base b can be due to

The samples with the numbers b-1 and b 1 particularly easily. For example, results in the hexadecimal system, the cross sum the 15er group and the alternating sum of the digits of the 17s radical. The 15er and the 17er sample will then see, for example, the bill A1F C02 as follows:

Swell

• Muhammad ibn Musa Alchwarizmi 's Algorismus: The earliest textbook to calculations with Indian numerals: After single ( Latin ) handwriting (. Ms. Ii.6.5 Cambridge Un Lib. ) In facsimile with transcription and commentary - Edited by Kurt Vogel - Otto Zeller: Aalen, 1963.

• Numeracy