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48-432: Sub-zero literally means "beneath zero". As such, it is usually used for negative numbers ; the most common usage refers to negative temperature . Sub-zero can also refer to: Negative number In mathematics , a negative number is the opposite (mathematics) of a positive real number . Equivalently, a negative number is a real number that is less than zero . Negative numbers are often used to represent

96-675: A r r y ⟵ M i n u e n d ⟵ S u b t r a h e n d ⟵ R e s t o r D i f f e r e n c e {\displaystyle {\begin{array}{rrrr}&\color {Red}-1\\&C&D&U\\&7&0&4\\&5&1&2\\\hline &1&9&2\\\end{array}}{\begin{array}{l}{\color {Red}\longleftarrow {\rm {carry}}}\\\\\longleftarrow \;{\rm {Minuend}}\\\longleftarrow \;{\rm {Subtrahend}}\\\longleftarrow {\rm {Rest\;or\;Difference}}\\\end{array}}} The minuend

144-508: A difference left-to-right, the result is the negative of the original result. Symbolically, if a and b are any two numbers, then Subtraction is non-associative , which comes up when one tries to define repeated subtraction. In general, the expression can be defined to mean either ( a − b ) − c or a − ( b − c ), but these two possibilities lead to different answers. To resolve this issue, one must establish an order of operations , with different orders yielding different results. In

192-453: A larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example, since 8 − 5 = 3 . The minus sign "−" signifies the operator for both the binary (two- operand ) operation of subtraction (as in y − z ) and the unary (one-operand) operation of negation (as in − x , or twice in −(− x ) ). A special case of unary negation occurs when it operates on

240-403: A negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that losing a debt is the same thing as gaining a credit.) Thus and When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules: Thus and The reason behind the first example

288-404: A number. Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication . All of these rules can be proven , starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that follow these patterns are studied in abstract algebra . In computability theory , considering subtraction

336-456: A positive number, in which case the result is a negative number (as in −5 ). The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize

384-403: A real number (the subtrahend) from another (the minuend) can then be defined as the addition of the minuend and the additive inverse of the subtrahend. For example, 3 − π = 3 + (− π ) . Alternatively, instead of requiring these unary operations, the binary operations of subtraction and division can be taken as basic. Subtraction is anti-commutative , meaning that if one reverses the terms in

432-548: A study—claiming that crutches were beneficial to students using this method. This system caught on rapidly, displacing the other methods of subtraction in use in America at that time. Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country. Both these methods break up

480-403: Is 704, the subtrahend is 512. The minuend digits are m 3 = 7 , m 2 = 0 and m 1 = 4 . The subtrahend digits are s 3 = 5 , s 2 = 1 and s 1 = 2 . Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one's place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9,

528-409: Is added to x and one is added to the sum. The leading digit "1" of the result is then discarded. The method of complements is especially useful in binary (radix 2) since the ones' complement is very easily obtained by inverting each bit (changing "0" to "1" and vice versa). And adding 1 to get the two's complement can be done by simulating a carry into the least significant bit. For example: becomes

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576-424: Is considered to be less than negative 5 : In the context of negative numbers, a number that is greater than zero is referred to as positive . Thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative,

624-401: Is different from other vertical subtraction methods because no borrowing or carrying takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller than the subtrahend. The sum of the partial differences is the total difference. Example: Instead of finding the difference digit by digit, one can count up the numbers between the subtrahend and

672-502: Is extended in either the fourth or fifth grade to include decimal representations of fractional numbers. Almost all American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after William A. Brownell published

720-551: Is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit m i +1 by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit s i +1 by one. Example: 704 − 512. − 1 C D U 7 0 4 5 1 2 1 9 2 ⟵ c

768-412: Is not well-defined over natural numbers , operations between numbers are actually defined using "truncated subtraction" or monus . Subtraction is usually written using the minus sign "−" between the terms; that is, in infix notation . The result is expressed with an equals sign . For example, There are also situations where subtraction is "understood", even though no symbol appears: Formally,

816-468: Is simple: adding three −2 's together yields −6 : The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six: The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the distributive law . In this case, we know that Since 2 × (−3) = −6 ,

864-413: Is usually ( but not always ) thought of as neither positive nor negative . The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign . Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while

912-421: Is very similar to addition of two positive numbers. For example, The idea is that two debts can be combined into a single debt of greater magnitude. When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example: In the first example, a credit of 8 is combined with a debt of 3 , which yields a total credit of 5 . If

960-426: Is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192. The Austrian method does not reduce

1008-461: The gerundive suffix -nd results in "subtrahend", "thing to be subtracted". Likewise, from minuere "to reduce or diminish", one gets "minuend", which means "thing to be diminished". Imagine a line segment of length b with the left end labeled a and the right end labeled c . Starting from a , it takes b steps to the right to reach c . This movement to the right is modeled mathematically by addition : From c , it takes b steps to

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1056-429: The left to get back to a . This movement to the left is modeled by subtraction: Now, a line segment labeled with the numbers 1 , 2 , and 3 . From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3 . It takes 2 steps to the left to get to position 1, so 3 − 2 = 1 . This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation,

1104-543: The magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative . Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that

1152-405: The natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers ( a , b ). We can extend addition and multiplication to these pairs with the following rules: We define an equivalence relation ~ upon these pairs with the following rule: This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be

1200-413: The quotient set N ²/~, i.e. we identify two pairs ( a , b ) and ( c , d ) if they are equivalent in the above sense. Note that Z , equipped with these operations of addition and multiplication, is a ring , and is in fact, the prototypical example of a ring. Subtraction In a sense, subtraction is the inverse of addition. That is, c = a − b if and only if c + b = a . In words:

1248-432: The relative change between the two quantities as a percentage, while percentage point change is simply the number obtained by subtracting the two percentages. As an example, suppose that 30% of widgets made in a factory are defective. Six months later, 20% of widgets are defective. The percentage change is ⁠ 20% − 30% / 30% ⁠ = − ⁠ 1 / 3 ⁠ = ⁠−33 + 1 / 3 ⁠ %, while

1296-414: The 7 to 6. Rather it increases the subtrahend hundreds digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundreds place. There is an additional subtlety in that the student always employs a mental subtraction table in

1344-444: The American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7. Example: Example: In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If

1392-425: The common-sense idea of an opposite is reflected in arithmetic. For example, − ‍ (−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three". Conversely, a number that is greater than zero is called positive ; zero

1440-400: The concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz held that negative numbers were invalid, but still used them in calculations. The relationship between negative numbers, positive numbers, and zero is often expressed in

1488-538: The context of integers, subtraction of one also plays a special role: for any integer a , the integer ( a − 1) is the largest integer less than a , also known as the predecessor of a . When subtracting two numbers with units of measurement such as kilograms or pounds , they must have the same unit. In most cases, the difference will have the same unit as the original numbers. Changes in percentages can be reported in at least two forms, percentage change and percentage point change. Percentage change represents

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1536-414: The definition of negation to include zero and negative numbers. Specifically: For example, the negation of −3 is +3 . In general, The absolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3 , and the absolute value of 0 is 0 . In a similar manner to rational numbers , we can extend

1584-449: The difference of two numbers is the number that gives the first one when added to the second one. Subtraction follows several important patterns. It is anticommutative , meaning that changing the order changes the sign of the answer. It is also not associative , meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Because 0 is the additive identity , subtraction of it does not change

1632-428: The form of a number line : Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are lesser. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example, even though (positive) 8 is greater than (positive) 5 , written negative 8

1680-510: The left from 3 to get to −1: Subtraction of natural numbers is not closed : the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches: The field of real numbers can be defined specifying only two binary operations, addition and multiplication, together with unary operations yielding additive and multiplicative inverses. The subtraction of

1728-466: The line must be extended. To subtract arbitrary natural numbers , one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0 . But 3 − 4 is still invalid, since it again leaves the line. The natural numbers are not a useful context for subtraction. The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). This way, it takes 4 steps to

1776-425: The negative number has greater magnitude, then the result is negative: Here the credit is less than the debt, so the net result is a debt. As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer: In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus and On the other hand, subtracting

1824-487: The number being subtracted is known as the subtrahend , while the number it is subtracted from is the minuend . The result is the difference . That is, All of this terminology derives from Latin . " Subtraction " is an English word derived from the Latin verb subtrahere , which in turn is a compound of sub "from under" and trahere "to pull". Thus, to subtract is to draw from below , or to take away . Using

1872-401: The percentage point change is −10 percentage points. The method of complements is a technique used to subtract one number from another using only the addition of positive numbers. This method was commonly used in mechanical calculators , and is still used in modern computers . To subtract a binary number y (the subtrahend) from another number x (the minuend), the ones' complement of y

1920-516: The period of the Chinese Han dynasty (202 BC – AD 220), but may well contain much older material. Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients . Prior to

1968-533: The positive and negative whole numbers (together with zero) are referred to as integers . (Some definitions of the natural numbers exclude zero.) In bookkeeping , amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers were used in the Nine Chapters on the Mathematical Art , which in its present form dates from

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2016-420: The product (−2) × (−3) must equal 6 . These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign of a as follows: The justification for why the product of two negative numbers is a positive number can be observed in the analysis of complex numbers . The sign rules for division are the same as for multiplication. For example, and If dividend and divisor have

2064-560: The same sign, the result is positive, if they have different signs the result is negative. The negative version of a positive number is referred to as its negation . For example, −3 is the negation of the positive number 3 . The sum of a number and its negation is equal to zero: That is, the negation of a positive number is the additive inverse of the number. Using algebra , we may write this principle as an algebraic identity : This identity holds for any positive number x . It can be made to hold for all real numbers by extending

2112-411: The subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of the subtrahend: from the minuend where each s i and m i is a digit, proceeds by writing down m 1 − s 1 , m 2 − s 2 , and so forth, as long as s i does not exceed m i . Otherwise, m i is increased by 10 and some other digit

2160-514: The sum: Dropping the initial "1" gives the answer: 01001110 (equals decimal 78) Methods used to teach subtraction to elementary school vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States as traditional mathematics , a specific process is taught to students at the end of the 1st year (or during the 2nd year) for use with multi-digit whole numbers, and

2208-417: The term nonnegative is sometimes used to refer to a number that is either positive or zero, while nonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number. Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: In general, the subtraction of

2256-415: The top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is "borrowed" from the top digit to the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example: A variant of the American method where all borrowing is done before all subtraction. Example: The partial differences method

2304-560: The unary "−" along with its operand. For example, the expression 7 + −5 may be clearer if written 7 + (−5) (even though they mean exactly the same thing formally). The subtraction expression 7 – 5 is a different expression that doesn't represent the same operations, but it evaluates to the same result. Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in Addition of two negative numbers

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