Overview
Operations with real numbers are the most familiar types in mathematics. Those numerical expressions with constants and operations include such actions as addition, subtraction, multiplication, and division. Many algebraic expressions are usually operations with real numbers.
Real Numbers
The set of real numbers include the entire set of rational numbers, such as positive and negative integers, whole numbers, and natural numbers. Positive and negative fractions are rational numbers, because the numerator and denominator of the fraction represents a ratio. Whole numbers are rational numbers, because they are ratios in simplified form. For example, the fraction 4/1 is usually represented as 4. Irrational numbers, such as pi π, the constant e, √7, or √2 are also real numbers, but imaginary numbers and the infinities are not.
Addition of Real Numbers
Addition of real numbers can be represented on the number line by moving right on the number line. For example, 4 +3 =7 can be modeled on the number line by starting at 4 spaces from 0 and then moving right 3 more spaces. Fractions can be added by finding their common denominator, then adding the numerators. For example, ½ +1/3 = 3/6 +2/6 = 5/6.
Subtraction of Real Numbers
Real numbers can be subtracted on the number line by moving to the left. Subtraction is the same as adding the additive inverse, so that 7-5 means the same thing as adding 7 + (-5). The additive inverse of a number is the opposite of that number, so that the additive inverse of 3 is -3, the additive inverse of 1/5 is -1/5, and the additive inverse of -12 is 12. Fractions can also be subtracted by adding the additive inverse, as long as the common denominators are found.
Multiplication and Division of Real Numbers
In the most common model of multiplication and division, multiplication is repeated addition and division is repeated subtraction. When multiplying two positive or two negative numbers, the product is positive, but if one number is positive and the other is negative, the product will be negative. Division by zero is undefined, the product of any number and 0 is zero, and the quotient of 0 and any nonzero number is 0.
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