33:
333:
598:
943:
3922:
201:
823:
2025:(gain)—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (
781:
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
2628:
Contributions to the
Geology of the Bering Sea Basin and Adjacent Regions: Selected Papers from the Symposium on the Geology and Geophysics of the Bering Sea Region, on the Occasion of the Inauguration of the C. T. Elvey Building, University of Alaska, June 26-28, 1970, and from the 2d International
2612:
Wind assistance is normally expressed in meters per second, either positive or negative. A positive measurement means that the wind is helping the runners and a negative measurement means that the runners had to work against the wind. So, for example, winds of −2.2m/s and +1.9m/s are legal, while a
1895:
carried out calculations with negative numbers, using "+" as a negative sign. The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century, Hoernle dates it between the third and fourth centuries, Ayyangar and
Pingree dates it to the 8th or 9th centuries, and George
1880:
and black rods for negative. This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. Liu Hui writes:
2556:
Delta time: A term used to describe the time difference between two different laps or two different cars. For example, there is usually a negative delta between a driver's best practice lap time and his best qualifying lap time because he uses a low fuel load and new
208:
Numbers appearing farther to the right on this line are greater (but not bigger), while numbers appearing farther to the left are lesser (but not smaller). Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left.
32:
2037:
gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."
1786:
For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false".
2530:
But in the third minute of stoppage time, the striker turned in Luke Murphy's cross from eight yards to earn a third straight League One win for Hill's side, who started the campaign on −12 points after going into administration in
1998:
1011:
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:
75:
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
2613:
wind of +2.1m/s is too much assistance and considered illegal. The terms "tailwind" and "headwind" are also frequently used. A tailwind pushes the runners forward (+) while a headwind pushes the runners backwards (−)
1843:, saying that the equation was absurd. For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others.
2220:
in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with
702:
if they have worked fewer total hours than contracted to that point. Employees may be able to take more than their annual holiday allowance in a year, and carry forward a negative balance to the next year.
411:
may be given as the difference compared to a previous lap (or sector) (such as the previous record, or the lap just completed by a driver in front), and will be positive if slower and negative if faster.
1726:
501:. When a battery is connected in reverse polarity, the voltage applied is said to be the opposite of its rated voltage. For example, a 6-volt battery connected in reverse applies a voltage of −6 volts.
1647:
The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero.
103:
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
391:: the difference in total goals scored for the team (+) and against the team (−) when a particular player is on the ice is the player's +/− rating. Players can have a negative (+/−) rating.
2742:
302:
In general, the subtraction of 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,
1885:
Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative.
1929:
1864:(c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese
625:, the screen display may show the time remaining as a negative number, which increases up to zero time remaining at the same rate as the time already played increases from zero.
1982:
3419:
3734:
3657:
3618:
3580:
3552:
3524:
3496:
3384:
3351:
3323:
3295:
658:
has a negative money score – contestants play for an amount of money and any incorrect answer that costs them more than what they have now can result in a negative score.
2186:
2144:
341:
2218:
1835:
115:. 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
1923:
were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.
804:
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
92:
scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −
3077:
1868:
made it easier for the
Chinese to accept the idea of negative numbers. The Chinese were able to solve simultaneous equations involving negative numbers. The
842:
When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:
404:
Clubs may be deducted points for breaches of the laws, and thus have a negative points total until they have earned at least that many points that season.
170:
considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like
2569:
263:
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
912:
On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that
2843:
17:
2321:
The convention that zero is neither positive nor negative is not universal. For example, in the French convention, zero is considered to be
691:, a negative number indicates loss of life, damage, a score penalty, or consumption of a resource, depending on the genre of the simulation.
2733:
2029:), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.
1669:
1228:. It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. Specifically:
137:, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.
2791:
530:
can include negative balances, indicated either by a minus sign or by enclosing the balance in parentheses. Examples include bank account
668:
s pricing game Buy or Sell, if an amount of money is lost that is more than the amount currently in the bank, it incurs a negative score.
2626:
774:). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in
681:
3111:
Nicolas
Chuquet, Renaissance Mathematician: a study with extensive translations of Chuquet's mathematical manuscript completed in 1484
2654:
3946:
2100:, provided the first satisfactory treatment of negative numbers in Europe. He did not allow negative numbers in his consideration of
2305:"Integers are the set of whole numbers and their opposites.", Richard W. Fisher, No-Nonsense Algebra, 2nd Edition, Math Essentials,
3245:
896:
In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus
154:(c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as
1993:
826:
A visual representation of the addition of positive and negative numbers. Larger balls represent numbers with greater magnitude.
2691:
2379:
1848:
142:
3434:
1152:
If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.
663:
3429:
2310:
1933:(from which the word "algebra" derives) did not use negative numbers or negative coefficients. But within fifty years,
1112:
The justification for why the product of two negative numbers is a positive number can be observed in the analysis of
3175:
3146:
3119:
3093:
3061:
3038:
2985:
2961:
2917:
2885:
2664:
2637:
2422:
1029:
The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the
122:
Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as
2345:
Struik, pages 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
947:
712:
are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one
3389:
582:
A negative number in financing is synonymous with "debt" and "deficit" which are also known as "being in the red".
3807:
3885:
782:
symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.
3951:
2999:
888:
As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:
3196:
1636:
1463:
This equivalence relation is compatible with the addition and multiplication defined above, and we may define
162:
further developed the rules of subtracting and multiplying negative numbers and solved problems with negative
3768:
2280:
962:
When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The
294:
of a larger number from a smaller. For example, negative three is the result of subtracting three from zero:
126:(i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as
3238:
2095:
2005:
By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve
801:
is a different expression that doesn't represent the same operations, but it evaluates to the same result.
597:
576:
212:
Note that a negative number with greater magnitude is considered less. For example, even though (positive)
3210:
3201:
2934:
2044:
allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of
2041:
193:
The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a
3394:
2255:
332:
2573:
2953:
Science and
Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth
2909:
Science and
Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth
1940:
3802:
3758:
171:
3400:
2502:
Mark McCall's side have subsequently dropped from third to bottom of the
Premiership with −22 points
3925:
3797:
2543:
1267:
of a number is the non-negative number with the same magnitude. For example, the absolute value of
830:
Addition of two negative numbers is very similar to addition of two positive numbers. For example,
61:
3717:
3640:
3601:
3563:
3535:
3507:
3479:
3367:
3334:
3306:
3278:
942:
3231:
2595:
951:
738:
384:
68:
3022:
2149:
2107:
2371:
1920:
1125:
730:
542:
159:
53:
3109:
2681:
2191:
3870:
3706:
3136:
1908:
1805:
705:
618:
492:
84:. Negative numbers are used to describe values on a scale that goes below zero, such as the
3623:
3356:
1891:
1422:
726:
646:
354:
100:
2877:
2870:
2330:
8:
3834:
3744:
3701:
3683:
3461:
3086:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures
3031:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures
2079:
2051:
2006:
709:
527:
514:
498:
255:
In the context of negative numbers, a number that is greater than zero is referred to as
282:
is used to refer to a number that is either negative or zero. Zero is a neutral number.
3739:
3451:
3054:
Lost
Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas
2978:
Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas
2472:
2364:
2275:
1900:
1899:
During the 7th century AD, negative numbers were used in India to represent debts. The
1865:
1781:
1495:
1203:
963:
568:
423:
250:
116:
2570:"BBC Sport - Olympic Games - London 2012 - Men's Long Jump : Athletics - Results"
1999:
A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen
1328:). We can extend addition and multiplication to these pairs with the following rules:
650:
have a negative score if their first answers are incorrect and interrupt the question.
200:
3897:
3860:
3824:
3763:
3749:
3444:
3424:
3171:
3142:
3115:
3089:
3057:
3034:
2981:
2957:
2913:
2881:
2687:
2660:
2633:
2418:
2375:
2326:
2306:
1913:
1912:(written c. AD 630), discussed the use of negative numbers to produce a general form
1896:
Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century,
1795:
1791:
415:
3218:
839:
The idea is that two debts can be combined into a single debt of greater magnitude.
378:
3956:
3915:
3844:
3819:
3753:
3662:
3628:
3469:
3439:
3361:
3264:
2486:
2260:
2240:
2091:
1563:
1192:
1167:
1161:
1030:
756:
674:
419:
394:
946:
A multiplication by a negative number can be seen as a change of direction of the
3792:
3696:
3328:
2951:
2907:
2245:
2222:
2060:
1302:
1113:
564:
435:
401:: the run differential is negative if the team allows more runs than they scored.
350:
2514:
2333:
mean the same as English "positive or zero" and "negative or zero" respectively.
3839:
3829:
3814:
3633:
3501:
3272:
2101:
2070:
2064:
2010:
1306:
1264:
955:
585:
572:
510:
466:
362:
123:
2844:"Popularity of Miliband and Clegg falls to lowest levels recorded by ICM poll"
2487:"Saracens salary cap breach: Premiership champions will not contest sanctions"
2034:
1296:
278:
is sometimes used to refer to a number that is either positive or zero, while
174:
held that negative numbers were invalid, but still used them in calculations.
3940:
3902:
3875:
3784:
3081:
3026:
2816:
2764:
2706:
2270:
1873:
867:. If the negative number has greater magnitude, then the result is negative:
112:
3865:
3667:
2956:(reprint ed.). Cambridge: Cambridge University Press. pp. 90–91.
2265:
1924:
1468:
1317:
614:
488:
366:
163:
1992:
that "negative quantities must be counted as terms". In the 10th century,
3691:
3473:
2442:
2235:
2046:
1903:
1877:
1857:
1839:
1592:
1502:
794:
742:
688:
452:
427:
291:
260:
188:
155:
147:
134:
108:
57:
45:
673:
The change in support for a political party between elections, known as
3672:
3529:
2912:(reprint ed.). Cambridge: Cambridge University Press. p. 90.
2437:
1799:
1235:
The negation of a negative number is the corresponding positive number.
970:
The product of one positive number and one negative number is negative.
629:
472:
388:
167:
89:
64:
37:
1494:, equipped with these operations of addition and multiplication, is a
1057:
These rules lead to another (equivalent) rule—the sign of any product
2629:
Symposium on Arctic Geology Held in San Francisco, February 1-4, 1971
2415:
The Development of Arabic Mathematics: Between Arithmetic and Algebra
1985:
1934:
1802:
in the 3rd century AD referred to an equation that was equivalent to
699:
654:
557:
553:
546:
531:
480:
462:
431:
408:
264:
880:
Here the credit is less than the debt, so the net result is a debt.
96:(−3) = 3 because the opposite of an opposite is the original value.
3780:
3711:
3557:
2820:
2768:
2250:
1179:
713:
695:
588:
can be negative, when the lender is charged to deposit their money.
535:
484:
458:
439:
398:
166:. Prior to the concept of negative numbers, mathematicians such as
3300:
3223:
1861:
1199:
734:
370:
151:
146:, which in its present form dates from the period of the Chinese
127:
85:
1937:
illustrated the rules of signs for expanding the multiplication
1166:
The negative version of a positive number is referred to as its
822:
3254:
1037:(−2) × (−3) + 2 × (−3) = (−2 + 2) × (−3) = 0 × (−3) = 0
607:
476:
358:
2792:"Think negative interest rates can't happen here? Think again"
1860:, but may well contain much older material. The mathematician
1746:. Using the law of cancellation for addition, it is seen that
1846:
Negative numbers appear for the first time in history in the
793:(even though they mean exactly the same thing formally). The
150:(202 BC – AD 220), but may well contain much older material.
994:
The reason behind the first example is simple: adding three
2867:
2074:
1290:
622:
504:
483:, which can be negative (e.g. the surface elevation of the
374:
72:
60:. Equivalently, a negative number is a real number that is
130:. (Some definitions of the natural numbers exclude zero.)
290:
Negative numbers can be thought of as resulting from the
2943:
720:
545:
might be negative, which is one indicator of being in a
274:
Because zero is neither positive nor negative, the term
3219:
Math Forum: Ask Dr. Math FAQ: Negative Times a Negative
2901:
2899:
2897:
2366:
A History of Mathematics: From Mesopotamia to Modernity
1490:) if they are equivalent in the above sense. Note that
637:
1498:, and is in fact, the prototypical example of a ring.
3720:
3643:
3604:
3566:
3538:
3510:
3482:
3403:
3370:
3337:
3309:
3281:
3170:. Berlin, Heidelberg, and New York: Springer-Verlag.
3088:, vol. 1 (2nd ed.), Springer, p. 115,
2194:
2152:
2110:
1943:
1808:
1721:{\displaystyle x+y'=0,\quad {\text{and}}\quad x+y=0.}
1672:
966:
of the product is determined by the following rules:
2894:
317:
2673:
67:. Negative numbers are often used to represent the
3728:
3651:
3612:
3574:
3546:
3518:
3490:
3413:
3378:
3345:
3317:
3289:
2869:
2732:
2363:
2212:
2180:
2138:
1976:
1829:
1720:
1191:That is, the negation of a positive number is the
1128:are the same as for multiplication. For example,
2460:. Oxford University Press, New York. p. 252.
2458:Mathematical Thought from Ancient to Modern Times
3938:
2596:"How Wind Assistance Works in Track & Field"
2386:Liu is explicit on this; at the point where the
973:The product of two negative numbers is positive.
438:is measured and recorded, and is positive for a
2679:
2656:Statistical Methods in the Atmospheric Sciences
1182:of a number and its negation is equal to zero:
285:
2939:. Cambridge University Press. pp. cxxiii.
2632:. Geological Society of America. p. 194.
507:have a positive or negative electrical charge.
3239:
3080:(2008), "Algebra in Islamic Mathematics", in
3002:. The MacTutor History of Mathematics archive
513:of an AM broadcast tower used in multi-tower
455:which are colder than 0 °C or 0 °F.
158:were describing the use of negative numbers.
3107:
2734:"First negative inflation figure since 1960"
2225:, but understandably liked them even less.)
1856:), which in its present form dates from the
1425:~ upon these pairs with the following rule:
1222:This identity holds for any positive number
541:The annual percentage growth in a country's
475:features of the earth's surface are given a
99:Negative numbers are usually written with a
3014:
2949:
2905:
1635:This construction is a special case of the
755:) and the unary (one-operand) operation of
27:Real number that is strictly less than zero
3921:
3246:
3232:
3141:, Greenwood Publishing Group, p. 56,
3076:
2341:
2339:
2067:but referred to them as "absurd numbers".
1758:is equal to any other additive inverse of
641:often finish with a negative points score.
517:arrays, which can be positive or negative.
36:This thermometer is indicating a negative
3722:
3645:
3606:
3568:
3540:
3512:
3484:
3372:
3339:
3311:
3283:
3114:, D. Reidel Publishing Co., p. 354,
3108:Flegg, Graham; Hay, C.; Moss, B. (1985),
3072:
3070:
2515:"Bolton Wanderers 1−0 Milton Keynes Dons"
2357:
2355:
2353:
2351:
3138:Famous Problems and Their Mathematicians
2470:
2408:
2406:
2404:
2402:
2400:
2398:
2325:positive and negative. The French words
2063:, a Frenchman, used negative numbers as
1996:considered debts as negative numbers in
1291:Formal construction of negative integers
941:
821:
596:
31:
3134:
3020:
2841:
2680:Carysforth, Carol; Neild, Mike (2002),
2390:give a detailed and helpful 'Sign Rule'
2361:
2336:
1174:is the negation of the positive number
560:), indicating a fall in average prices.
232:is considered to be less than negative
14:
3939:
3168:Elements of the History of Mathematics
3067:
2997:
2868:Grant P. Wiggins; Jay McTighe (2005).
2817:"Swiss interest rate to turn negative"
2624:
2412:
2348:
1583:), a multiplicative unit of the form (
1006:(−2) × 3 = (−2) + (−2) + (−2) = −6
601:Negative story numbers in an elevator.
344:Negative golf scores relative to par.
3227:
3206:, on "Negative Numbers", 9 March 2006
3033:, vol. 1, Springer, p. B2,
2950:Needham, Joseph; Wang, Ling (1995) .
2932:
2906:Needham, Joseph; Wang, Ling (1995) .
2652:
2455:
2395:
1849:Nine Chapters on the Mathematical Art
721:Arithmetic involving negative numbers
698:may have a negative balance on their
610:in a building below the ground floor.
143:Nine Chapters on the Mathematical Art
111:) thought of as neither positive nor
2765:"ECB imposes negative interest rate"
2625:Forbes, Robert B. (6 January 1975).
1202:, we may write this principle as an
3435:Set-theoretically definable numbers
2653:Wilks, Daniel S. (6 January 2018).
2370:. Oxford University Press. p.
2104:, so he had to treat, for example,
2073:dealt with negative numbers in his
1837:(which has a negative solution) in
24:
3406:
3253:
3211:Endless Examples & Exercises:
3056:. New York: Simon & Schuster.
2980:. New York: Simon & Schuster.
2707:"UK economy shrank at end of 2012"
1766:is the unique additive inverse of
855:In the first example, a credit of
199:
182:
140:Negative numbers were used in the
25:
3968:
3197:Maseres' biographical information
3190:
2842:Wintour, Patrick (17 June 2014).
2745:from the original on 18 June 2022
2017:the product of a negative number—
1916:similar to the one in use today.
1658:be its additive inverse. Suppose
1474:²/~, i.e. we identify two pairs (
937:
863:, which yields a total credit of
318:Everyday uses of negative numbers
244:
3947:Chinese mathematical discoveries
3920:
3183:A Concise History of Mathematics
2857:– via www.theguardian.com.
2789:
2050:, 1202) and later as losses (in
1977:{\displaystyle (a\pm b)(c\pm d)}
331:
3213:Operations with Signed Integers
3185:. New York: Dover Publications.
3159:
3128:
3101:
3046:
2991:
2970:
2926:
2861:
2835:
2809:
2783:
2757:
2725:
2699:
2646:
2618:
2588:
2572:. 5 August 2012. Archived from
2562:
2536:
2507:
1702:
1696:
1662:is another additive inverse of
1104:is the opposite of the sign of
908:(−3) − 5 = (−3) + (−5) = −8
177:
3414:{\displaystyle {\mathcal {P}}}
2659:. Academic Press. p. 17.
2479:
2464:
2449:
2431:
2315:
2299:
2033:In the 12th century in India,
1971:
1959:
1956:
1944:
1096:is negative, then the sign of
1077:is positive, then the sign of
1033:. In this case, we know that
958:of the product of the factors.
932:(−5) − (−8) = (−5) + 8 = 3
883:
534:and business losses (negative
13:
1:
3769:Plane-based geometric algebra
2876:. ACSD Publications. p.
2473:"History of Negative Numbers"
2287:
2281:Signed number representations
1782:Complex number § History
1642:
1239:For example, the negation of
18:Positive and negative numbers
3729:{\displaystyle \mathbb {S} }
3652:{\displaystyle \mathbb {C} }
3613:{\displaystyle \mathbb {R} }
3575:{\displaystyle \mathbb {O} }
3547:{\displaystyle \mathbb {H} }
3519:{\displaystyle \mathbb {C} }
3491:{\displaystyle \mathbb {R} }
3379:{\displaystyle \mathbb {A} }
3346:{\displaystyle \mathbb {Q} }
3318:{\displaystyle \mathbb {Z} }
3290:{\displaystyle \mathbb {N} }
2417:. Springer. pp. 36–37.
2292:
2083:, where he also called them
2021:(loss)—by a positive number—
1279:, and the absolute value of
916:a debt is the same thing as
785:For example, the expression
442:and negative for a headwind.
286:As the result of subtraction
7:
2413:Rashed, R. (30 June 1994).
2256:Positive and negative parts
2228:
1316:by defining integers as an
1297:Integer § Construction
1232:The negation of 0 is 0, and
1155:
1119:
1085:is the same as the sign of
859:is combined with a debt of
817:
216:is greater than (positive)
71:of a loss or deficiency. A
10:
3973:
3166:Bourbaki, Nicolas (1998).
3000:"The Bakhshali manuscript"
2686:, Heinemann, p. 375,
2181:{\displaystyle x^{3}=ax+b}
2139:{\displaystyle x^{3}+ax=b}
1779:
1773:
1294:
1271:and the absolute value of
1159:
789:may be clearer if written
521:
491:, or the elevation of the
446:
271:denotes a positive three.
248:
186:
3911:
3853:
3779:
3759:Algebra of physical space
3681:
3589:
3460:
3262:
2933:Heath, Thomas L. (1897).
1637:Grothendieck construction
900:5 − 8 = 5 + (−8) = −3
875:2 + (−7) = 2 − 7 = −5
871:(−8) + 3 = 3 − 8 = −5
733:for both the binary (two-
461:south of the equator and
407:Lap (or sector) times in
40:temperature (−4 °F).
3815:Extended complex numbers
3798:Extended natural numbers
3181:Struik, Dirk J. (1987).
2998:Pearce, Ian (May 2002).
2213:{\displaystyle a,b>0}
924:3 − (−5) = 3 + 5 = 8
850:(−2) + 7 = 7 − 2 = 5
846:8 + (−3) = 8 − 3 = 5
592:
552:Occasionally, a rate of
322:
3021:Hayashi, Takao (2008),
2936:The works of Archimedes
2872:Understanding by design
1994:Abū al-Wafā' al-Būzjānī
1909:Brahma-Sphuta-Siddhanta
1830:{\displaystyle 4x+20=4}
1591:), and a definition of
1301:In a similar manner to
1065:depends on the sign of
361:; points difference in
3871:Transcendental numbers
3730:
3707:Hyperbolic quaternions
3653:
3614:
3576:
3548:
3520:
3492:
3415:
3380:
3347:
3319:
3291:
3052:Teresi, Dick. (2002).
3023:"Bakhshālī Manuscript"
2976:Teresi, Dick. (2002).
2456:Kline, Morris (1972).
2362:Hodgkin, Luke (2005).
2214:
2182:
2140:
2031:
1978:
1921:Islamic mathematicians
1887:
1831:
1722:
959:
827:
696:flexible working hours
602:
563:The daily change in a
205:
160:Islamic mathematicians
54:opposite (mathematics)
41:
3952:Elementary arithmetic
3803:Extended real numbers
3731:
3654:
3624:Split-complex numbers
3615:
3577:
3549:
3521:
3493:
3416:
3381:
3357:Constructible numbers
3348:
3320:
3292:
3135:Johnson, Art (1999),
2215:
2183:
2141:
2059:In the 15th century,
2015:
1979:
1930:Al-jabr wa'l-muqabala
1883:
1832:
1723:
1549:This will lead to an
1501:We can also define a
945:
825:
619:portable media player
600:
493:Thames Tideway Tunnel
203:
35:
3835:Supernatural numbers
3745:Multicomplex numbers
3718:
3702:Dual-complex numbers
3641:
3602:
3564:
3536:
3508:
3480:
3462:Composition algebras
3430:Arithmetical numbers
3401:
3368:
3335:
3307:
3279:
3078:Bin Ismail, Mat Rofa
2192:
2150:
2108:
2007:polynomial divisions
1941:
1919:In the 9th century,
1901:Indian mathematician
1892:Bakhshali Manuscript
1806:
1670:
1654:be a number and let
1423:equivalence relation
1320:of natural numbers (
1305:, we can extend the
647:University Challenge
528:Financial statements
355:association football
3740:Split-biquaternions
3452:Eisenstein integers
3390:Closed-form numbers
3202:BBC Radio 4 series
2080:Arithmetica Integra
1889:The ancient Indian
1876:to denote positive
1124:The sign rules for
998:'s together yields
710:electronic keyboard
515:directional antenna
499:Electrical circuits
377:scores relative to
107:; zero is usually (
3898:Profinite integers
3861:Irrational numbers
3726:
3649:
3610:
3572:
3544:
3516:
3488:
3445:Gaussian rationals
3425:Computable numbers
3411:
3376:
3343:
3315:
3287:
2823:. 18 December 2014
2276:Sign (mathematics)
2210:
2178:
2136:
2087:(absurd numbers).
1974:
1866:natural philosophy
1854:Jiǔ zhāng suàn-shù
1827:
1718:
1275:are both equal to
1204:algebraic identity
960:
834:(−3) + (−5) = −8
828:
729:"−" signifies the
664:The Price Is Right
603:
569:stock market index
251:Sign (mathematics)
206:
42:
3934:
3933:
3845:Superreal numbers
3825:Levi-Civita field
3820:Hyperreal numbers
3764:Spacetime algebra
3750:Geometric algebra
3663:Bicomplex numbers
3629:Split-quaternions
3470:Division algebras
3440:Gaussian integers
3362:Algebraic numbers
3265:definable numbers
2741:. 21 April 2009.
2713:. 25 January 2013
2693:978-0-435-44746-5
2381:978-0-19-152383-0
1914:quadratic formula
1792:Hellenistic Egypt
1700:
1666:. By definition,
1529:) if and only if
1444:) if and only if
1147:(−8) ÷ (−2) = 4
989:(−2) × (−3) = 6
810: gives
606:The numbering of
556:may be negative (
16:(Redirected from
3964:
3924:
3923:
3891:
3881:
3793:Cardinal numbers
3754:Clifford algebra
3735:
3733:
3732:
3727:
3725:
3697:Dual quaternions
3658:
3656:
3655:
3650:
3648:
3619:
3617:
3616:
3611:
3609:
3581:
3579:
3578:
3573:
3571:
3553:
3551:
3550:
3545:
3543:
3525:
3523:
3522:
3517:
3515:
3497:
3495:
3494:
3489:
3487:
3420:
3418:
3417:
3412:
3410:
3409:
3385:
3383:
3382:
3377:
3375:
3352:
3350:
3349:
3344:
3342:
3329:Rational numbers
3324:
3322:
3321:
3316:
3314:
3296:
3294:
3293:
3288:
3286:
3248:
3241:
3234:
3225:
3224:
3153:
3151:
3132:
3126:
3124:
3105:
3099:
3098:
3074:
3065:
3050:
3044:
3043:
3018:
3012:
3011:
3009:
3007:
2995:
2989:
2974:
2968:
2967:
2947:
2941:
2940:
2930:
2924:
2923:
2903:
2892:
2891:
2875:
2865:
2859:
2858:
2856:
2854:
2839:
2833:
2832:
2830:
2828:
2813:
2807:
2806:
2804:
2802:
2787:
2781:
2780:
2778:
2776:
2761:
2755:
2754:
2752:
2750:
2736:
2729:
2723:
2722:
2720:
2718:
2703:
2697:
2696:
2677:
2671:
2670:
2650:
2644:
2643:
2622:
2616:
2615:
2609:
2607:
2592:
2586:
2585:
2583:
2581:
2576:on 5 August 2012
2566:
2560:
2559:
2553:
2551:
2540:
2534:
2533:
2527:
2525:
2511:
2505:
2504:
2499:
2497:
2483:
2477:
2476:
2468:
2462:
2461:
2453:
2447:
2435:
2429:
2428:
2410:
2393:
2392:
2369:
2359:
2346:
2343:
2334:
2319:
2313:
2303:
2261:Rational numbers
2241:Additive inverse
2219:
2217:
2216:
2211:
2187:
2185:
2184:
2179:
2162:
2161:
2146:separately from
2145:
2143:
2142:
2137:
2120:
2119:
2092:Gerolamo Cardano
2027:martaba khāliyya
1983:
1981:
1980:
1975:
1836:
1834:
1833:
1828:
1727:
1725:
1724:
1719:
1701:
1698:
1686:
1631:
1564:additive inverse
1545:
1418:
1365:
1312:to the integers
1303:rational numbers
1286:
1282:
1278:
1274:
1270:
1259:
1246:
1242:
1227:
1218:
1193:additive inverse
1187:
1177:
1173:
1162:Additive inverse
1148:
1139:
1133:
1053:
1049:
1045:
1038:
1031:distributive law
1025:
1021:
1017:
1007:
1001:
997:
990:
982:
933:
925:
920:a credit.) Thus
909:
901:
893:
876:
872:
866:
862:
858:
851:
847:
835:
813:
809:
800:
792:
788:
777:
773:
765:
754:
635:Participants on
613:When playing an
418:events, such as
395:Run differential
387:differential in
335:
313:
307:
299:
270:
241:
235:
231:
225:
219:
215:
95:
21:
3972:
3971:
3967:
3966:
3965:
3963:
3962:
3961:
3937:
3936:
3935:
3930:
3907:
3886:
3876:
3849:
3840:Surreal numbers
3830:Ordinal numbers
3775:
3721:
3719:
3716:
3715:
3677:
3644:
3642:
3639:
3638:
3636:
3634:Split-octonions
3605:
3603:
3600:
3599:
3591:
3585:
3567:
3565:
3562:
3561:
3539:
3537:
3534:
3533:
3511:
3509:
3506:
3505:
3502:Complex numbers
3483:
3481:
3478:
3477:
3456:
3405:
3404:
3402:
3399:
3398:
3371:
3369:
3366:
3365:
3338:
3336:
3333:
3332:
3310:
3308:
3305:
3304:
3282:
3280:
3277:
3276:
3273:Natural numbers
3258:
3252:
3193:
3188:
3162:
3157:
3156:
3149:
3133:
3129:
3122:
3106:
3102:
3096:
3075:
3068:
3051:
3047:
3041:
3019:
3015:
3005:
3003:
2996:
2992:
2975:
2971:
2964:
2948:
2944:
2931:
2927:
2920:
2904:
2895:
2888:
2866:
2862:
2852:
2850:
2840:
2836:
2826:
2824:
2815:
2814:
2810:
2800:
2798:
2790:Lynn, Matthew.
2788:
2784:
2774:
2772:
2763:
2762:
2758:
2748:
2746:
2739:The Independent
2731:
2730:
2726:
2716:
2714:
2705:
2704:
2700:
2694:
2678:
2674:
2667:
2651:
2647:
2640:
2623:
2619:
2605:
2603:
2594:
2593:
2589:
2579:
2577:
2568:
2567:
2563:
2549:
2547:
2542:
2541:
2537:
2523:
2521:
2513:
2512:
2508:
2495:
2493:
2485:
2484:
2480:
2469:
2465:
2454:
2450:
2436:
2432:
2425:
2411:
2396:
2382:
2360:
2349:
2344:
2337:
2320:
2316:
2304:
2300:
2295:
2290:
2285:
2246:History of zero
2231:
2223:complex numbers
2193:
2190:
2189:
2157:
2153:
2151:
2148:
2147:
2115:
2111:
2109:
2106:
2105:
2102:cubic equations
2061:Nicolas Chuquet
1942:
1939:
1938:
1807:
1804:
1803:
1784:
1776:
1697:
1679:
1671:
1668:
1667:
1645:
1633:
1597:
1575:) of the form (
1547:
1512:
1461:
1419:
1368:
1366:
1331:
1307:natural numbers
1299:
1293:
1284:
1280:
1276:
1272:
1268:
1261:
1250:
1247:. In general,
1244:
1240:
1223:
1220:
1209:
1195:of the number.
1189:
1185:
1175:
1171:
1170:. For example,
1164:
1158:
1150:
1146:
1141:
1138:(−8) ÷ 2 = −4
1137:
1135:
1132:8 ÷ (−2) = −4
1131:
1122:
1114:complex numbers
1051:
1047:
1043:
1040:
1036:
1027:
1023:
1019:
1015:
1009:
1005:
999:
995:
992:
988:
983:
981:(−2) × 3 = −6
980:
940:
935:
931:
926:
923:
910:
907:
902:
899:
894:
891:
886:
878:
874:
873: and
870:
864:
860:
856:
853:
849:
848: and
845:
837:
833:
820:
815:
811:
807:
798:
790:
786:
775:
767:
760:
746:
723:
694:Employees with
682:approval rating
680:A politician's
595:
524:
449:
436:wind assistance
351:Goal difference
347:
346:
345:
343:
338:
337:
336:
325:
320:
311:
308:
305:
300:
297:
288:
268:
267:in front, e.g.
253:
247:
242:
239:
233:
229:
226:
223:
217:
213:
204:The number line
191:
185:
183:The number line
180:
124:natural numbers
93:
50:negative number
28:
23:
22:
15:
12:
11:
5:
3970:
3960:
3959:
3954:
3949:
3932:
3931:
3929:
3928:
3918:
3916:Classification
3912:
3909:
3908:
3906:
3905:
3903:Normal numbers
3900:
3895:
3873:
3868:
3863:
3857:
3855:
3851:
3850:
3848:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3811:
3810:
3800:
3795:
3789:
3787:
3785:infinitesimals
3777:
3776:
3774:
3773:
3772:
3771:
3766:
3761:
3747:
3742:
3737:
3724:
3709:
3704:
3699:
3694:
3688:
3686:
3679:
3678:
3676:
3675:
3670:
3665:
3660:
3647:
3631:
3626:
3621:
3608:
3595:
3593:
3587:
3586:
3584:
3583:
3570:
3555:
3542:
3527:
3514:
3499:
3486:
3466:
3464:
3458:
3457:
3455:
3454:
3449:
3448:
3447:
3437:
3432:
3427:
3422:
3408:
3392:
3387:
3374:
3359:
3354:
3341:
3326:
3313:
3298:
3285:
3269:
3267:
3260:
3259:
3251:
3250:
3243:
3236:
3228:
3222:
3221:
3216:
3208:
3199:
3192:
3191:External links
3189:
3187:
3186:
3179:
3163:
3161:
3158:
3155:
3154:
3147:
3127:
3120:
3100:
3094:
3066:
3045:
3039:
3013:
2990:
2969:
2962:
2942:
2925:
2918:
2893:
2886:
2860:
2834:
2808:
2782:
2756:
2724:
2698:
2692:
2672:
2665:
2645:
2638:
2617:
2587:
2561:
2546:. Formula1.com
2535:
2506:
2478:
2471:Martha Smith.
2463:
2448:
2430:
2423:
2394:
2380:
2347:
2335:
2314:
2311:978-0999443330
2297:
2296:
2294:
2291:
2289:
2286:
2284:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2232:
2230:
2227:
2209:
2206:
2203:
2200:
2197:
2177:
2174:
2171:
2168:
2165:
2160:
2156:
2135:
2132:
2129:
2126:
2123:
2118:
2114:
2085:numeri absurdi
2071:Michael Stifel
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1826:
1823:
1820:
1817:
1814:
1811:
1798:mathematician
1775:
1772:
1717:
1714:
1711:
1708:
1705:
1695:
1692:
1689:
1685:
1682:
1678:
1675:
1644:
1641:
1596:
1511:
1427:
1367:
1330:
1292:
1289:
1265:absolute value
1249:
1237:
1236:
1233:
1208:
1186:3 + (−3) = 0
1184:
1160:Main article:
1157:
1154:
1145:
1136:
1130:
1121:
1118:
1110:
1109:
1090:
1046:, the product
1035:
1014:
1004:
987:
979:
975:
974:
971:
956:absolute value
939:
938:Multiplication
936:
930:
922:
906:
898:
890:
885:
882:
869:
844:
832:
819:
816:
806:
766:, or twice in
722:
719:
718:
717:
703:
692:
685:
678:
671:
670:
669:
659:
651:
642:
626:
611:
594:
591:
590:
589:
586:Interest rates
583:
580:
571:, such as the
561:
550:
539:
523:
520:
519:
518:
508:
502:
496:
470:
467:prime meridian
456:
448:
445:
444:
443:
412:
405:
402:
392:
382:
363:rugby football
340:
339:
330:
329:
328:
327:
326:
324:
321:
319:
316:
304:
296:
287:
284:
249:Main article:
246:
245:Signed numbers
243:
238:
222:
187:Main article:
184:
181:
179:
176:
109:but not always
56:of a positive
26:
9:
6:
4:
3:
2:
3969:
3958:
3955:
3953:
3950:
3948:
3945:
3944:
3942:
3927:
3919:
3917:
3914:
3913:
3910:
3904:
3901:
3899:
3896:
3893:
3889:
3883:
3879:
3874:
3872:
3869:
3867:
3866:Fuzzy numbers
3864:
3862:
3859:
3858:
3856:
3852:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3809:
3806:
3805:
3804:
3801:
3799:
3796:
3794:
3791:
3790:
3788:
3786:
3782:
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3762:
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3713:
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3698:
3695:
3693:
3690:
3689:
3687:
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3680:
3674:
3671:
3669:
3668:Biquaternions
3666:
3664:
3661:
3635:
3632:
3630:
3627:
3625:
3622:
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3594:
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3214:
3209:
3207:
3205:
3200:
3198:
3195:
3194:
3184:
3180:
3177:
3176:3-540-64767-8
3173:
3169:
3165:
3164:
3150:
3148:9781563084461
3144:
3140:
3139:
3131:
3123:
3121:9789027718723
3117:
3113:
3112:
3104:
3097:
3095:9781402045592
3091:
3087:
3083:
3082:Helaine Selin
3079:
3073:
3071:
3064:. Page 65–66.
3063:
3062:0-684-83718-8
3059:
3055:
3049:
3042:
3040:9781402045592
3036:
3032:
3028:
3027:Helaine Selin
3024:
3017:
3001:
2994:
2987:
2986:0-684-83718-8
2983:
2979:
2973:
2965:
2963:0-521-05801-5
2959:
2955:
2954:
2946:
2938:
2937:
2929:
2921:
2919:0-521-05801-5
2915:
2911:
2910:
2902:
2900:
2898:
2889:
2887:1-4166-0035-3
2883:
2879:
2874:
2873:
2864:
2849:
2845:
2838:
2822:
2818:
2812:
2797:
2793:
2786:
2771:. 5 June 2014
2770:
2766:
2760:
2744:
2740:
2735:
2728:
2712:
2708:
2702:
2695:
2689:
2685:
2684:
2676:
2668:
2666:9780123850225
2662:
2658:
2657:
2649:
2641:
2639:9780813721514
2635:
2631:
2630:
2621:
2614:
2602:. 3 July 2008
2601:
2600:elitefeet.com
2597:
2591:
2575:
2571:
2565:
2558:
2545:
2539:
2532:
2520:
2516:
2510:
2503:
2492:
2488:
2482:
2474:
2467:
2459:
2452:
2445:
2444:
2439:
2434:
2426:
2424:9780792325659
2420:
2416:
2409:
2407:
2405:
2403:
2401:
2399:
2391:
2389:
2388:Nine Chapters
2383:
2377:
2373:
2368:
2367:
2358:
2356:
2354:
2352:
2342:
2340:
2332:
2328:
2324:
2318:
2312:
2308:
2302:
2298:
2282:
2279:
2277:
2274:
2272:
2271:Sign function
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2233:
2226:
2224:
2207:
2204:
2201:
2198:
2195:
2175:
2172:
2169:
2166:
2163:
2158:
2154:
2133:
2130:
2127:
2124:
2121:
2116:
2112:
2103:
2099:
2098:
2093:
2088:
2086:
2082:
2081:
2076:
2072:
2068:
2066:
2062:
2057:
2055:
2054:
2049:
2048:
2043:
2039:
2036:
2030:
2028:
2024:
2020:
2014:
2012:
2008:
2003:
2001:
2000:
1995:
1991:
1988:wrote in his
1987:
1968:
1965:
1962:
1953:
1950:
1947:
1936:
1932:
1931:
1926:
1922:
1917:
1915:
1911:
1910:
1905:
1902:
1897:
1894:
1893:
1886:
1882:
1879:
1875:
1874:counting rods
1871:
1870:Nine Chapters
1867:
1863:
1859:
1855:
1851:
1850:
1844:
1842:
1841:
1824:
1821:
1818:
1815:
1812:
1809:
1801:
1797:
1793:
1788:
1783:
1778:
1771:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1728:
1715:
1712:
1709:
1706:
1703:
1693:
1690:
1687:
1683:
1680:
1676:
1673:
1665:
1661:
1657:
1653:
1648:
1640:
1638:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1595:
1594:
1590:
1586:
1582:
1578:
1574:
1570:
1566:
1565:
1560:
1556:
1553:of the form (
1552:
1551:additive zero
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1510:
1508:
1504:
1499:
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1470:
1466:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1426:
1424:
1421:We define an
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1376:
1372:
1363:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1329:
1327:
1323:
1319:
1315:
1311:
1308:
1304:
1298:
1288:
1266:
1258:
1254:
1248:
1234:
1231:
1230:
1229:
1226:
1216:
1212:
1207:
1205:
1201:
1196:
1194:
1183:
1181:
1169:
1163:
1153:
1144:
1129:
1127:
1117:
1115:
1107:
1103:
1099:
1095:
1091:
1088:
1084:
1080:
1076:
1072:
1071:
1070:
1068:
1064:
1060:
1055:
1044:2 × (−3) = −6
1034:
1032:
1013:
1003:
986:
978:
972:
969:
968:
967:
965:
957:
954:equal to the
953:
949:
944:
929:
921:
919:
915:
905:
897:
889:
881:
868:
843:
840:
831:
824:
805:
802:
796:
783:
779:
771:
764:
758:
753:
749:
744:
740:
736:
732:
728:
715:
711:
707:
704:
701:
697:
693:
690:
686:
683:
679:
676:
672:
667:
665:
660:
657:
656:
652:
649:
648:
643:
640:
639:
634:
633:
631:
627:
624:
621:, such as an
620:
616:
612:
609:
605:
604:
599:
587:
584:
581:
578:
574:
570:
566:
562:
559:
555:
551:
548:
544:
540:
537:
533:
529:
526:
525:
516:
512:
509:
506:
503:
500:
497:
494:
490:
486:
482:
478:
474:
473:Topographical
471:
468:
464:
460:
457:
454:
451:
450:
441:
437:
433:
429:
425:
421:
417:
413:
410:
406:
403:
400:
396:
393:
390:
386:
383:
380:
376:
372:
368:
364:
360:
356:
352:
349:
348:
342:
334:
315:
303:
298:0 − 3 = −3.
295:
293:
283:
281:
277:
272:
266:
262:
259:. Thus every
258:
252:
237:
221:
210:
202:
198:
196:
190:
175:
173:
169:
165:
161:
157:
153:
149:
145:
144:
138:
136:
131:
129:
125:
120:
118:
114:
110:
106:
102:
97:
91:
87:
83:
79:
74:
70:
66:
63:
59:
55:
51:
47:
39:
34:
30:
19:
3887:
3877:
3692:Dual numbers
3684:hypercomplex
3474:Real numbers
3212:
3203:
3182:
3167:
3160:Bibliography
3137:
3130:
3110:
3103:
3085:
3053:
3048:
3030:
3016:
3004:. Retrieved
2993:
2977:
2972:
2952:
2945:
2935:
2928:
2908:
2871:
2863:
2851:. Retrieved
2848:The Guardian
2847:
2837:
2825:. Retrieved
2811:
2799:. Retrieved
2795:
2785:
2773:. Retrieved
2759:
2747:. Retrieved
2738:
2727:
2715:. Retrieved
2710:
2701:
2683:Double Award
2682:
2675:
2655:
2648:
2627:
2620:
2611:
2604:. Retrieved
2599:
2590:
2578:. Retrieved
2574:the original
2564:
2555:
2548:. Retrieved
2538:
2529:
2522:. Retrieved
2518:
2509:
2501:
2494:. Retrieved
2490:
2481:
2466:
2457:
2451:
2441:
2433:
2414:
2387:
2385:
2365:
2322:
2317:
2301:
2266:Real numbers
2096:
2089:
2084:
2078:
2069:
2058:
2052:
2045:
2040:
2032:
2026:
2022:
2018:
2016:
2004:
1997:
1989:
1928:
1925:Al-Khwarizmi
1918:
1907:
1898:
1890:
1888:
1884:
1878:coefficients
1869:
1853:
1847:
1845:
1838:
1789:
1785:
1777:
1767:
1763:
1759:
1755:
1751:
1747:
1743:
1739:
1735:
1731:
1729:
1663:
1659:
1655:
1651:
1649:
1646:
1634:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1588:
1584:
1580:
1576:
1572:
1568:
1562:
1558:
1554:
1550:
1548:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1514:
1506:
1500:
1491:
1487:
1483:
1479:
1475:
1471:
1469:quotient set
1464:
1462:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1420:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1370:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1325:
1321:
1318:ordered pair
1313:
1309:
1300:
1262:
1256:
1252:
1238:
1224:
1221:
1214:
1210:
1197:
1190:
1165:
1151:
1142:
1123:
1111:
1105:
1101:
1097:
1093:
1086:
1082:
1078:
1074:
1069:as follows:
1066:
1062:
1058:
1056:
1041:
1028:
1010:
993:
984:
976:
961:
927:
917:
913:
911:
903:
895:
892:5 − 8 = −3
887:
879:
854:
841:
838:
829:
803:
784:
780:
769:
762:
751:
747:
724:
708:notes on an
662:
653:
645:
636:
489:Death Valley
465:west of the
453:Temperatures
420:sprint races
367:net run rate
309:
306:5 − 8 = −3
301:
289:
279:
275:
273:
256:
254:
227:
211:
207:
194:
192:
178:Introduction
164:coefficients
141:
139:
132:
121:
104:
98:
81:
77:
49:
43:
29:
3854:Other types
3673:Bioctonions
3530:Quaternions
3204:In Our Time
2796:MarketWatch
2606:18 November
2550:30 November
2524:30 November
2496:18 November
2443:Arithmetica
2236:Signed zero
2047:Liber Abaci
2035:Bhāskara II
2011:al-Samaw'al
1904:Brahmagupta
1840:Arithmetica
1762:. That is,
1593:subtraction
1509:by writing
1503:total order
1050:must equal
1048:(−2) × (−3)
884:Subtraction
797:expression
795:subtraction
743:subtraction
706:Transposing
689:video games
628:Television
428:triple jump
292:subtraction
280:nonpositive
276:nonnegative
261:real number
240:−8 < −5.
195:number line
189:Number line
156:Brahmagupta
148:Han dynasty
135:bookkeeping
58:real number
46:mathematics
3941:Categories
3808:Projective
3781:Infinities
2988:. Page 65.
2853:5 December
2827:5 December
2801:5 December
2775:5 December
2749:5 December
2717:5 December
2580:5 December
2544:"Glossary"
2438:Diophantus
2288:References
1858:Han period
1800:Diophantus
1780:See also:
1643:Uniqueness
1467:to be the
1295:See also:
727:minus sign
630:game shows
617:file on a
532:overdrafts
463:longitudes
389:ice hockey
385:Plus-minus
220:, written
168:Diophantus
101:minus sign
90:Fahrenheit
38:Fahrenheit
3892:solenoids
3712:Sedenions
3558:Octonions
2519:BBC Sport
2491:BBC Sport
2293:Citations
2097:Ars Magna
2094:, in his
2090:In 1545,
2065:exponents
2056:, 1225).
2042:Fibonacci
1990:al-Fakhrī
1986:al-Karaji
1966:±
1951:±
1935:Abu Kamil
1872:used red
952:magnitude
739:operation
700:timesheet
655:Jeopardy!
644:Teams on
577:Dow Jones
567:price or
558:deflation
554:inflation
547:recession
511:Impedance
481:sea level
459:Latitudes
432:long jump
416:athletics
409:Formula 1
312:8 − 5 = 3
265:plus sign
228:negative
69:magnitude
62:less than
3301:Integers
3263:Sets of
2821:BBC News
2769:BBC News
2743:Archived
2711:BBC News
2251:Integers
2229:See also
2023:al-zāʾid
2019:al-nāqiṣ
2013:writes:
1730:And so,
1684:′
1168:negation
1156:Negation
1126:division
1120:Division
1024:) = +6
818:Addition
791:7 + (−5)
757:negation
731:operator
714:semitone
573:FTSE 100
536:earnings
485:Dead Sea
440:tailwind
430:and the
414:In some
399:baseball
257:positive
224:8 > 5
128:integers
113:negative
105:positive
82:negative
78:positive
3957:Numbers
3882:numbers
3714: (
3560: (
3532: (
3504: (
3476: (
3397: (
3395:Periods
3364: (
3331: (
3303: (
3275: (
3257:systems
3084:(ed.),
3029:(ed.),
3006:24 July
2331:négatif
2327:positif
1927:in his
1862:Liu Hui
1852:(九章算術,
1774:History
1754:. Thus
1482:) and (
1200:algebra
1178:. The
1026:credit.
1020:) × (−3
918:gaining
759:(as in
745:(as in
735:operand
608:stories
575:or the
522:Finance
447:Science
424:hurdles
371:cricket
172:Leibniz
152:Liu Hui
94:
86:Celsius
52:is the
3682:Other
3255:Number
3174:
3145:
3118:
3092:
3060:
3037:
2984:
2960:
2916:
2884:
2690:
2663:
2636:
2557:tyres.
2421:
2378:
2309:
2188:(with
1984:, and
1794:, the
1561:), an
1255:) =
1217:) = 0
1198:Using
1042:Since
1018:debts
948:vector
914:losing
787:7 + −5
479:above
477:height
434:, the
426:, the
422:, the
359:hockey
310:since
3890:-adic
3880:-adic
3637:Over
3598:Over
3592:types
3590:Split
3025:, in
2009:. As
1906:, in
1796:Greek
1614:) = (
1606:) − (
1587:+ 1,
1521:) ≤ (
1436:) ~ (
1385:) = (
1377:) × (
1348:) = (
1340:) + (
1089:, and
977:Thus
808:2 + 5
799:7 – 5
716:down.
675:swing
615:audio
593:Other
565:share
323:Sport
3926:List
3783:and
3172:ISBN
3143:ISBN
3116:ISBN
3090:ISBN
3058:ISBN
3035:ISBN
3008:2007
2982:ISBN
2958:ISBN
2914:ISBN
2882:ISBN
2855:2018
2829:2018
2803:2018
2777:2018
2751:2018
2719:2018
2688:ISBN
2661:ISBN
2634:ISBN
2608:2019
2582:2018
2552:2019
2531:May.
2526:2019
2498:2019
2419:ISBN
2376:ISBN
2329:and
2323:both
2307:ISBN
2205:>
2075:1544
2053:Flos
1650:Let
1567:of (
1496:ring
1263:The
1213:+ (−
1143:and
1022:each
985:and
964:sign
928:and
904:and
725:The
623:iPod
505:Ions
375:golf
357:and
117:sign
88:and
80:and
73:debt
65:zero
48:, a
2878:210
2077:AD
1790:In
1699:and
1505:on
1283:is
1251:−(−
1243:is
1180:sum
1092:if
1073:if
1016:(−2
950:of
778:).
768:−(−
741:of
687:In
661:In
543:GDP
487:or
397:in
379:par
369:in
353:in
133:In
44:In
3943::
3472::
3069:^
2896:^
2880:.
2846:.
2819:.
2794:.
2767:.
2737:.
2709:.
2610:.
2598:.
2554:.
2528:.
2517:.
2500:.
2489:.
2440:,
2397:^
2384:.
2374:.
2372:88
2350:^
2338:^
2002:.
1819:20
1770:.
1750:=
1748:y′
1742:+
1738:=
1736:y′
1734:+
1716:0.
1660:y′
1639:.
1626:+
1622:,
1618:+
1610:,
1602:,
1579:,
1571:,
1557:,
1541:+
1537:≤
1533:+
1525:,
1517:,
1486:,
1478:,
1456:+
1452:=
1448:+
1440:,
1432:,
1413:×
1409:+
1405:×
1401:,
1397:×
1393:+
1389:×
1381:,
1373:,
1360:+
1356:,
1352:+
1344:,
1336:,
1324:,
1287:.
1269:−3
1245:+3
1241:−3
1206::
1172:−3
1116:.
1100:×
1081:×
1061:×
1054:.
1002::
1000:−6
996:−2
776:−5
750:−
737:)
638:QI
632::
538:).
495:).
373:;
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314:.
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