MATHEMATICS  -   High School to University Level
ALGEBRA
REAL NUMBERS
 Real numbers are either rational or irrational. A rational number is a ratio or quotient of two integers. Rational numbers can be represented as integers, fractions, terminating decimals and recurring or repeating decimals. Irrational numbers are the numbers that cannot be expressed as the ratio of two integers. An irrational number expressed as a decimal never repeat or terminate. The real number system is the set of all decimal numbers. All rational and all irrational numbers are real numbers. The real numbers are all numbers on the number line. R = Q U I,   R is the set of real numbers. :: The absolute value of a real number a is its distance from the origin
 Properties of absolute value Examples 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 ::  Properties of real numbers
 Properties Examples 1 a + b = b + a 1 −1 + 4 = 4 + (−1) = 3 2 (a + b) + c = a + (b + c) 2 (−3 + 2) + 7 = −3 + (2 + 7) = 6 3 a + 0 = a 3 − 8 + 0 = − 8 4 a + (− a) = 0 4 9 + (- 9) = 0 5 a · (b + c) = a · b + a · c 5 −3 · (−1 + 7) = −3 · (−1) + (−3) · 7 = − 18 6 a · b = b · a 6 (− 4) · 2 = 2 · (− 4) = − 8 7 a + (− b) = a − b 7 9 + (- 4) = 9 - 4 = 5 8 − (a + b) = − a − b 8 − (−3 + 2) = − (−3) − 2 = 3 − 2 = 1 9 b − a = − (a − b) 9 5 - 7 = - (7 - 5) = - 2 10 a · (− b) = − a · b 10 3 · (- 5) = -3 · 5 = -15 11 (− a) · (− b) = a · b 11 (- 3) · (- 6) = 3 · 6 = 18 12 − (− a) = a 12 - (- 7) = 7 13 a · 0 = 0 13 (-11) · 0 = 0 14 14 15 a · 1 = 1 · a = a 15 - 5 · 1 = 1 · (- 5) = - 5 16 (−1) · a = − a 16 (-1) · 4 = - 4 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 if 28 if 29 if 29 if ::  Decimal fractions are vulgar fractions whose denominator is a power of ten.
 Examples: ::  Recurring decimal to fraction conversion
 Rational number which cannot be expressed as a decimal fraction converts to recurring decimal. The vulgar fractions in lowest terms whose prime factors in the denominator are other than 2 or 5 convert to the purely recurring decimals, while those fractions whose denominator is a product of 2's and/or 5's besides the prime numbers convert to the mixed recurring decimals. Examples:  Number of 9’s in the denominator equals the number of digits of the period while 0’s equals to the number of digits in the non-recurring pattern.   Contents A 