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Lesson 27  Section 3


A proof of the order property

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Section 2

HERE IS THE most general definition of multiplication.

Whatever ratio the multiplier has to 1
the product shall have to the multiplicand.

Consider this multiplication of whole numbers:

3 × 8 = 24

The multiplier 3 is three times 1; therefore the product will be three times the multiplicand; it will be three times 8.

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.


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.

Euclid, VII, 16.

The proof will depend on this elementary proposition:

The fourth proportional to three given numbers is unique.

  a : b  =  c : x
  a : b  =  c : y,
  x  =  y.

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 a, then, multiply a number b, and produce c.

Let b multiply a and produce d.

Then c = d.

For, if a multiplies b, then

1 : a = b : c.

Therefore, alternately,

1 : b = a : c.

If b multiplies a, then

1 : b = a : d.

But the fourth proportional to three given numbers is unique.  Therefore

c = d.

Which is what we wanted to prove.

(For a proof that follows from the distributive property, see Appendix 3.)

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