Subtraction
The subtraction, colloquially called minus arithmetic, is one of the four basic operations of arithmetic. Under the subtraction means the withdrawal of a number from another. Mathematically, it is in the subtraction by a two-digit shortcut. The subtraction is the inverse operation of addition. The arithmetic sign of the subtraction is the minus sign "-".
- 4.2.1 Part differences
- 4.3.1 striding the difference
- 4.3.2 dissection of the subtrahend
- 4.3.3 In the same change
Language rules and basic properties
For the elements of a subtraction there are the following symbols and ways of speaking:
- The arithmetic sign of the subtraction is the minus sign "-". It was introduced in 1489 by Johannes Widmann.
- The number from which something is subtracted, is called the minuend ( Latin for "the to be reduced ").
- The number that is subtracted is called the subtrahend ( Latin for "the subtracted ").
- The arithmetic expression ( term), which includes the minuend, the minus sign and the subtrahend, ie difference.
- The result of subtracting the value of the difference ( difference value or too short only difference ).
- The symbol for differences than Terme is the Greek letter Delta, " Δ ", which is also used as an operator for the difference ( see below). It is often the difference - especially in everyday language - but only the result of this " negative invoice ", more commonly referred to the amount of this result. Example: The difference between 7 and 9 and the difference 5-3 is 2 in the example, this is " is " by the verb emphasized.
Memory aids ( with consideration of the sign! )
- Minuend minus the subtrahend equal value of the difference.
- Value of the difference = minuend - subtrahend or
- Minuend - subtrahend = value of the difference
- ( Mnemonic: minuend comes in the alphabet before subtrahend )
Examples ( with consideration of the sign! )
The set of natural numbers is not completed with respect to the subtraction, that is, the subtraction may be obtained a result that exceeds the range of natural numbers.
- Example: 1 - 4 = -3
Mathematical definition
The subtraction is the inverse operation of addition. In groups can be at any given and find exactly one, so that:
The provision of means subtraction. can be determined by subtracting from ( " subtract " ):
Is called the minuend, the subtrahend. The result of a subtraction, here called value of the difference. A subtraction is listed with a minus sign:
Calculation aid
Because the subtraction is an addition of the inverse element, a subtraction can also be written in the form of an addition by the subtrahend is previously multiplied by a factor -1:
Written subtraction
The written subtraction is written next to the addition of a basic culture techniques that are learned in the first years of primary school education. Mastery of written subtraction is a prerequisite for learning the long division.
Vertical Subtraction with carries
In primary schools, most procedures are taught today, in which the corresponding positions of the minuend and subtrahend are one above the other. The locations are sequential, mostly from right to left.
When the subtrahend is greater than the minuend, transfers have to be handled. That is, the minuend is to enable the subtraction, plus 10; ( subtracting from right to left addition method ); to compensate for this, either the minuend decreased ( unbundling procedures preliminary calculation of the unearned premiums) must be in the adjacent column to the left or the subtrahend be increased. In German-speaking countries with the addition method, the latter procedure enforced. In 2000 occurred in some states, a new curriculum in force, which now instead of supplementing requires unbundling as standard.
Supplementary Procedures
In addition procedure, which is also called Refill technique or ( in the U.S.) Austrian method ( "Austrian method" ), the subtrahend no subtraction is made, but conversely increased to the minuend. If this is not possible, the minuend is increased by 10. The 10 is not " borrowed", but added than 1 subtrahend to the next part of the calculation. In German-speaking countries, this method is taught in primary schools as a standard method. One of the advantages of the method is that it prepares the handling of tasks in which a plurality minuend subtrahend should be removed.
Example:
The result is written under the dash.
9 ... = 5 The envisaged sum ( 5) is too small!
It is therefore increased by 10. The 1 is written under the next subtrahend.
9 ... = 15 The calculation can now be performed, the result is written under the dash.
(4 1) ... = 7
The result is written under the dash.
The overall results.
Subtraction from left to right
The subtraction can also be done from left to right. In this unusual method which is a variant of the complement method, the carries are processed before the difference is exactly calculated. Since the transfers have to be written down or noticed the method is not only relatively resistant to oversight, but also very fast and even suitable for mental arithmetic.
Example:
Since, in the next column of the minuend is smaller than the subtrahend, the recently calculated value is decremented by 1.
15-9 = 6
Since, in the next column of the minuend is not smaller than the subtrahend, it remains at this value.
3 - 1 = 2
There is a column or a sequence of several columns in which are two of the same numbers, and right next to a column with a minuend, which is smaller than the subtrahend, so must the routine in this process " perspective " not only the same two digits, but also include the subsequent columns. Each column with the same digit then receives a nine instead of a zero as a result.
Ahead across multiple columns in the cases described above is a weak point of this method.
Unbundling procedures
Stripping with " unbundling " means that the too small minuend a " bond " makes at his left-hand neighbor. The minuend is increased by 10, and the left neighbor is decremented by 1. The method is taught in primary schools, for example, the United States as the standard method. Pure computation is similar to the addition method; if it can be " borrowed" from a zero, it must, however, make their own neighbors left a " bond " - a technique that may need to be learned ( the complement method it is not needed ). It must also be written in the unbundling more.
Example:
The result is written under the dash.
5-9 = ... The minuend ( 5) is too small!
It is therefore increased by 10. This 10 is from the left standing next digit ( 7) " borrowed"; it is decremented by 1.
15-9 = ... The subtraction can now be performed. The result is written under the dash.
6-4 = ...
The result is written under the dash.
The overall results.
Pre - unbundling
A variant of the unbundling process is that all sites are fully unbundled in a first operation, so for the second operation, in which only is subtracted yet, sufficiently large minuend are available.
Example:
4-9 = not possible. Therefore, the same procedure as in step 1
Execution of places: 11-3 = 8
14-9 = 5
6 - 4 = 2
Vertical subtraction without transfers
Part differences
The Partial Differences method differs from other vertical subtraction methods in that no transfers are used. They replace part differences, which - depending on whether is greater in a column of the minuend or the subtrahend - get a plus or a minus sign. The sum of the partial differentials results in the total difference.
Example:
The smaller number is subtracted from the larger: 90-50 = 40 Since the subtrahend is greater than the minuend, obtains the difference of a minus sign.
The smaller number is subtracted from the larger: 3 - 1 = 2 Because of the minuend is greater than the subtrahend, the difference given a positive sign.
300-40 2 = 262
Non-vertical method
Striding the difference
The calculation of a difference does not have to be done digit by digit. Most awkward, but it is also possible, auszuschreiten lying between a subtrahend and minuend a number space.
Example:
1234 - 567 = can be calculated using the following steps:
- 567 3 = 570
- 570 30 = 600
- 600 400 = 1000
- 1000 234 = 1234
To determine the difference, the values of the individual steps will be added: 3 30 400 234 = 667
Dissection of the subtrahend
Another approach, which is equally suitable for written subtraction as for the mental arithmetic, is the decomposition of the subtrahend is subtracted from the minuend in single steps.
Example:
" 1234 - 567 =" can be calculated using the following steps:
Same change
Basis of seed change subtraction is the observation that a subtraction is easy to perform if there are one or more zeros at the end of the subtrahend. The subtrahend is therefore increased in this method to the nearest ten or decreased; as the minuend is increased or decreased by the same difference that manipulation does not affect the difference. If the task then is still too difficult, the operation can be repeated.
Example:
" 1234 - 567 =" can be calculated using the following steps: