Lesson 27 Section 3 ## THE GENERAL DEFINITION OF MULTIPLICATION## A proof of the order propertyHERE IS THE most general definition of multiplication. Whatever ratio the
multiplier has to 1 Consider this multiplication of whole numbers: 3 × 8 = 24 The multiplier 3 is Similarly, to make sense out of ½ × 8, the multiplier ½ is half of 1 (Lesson 16, Question 8); therefore the product will be half of 8. Fractions have the same ratio to one another as natural numbers. (Lesson 23.) Therefore the definition is valid for fractions and natural numbers. Proportionally, As the Multiplier is to 1, so the Product is to the Multiplicand. Or inversely, As 1 is to the Multiplier, so the Multiplicand is to the Product. The Product, then, is the fourth proportional to 1, the Multiplier, and the Multiplicand. Using the conventional symbols for ratio: 1 : Multiplier = Multiplicand : Product. The order property of multiplication Theorem. If two numbers multiply one another, the products will equal one another. The proof will depend on this elementary proposition: The fourth proportional to three given numbers is unique.
For example, if the first three numbers are 1, 3, 5 -- 1 : 3 = 5 : ? -- then since 1 is the third part of 3, that fourth proportional must be 15. That fourth proportional is unique. The proof also depends on the theorem of the alternate proportion. Let a number Let Then For, if 1 : Therefore, alternately, 1 : If 1 : But the fourth proportional to three given numbers is unique. Therefore
Which is what we wanted to prove. (For a proof that follows from the distributive property, see Appendix 3.) Introduction | Home | Table of Contents Please make a donation to keep TheMathPage online. Copyright © 2015 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |