Arithmetic Tutorial Rules for Zero Below are some of the basic rules for the number zero. For every number x, you'll find that: x + 0 = x x - 0 = x x * 0 = 0 (x / 0) = undefined (0 / x) = 0 For every integer n, you'll find that 0n = 0. Zero is the only number equal to its opposite: 0 = -0. If the product of two or more numbers equals zero, then at least one of those numbers must be zero. Zero is smaller than every positive number and greater than every negative number. The only number that is neither positive nor negative is zero. Zero is an even integer. Rules of One Below are some of the basic rules of the number one. For any number x, the following applies: 1 * x = x x / 1 = x 1 is the smallest positive integer. 1 is a divisor of every integer. 1 is an odd integer. 1 is not a prime number. 1 is the only integer with only one divisor. For every integer n, you'll find that 1n = 1. Absolute Value The absolute value of x is written as | x |. The absolute value is the distance between x and 0 on the number line. If x = -5, the absolute value of x = 5. If x = 5, the absolute value of x = 5. Here are some examples: If x = -44 then | x | = 44 If x = 44 then | x | = 44 If x = 0 then | x | = 0 If x = -1 then | x | = 1 Opposites Opposites are two unequal numbers that have the same absolute value. For example, 7 and -7 are opposites. The number 0 is the only number that is equal to its opposite. The sum of any two opposites is always 0. For example: x + (-x) = 0 Subtracting a number is the same as adding its opposite. For example: x - y is the same as x + (-y) Multiplying/Dividing Positive and Negative Numbers If you multiply or divide two positive or two negative numbers, the result is a positive number. For example: 5 * 5 = 25 -5 * (-5) = 25 If you multiply or divide a positive and a negative number, the result is a negative number. For example: -5 * 5 = -25 -5 / 5 = -1 The product of an even number of negative factors is always positive. The product of an odd number of negative factors is always negative. For example: -2 * (-2) * (-2) * (-2) = 16 -2 * (-2) * (-2) = -8 Reciprocals If x represents any number other than 0, the reciprocal of x is 1 / x. The product of any number and its reciprocal is always 1. For example: x * (1 / x) = 1 Dividing by a number is the same as multiplying by its reciprocal. For example: x / y is the same as x * (1 / y) Positive and Negative Addition The sum of any two positive numbers is always positive. The sum of any two negative numbers is always negative. For example: 34 + 4 = 38 -34 + (-4) = -38 The best way to determine the sum of a positive and a negative number is to calculate the difference between their absolute values and then use the sign of the number with the larger absolute value. For example, the difference between -34 and 4 is |-34| - |4| = 30 and then take the sign of the number with the larger absolute value, yielding an answer of -30. Distributive Law For any real numbers x, y and z. x (y + z) = xy + xz x (y - z) = xy - xz If x is not equal to 0 then (y + z) / x = (y / x) + (z / x) (y - z) / x = (y / x) - (z / x) Integers and Prime Numbers Every integer has a finite set of factors and infinite set of multiples. For example, the factors of 3 are -1, -3, 1 , 3, while the multiples of 3 are ....-12, -9, -6, -3, 0, 3, 6, 9, 12... Every positive integer but 1 has at least two possible divisors, 1 and itself. The only positive divisor of 1 is 1. Positive integers that have exactly two positive divisors are called prime numbers. For example: 2, 3, 5, 7, 11, 13, 17... The number 1 is not a prime number. Every integer greater than 1 that is not a prime number can be written as a product of prime numbers. If two integers are both odd or both even, their sum and difference are even. For example: 7 + 5 = 12 6 + 4 = 10 7 - 5 = 2 6 - 4 = 2 etc.. If one integer is odd and the other is even, their sum and difference are odd. For example: 7 + 4 = 11 7 - 4 = 3 etc... The product of two integers is even, unless both of them are odd. For example: 5 * 2 = 10 5 * 4 = 20 5 * 3 = 15 Exponents Where m and n are positive integers, the following rules apply: xmxn = xm+n xm / xn = xm-n (xm)n = xmn xmym = (xy)m 0n = 0 If x is positive, then xn is always positive. If x is negative, then xn is always positive if n is even and negative if n is odd. Inequalities The inequalities will remain unchanged if you add or subtract the same number to both sides of the inequality. For example: 5 > 3 and 5 + 5 > 3 + 5, which is the same as saying 10 > 8 5 > 3 and 5 - 5 > 3 - 5, which is the same as saying 0 > (-2) On the same token, adding inequalities in the same direction preserves the inequality. If x < w and y < z, then x + y < w + z. For example: 1 < 3 and 9 < 11, so 1 + 9 < 3 + 11, which is the same as saying 10 < 14. If x is greater than y, then x - y is always positive. If x is less than y, then x - y is always negative. One of the following is always true: either x < y or x = y or x > y. Multiplying or dividing an inequality by a positive number preserves the inequality. If x > y, then 5 * x > 5 * y. Or if x > y, then (x / 5) > (y / 5). Multiplying or dividing an inequality by a negative number reverses the inequality. If x > y, then (-1) * x < (-1) * y, which is the same as saying -x < -y. If the numbers on either side of the inequality are positive, then taking their reciprocals reverses the inequality. If x > y, then (1 / x) < (1 / y). If 0 < y < 1 and x is positive, then xy < x. If 0 < y < 1, then ( 1 / y) > y and (1 / y) > 1. Zero Any number multiplied by 0 is 0. For example: 5 * 0 = 0 -499 * 0 = 0 334.44 * (-2000) * 0 * 2 = 0 x * 0 = 0 If x * y = 0, then either x or y must be equal to zero.