College Algebra (Abridged)

Real numbers are used extensively in mathematics and are represented on a continuous number line. The number line includes the set of all the whole numbers {0, 1, 2, 3, 4…}, natural numbers{1, 2, 3, 4…}, integers {…-3, -2, -1, 0, 1, 2, 3, …}, fractions (numbers written in the form of , such that the ratio is not equal to zero) and irrational numbers (like square root of 2, given by √ 2 ). On the number line, negative values are on the left side of zero while positive values are on the right side of zero, as shown below: The properties of real numbers deal with four major operations: addition (+), subtraction ( − ), multiplication ( ×, ∙ ) and division ( ÷,/ ). Addition, subtraction, and multiplication of real numbers will result in a real number. Division of real numbers is also possible, provided that the denominator of the fraction is not equal to zero. If the denominator is equal to zero, then the result is undefined. The Properties of Real Number are listed below (assume , , and are real numbers): 1. Cumulative l w states that when we add or multiply two real numbers, their order does not matter. That is, + = + and × = × 2. Associative law states that while adding or multiplying more than two real numbers then the effect of parentheses does not matter. That is, ( + ) + = + ( + ) and ( × ) × = × ( × ) 3. Distributive law applies to the cases where one makes use of both addition and multiplication operations. × ( + ) = × + × and ( + ) × = × + × 4. The additive identity is known as 0 and 1 is a multiplicative identity . That is, + 0 = and × 1 = 5. Additive inverse is denoted by a negative sign. That is, the additive inverse of would be ( − ). Addition of a real number with its additive inverse would be 0. So, + ( − ) = 0 6. Multiplicative inverse of would be −1 -1 or 1 . The multiplication of a real number with its multiplicative inverse would be equal to 1, Therefore, × −1 = 1 7. Cancellation law for addition: if + = + , then = . 8. Cancellation law for multiplication: if × = × , then = , such that ≠ 0 .

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