Algebra 1, Chapter 1, part 12

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Adding two more postulates as follows: -If you add together two NATURAL NUMBERS of (123456...), the result will always be a Natural Number. -If you multiply together two Natural Numbers, the result will always be a Natural Number. "These two properties are called CLOSURE Properties. Imagine that you are fenced in with a set of natural numbers and you are trying to get out. Whenever you add or multiply together two natural numbers, you are still trapped within the set of natural numbers (as if you were closed in). On the other hand, if you divide two natural numbers or subtract two natural numbers, you can sometimes escape outside the set of natural numbers. For example, 6 - 8 and 4 / 3 are not natural numbers. Therefore, the natural numbers do not obey the property of closure with respect to subtraction or division. We added one more postulate. Ms. O' Reilly asked us to calculate a holiday bonus for an employee who was to receive $3 for every hour he worked during the two weeks prior to the holiday. The first week he worked 26 hours and the second week he worked 24 hours, so Recordis figured that his bonus was 3 x 26 + 3 x 24 = 78 + 72 =150. The professor suggested that it would be easier to add the number of hours first and then multiply by 3, like tis: 3 x (26 + 24) = 3 x 50 = 150. "Does that always work?" Recordis asked. We experimented with some numbers, and this property seemed to be true, so we added it to our list of postulates. This property has become known as the DISTRIBUTIVE property. If a, b, and c are any three numbers, then a x (b + c) = (a x b + (a x c) = a x b + a x c. The distributive property defines the relationship between addition and multiplication . Note that a x (1+1) equals a x 2 for any value of a. According to the distributive property: a x (1 + 1) equals a x 1 + a x 1 Combining these results, we get a x 2 equals a x 1 + a x 1 Since a x 1 is equal to a, it follows that a x 2 equals a + a which matches our intuitive idea of the connection between multiplication and addition. (Chapter 1 to be continued)