arithmetic+operators

= =

=Arithmetic operations=

The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, and logarithmic functions. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field. Addition (+) ** Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers, the //addends // or //terms // , into a single number, the //sum // of the numbers. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting. Addition is commutative and associative so the order in which the terms are added does not matter. The identity element of addition (the additive identity) is 0, that is, adding zero to any number will yield that same number. Also, the inverse element of addition (the additive inverse) is the opposite of any number, that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is -7, so 7 + (-7) = 0. Addition can be given geometrically as follows: If //a // and //b // are the lengths of two sticks, then if we place the sticks one after the other, the length of the stick thus formed will be //a // + //b //. Subtraction (−) ** <span style="font-family: Arial,Helvetica,sans-serif;">Subtraction is the opposite of addition. Subtraction finds the //<span style="font-family: Arial,Helvetica,sans-serif;">difference //<span style="font-family: Arial,Helvetica,sans-serif;"> between two numbers, the //<span style="font-family: Arial,Helvetica,sans-serif;">minuend //<span style="font-family: Arial,Helvetica,sans-serif;"> minus the //<span style="font-family: Arial,Helvetica,sans-serif;">subtrahend //<span style="font-family: Arial,Helvetica,sans-serif;">. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero. ====<span style="font-family: Arial,Helvetica,sans-serif;"> Subtraction is neither commutative nor associative. For that reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is //<span style="font-family: Arial,Helvetica,sans-serif;">a //<span style="font-family: Arial,Helvetica,sans-serif; font-size: 8.5pt;"> − <span style="font-family: Arial,sans-serif; font-size: 8.5pt;">//<span style="font-family: Arial,Helvetica,sans-serif;">b // <span style="font-family: Arial,Helvetica,sans-serif; font-size: 8.5pt;"> = <span style="font-family: Arial,sans-serif; font-size: 8.5pt;">//<span style="font-family: Arial,Helvetica,sans-serif;">a // <span style="font-family: Arial,Helvetica,sans-serif; font-size: 8.5pt;"> + (− <span style="font-family: Arial,sans-serif; font-size: 8.5pt;">//<span style="font-family: Arial,Helvetica,sans-serif;">b // <span style="font-family: Arial,Helvetica,sans-serif; font-size: 8.5pt;">). When written as a sum, all the properties of addition hold. ==== <span style="font-family: Arial,Helvetica,sans-serif;">There are several methods for calculating results, some of which are particularly advantageous to machine calculation. For example, digital computers employ the method of two's complement. Of great importance is the counting up method by which change is made. Suppose an amount P is given to pay the required amount Q, with P greater than Q. Rather than performing the subtraction P-Q and counting out that amount in change, money is counted out starting at Q and continuing until reaching P. Curiously, although the amount counted out must equal the result of the subtraction P-Q, the subtraction was never really done and the value of P-Q might still be unknown to the change-maker. Multiplication (×, ·, or *) ** <span style="font-family: Arial,Helvetica,sans-serif;">Multiplication is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the //<span style="font-family: Arial,Helvetica,sans-serif;">product //<span style="font-family: Arial,Helvetica,sans-serif;">. The two original numbers are called the //<span style="font-family: Arial,Helvetica,sans-serif;">multiplier //<span style="font-family: Arial,Helvetica,sans-serif;"> and the //<span style="font-family: Arial,Helvetica,sans-serif;">multiplicand //<span style="font-family: Arial,Helvetica,sans-serif;">, sometimes both simply called //<span style="font-family: Arial,Helvetica,sans-serif;">factors //<span style="font-family: Arial,Helvetica,sans-serif;">. Multiplication is best viewed as a scaling operation. If the real numbers are imagined as lying in a line, multiplication by a number, say //<span style="font-family: Arial,Helvetica,sans-serif;">x //<span style="font-family: Arial,Helvetica,sans-serif;">, greater than 1 is the same as stretching everything away from zero uniformly, in such a way that the number 1 itself is stretched to where //<span style="font-family: Arial,Helvetica,sans-serif;">x was //<span style="font-family: Arial,Helvetica,sans-serif;">. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards zero. (Again, in such a way that 1 goes to the multiplicand.) Multiplication is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number (except zero; zero is the only number without a multiplicative inverse), that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity. Division (÷ or /) ** <span style="font-family: Arial,Helvetica,sans-serif;">Division is essentially the opposite of multiplication. Division finds the //<span style="font-family: Arial,Helvetica,sans-serif;">quotient //<span style="font-family: Arial,Helvetica,sans-serif;"> of two numbers, the //<span style="font-family: Arial,Helvetica,sans-serif;">dividend //<span style="font-family: Arial,Helvetica,sans-serif;"> divided by the //<span style="font-family: Arial,Helvetica,sans-serif;">divisor //<span style="font-family: Arial,Helvetica,sans-serif;">. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers). The quotient multiplied by the divisor always yields the dividend. Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a/b = a*(1/b)
 * <span style="font-family: Arial,Helvetica,sans-serif;">
 * <span style="font-family: Arial,Helvetica,sans-serif;">
 * <span style="font-family: Arial,Helvetica,sans-serif;">
 * <span style="font-family: Arial,Helvetica,sans-serif;">

When written as a product, it will obey all the properties of multiplication.
Examples **
 * <span style="font-family: Arial,Helvetica,sans-serif;">

Multiplication table


 * ** × ** || ** 1 ** || ** 2 ** || ** 3 ** || ** 4 ** || ** 5 ** || ** 6 ** || ** 7 ** || ** 8 ** || ** 9 ** || ** 10 ** || ** 11 ** || ** 12 ** || ** 13 ** || ** 14 ** || ** 15 ** || ** 16 ** || ** 17 ** || ** 18 ** || ** 19 ** || ** 20 ** || ** 21 ** || ** 22 ** || ** 23 ** || ** 24 ** || ** 25 ** ||
 * ** 1 ** || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20 || 21 || 22 || 23 || 24 || 25 ||
 * ** 2 ** || 2 || 4 || 6 || 8 || 10 || 12 || 14 || 16 || 18 || 20 || 22 || 24 || 26 || 28 || 30 || 32 || 34 || 36 || 38 || 40 || 42 || 44 || 46 || 48 || 50 ||
 * ** 3 ** || 3 || 6 || 9 || 12 || 15 || 18 || 21 || 24 || 27 || 30 || 33 || 36 || 39 || 42 || 45 || 48 || 51 || 54 || 57 || 60 || 63 || 66 || 69 || 72 || 75 ||
 * ** 4 ** || 4 || 8 || 12 || 16 || 20 || 24 || 28 || 32 || 36 || 40 || 44 || 48 || 52 || 56 || 60 || 64 || 68 || 72 || 76 || 80 || 84 || 88 || 92 || 96 || 100 ||
 * ** 5 ** || 5 || 10 || 15 || 20 || ** 25 ** || 30 || 35 || 40 || 45 || ** 50 ** || 55 || 60 || 65 || 70 || ** 75 ** || 80 || 85 || 90 || 95 || ** 100 ** || 105 || 110 || 115 || 120 || 125 ||
 * ** 6 ** || 6 || 12 || 18 || 24 || 30 || 36 || 42 || 48 || 54 || 60 || 66 || 72 || 78 || 84 || 90 || 96 || 102 || 108 || 114 || 120 || 126 || 132 || 138 || 144 || 150 ||
 * ** 7 ** || 7 || 14 || 21 || 28 || 35 || 42 || 49 || 56 || 63 || 70 || 77 || 84 || 91 || 98 || 105 || 112 || 119 || 126 || 133 || 140 || 147 || 154 || 161 || 168 || 175 ||
 * ** 8 ** || 8 || 16 || 24 || 32 || 40 || 48 || 56 || 64 || 72 || 80 || 88 || 96 || 104 || 112 || 120 || 128 || 136 || 144 || 152 || 160 || 168 || 176 || 184 || 192 || 200 ||
 * ** 9 ** || 9 || 18 || 27 || 36 || 45 || 54 || 63 || 72 || 81 || 90 || 99 || 108 || 117 || 126 || 135 || 144 || 153 || 162 || 171 || 180 || 189 || 198 || 207 || 216 || 225 ||
 * ** 10 ** || 10 || 20 || 30 || 40 || ** 50 ** || 60 || 70 || 80 || 90 || ** 100 ** || 110 || 120 || 130 || 140 || ** 150 ** || 160 || 170 || 180 || 190 || ** 200 ** || 210 || 220 || 230 || 240 || 250 ||
 * ** 11 ** || 11 || 22 || 33 || 44 || 55 || 66 || 77 || 88 || 99 || 110 || 121 || 132 || 143 || 154 || 165 || 176 || 187 || 198 || 209 || 220 || 231 || 242 || 253 || 264 || 275 ||
 * ** 12 ** || 12 || 24 || 36 || 48 || 60 || 72 || 84 || 96 || 108 || 120 || 132 || 144 || 156 || 168 || 180 || 192 || 204 || 216 || 228 || 240 || 252 || 264 || 276 || 288 || 300 ||
 * ** 13 ** || 13 || 26 || 39 || 52 || 65 || 78 || 91 || 104 || 117 || 130 || 143 || 156 || 169 || 182 || 195 || 208 || 221 || 234 || 247 || 260 || 273 || 286 || 299 || 312 || 325 ||
 * ** 14 ** || 14 || 28 || 42 || 56 || 70 || 84 || 98 || 112 || 126 || 140 || 154 || 168 || 182 || 196 || 210 || 224 || 238 || 252 || 266 || 280 || 294 || 308 || 322 || 336 || 350 ||
 * ** 15 ** || 15 || 30 || 45 || 60 || ** 75 ** || 90 || 105 || 120 || 135 || ** 150 ** || 165 || 180 || 195 || 210 || ** 225 ** || 240 || 255 || 270 || 285 || ** 300 ** || 315 || 330 || 345 || 360 || 375 ||
 * ** 16 ** || 16 || 32 || 48 || 64 || 80 || 96 || 112 || 128 || 144 || 160 || 176 || 192 || 208 || 224 || 240 || 256 || 272 || 288 || 304 || 320 || 336 || 352 || 368 || 384 || 400 ||
 * ** 17 ** || 17 || 34 || 51 || 68 || 85 || 102 || 119 || 136 || 153 || 170 || 187 || 204 || 221 || 238 || 255 || 272 || 289 || 306 || 323 || 340 || 357 || 374 || 391 || 408 || 425 ||
 * ** 18 ** || 18 || 36 || 54 || 72 || 90 || 108 || 126 || 144 || 162 || 180 || 198 || 216 || 234 || 252 || 270 || 288 || 306 || 324 || 342 || 360 || 378 || 396 || 414 || 432 || 450 ||
 * ** 19 ** || 19 || 38 || 57 || 76 || 95 || 114 || 133 || 152 || 171 || 190 || 209 || 228 || 247 || 266 || 285 || 304 || 323 || 342 || 361 || 380 || 399 || 418 || 437 || 456 || 475 ||
 * ** 20 ** || 20 || 40 || 60 || 80 || ** 100 ** || 120 || 140 || 160 || 180 || ** 200 ** || 220 || 240 || 260 || 280 || ** 300 ** || 320 || 340 || 360 || 380 || ** 400 ** || 420 || 440 || 460 || 480 || 500 ||