Mathematics professors have told me I am wrong when I have said multiplication is repeated subtraction.

The reason they tell me I am wrong is simple. Mathematics professors have been sucked in by the MIRA meme that multiplication IS repeated addition.

If multiplication is repeated addition, then division is repeated subtraction.

End of story. Or is it?

Integral multiplication of a multiplicand by a POSITIVE multiplier, can be calculated by the repeated ADDITION of the multiplicand from zero, as many times as the multiplier says.

Integral multiplication of a multiplicand by a NEGATIVE multiplier, can be calculated by the repeated SUBTRACTION of the multiplicand from zero, as many times as the multiplier says.

It seems people have either forgotten, or never knew, multiplication distributes over subtraction as well as addition.

So just as two multiplied by positive three 2 × (

^{+}3) may be calculated as 0 + 2 + 2 + 2, which equals positive six, two multiplied by negative three 2 × (^{–}3) may be calculated as 0 - 2 - 2 - 2, which equals negative six.

Thus multiplication is repeated subtraction, as well as repeated addition. It just depends on whether your integral multiplier is positive or negative.

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EXTENDED ARTICLE

The laws of mathematics require multiplication be undertaken via repeated subtraction just as they require multiplication be undertaken via repeated addition.

This is because the ** Distributive Law** does not discriminate between subtraction and addition. Integral multiplication distributes over both addition and subtraction.

Brahmagupta wrote Brāhma Sphuta-siddhānta (BSS) in 628 CE. In this book, the first to document the rules of zero, Brahmagupta gave the rules of ‘saṅkalana’, or addition. Here, he defined zero to be the sum of a positive number and negative number of equal magnitude, सम-ऐक्यम् खम् (BSS, Chapter 18:30a).

Thus in modern notation, 0 = ^{+}* n* +

^{–}

*. So given zero is described as, for example,*

**n**^{+}3 +

^{–}3, if we take away

^{–}3 then

^{+}3 remains and if from

^{+}3 +

^{–}3 (zero), we take away

^{+}3 then

^{–}3 remains. Thus based on what may be implied from the mnemonic Sanskrit verses (shlokas) of Brahmagupta, a negative number may be defined as 0 –

^{+}

*=*

**n**^{–}

*.*

**n**An example may be 0 – ^{+}3 = ^{–}3. Therefore, we can write the statements, ^{+}3 = 0 + 1 + 1 + 1 and ^{–}3 = 0 – 1 – 1 – 1. Given we know multiplication distributes over both addition and subtraction, 2 × ^{+}3 = 2 × (0 + 1 + 1 + 1) and this in turn becomes the equation 0 + 2 + 2 + 2 = * +*6.

By the ** Distributive Law**, 2 ×

^{–}3 = 2 × (0 – 1 – 1 – 1) and so as ‘multiplication

*is*repeated addition’, we must say ‘multiplication

*is*repeated subtraction’ as 2 ×

^{–}3 = 0 – 2 – 2 – 2 =

**–**6.

Thus, based on the Sanskrit of Brahmagupta, our definition of multiplication on the integers * ab*, is better described as: (the total of)

*either added to or subtracted from ZERO,*

**a***times, as per sign of*

**b**

**b.**As at 16 June, 2014, there are only 3 Google search results for the phrase *“multiplication is repeated subtraction”.* One is an error and should read “division is repeated subtraction”, while the other two are the author’s.

See www.google.com/search?q=%22multiplication+is+repeated+subtraction%22

This contrasts with more than 74,000 Google search results for the phrase *“multiplication is repeated addition”. www.google.com/search?q=%22multiplication+is+repeated+addition%22*

There has been a lot of fuss in recent years about whether Multiplication IS Repeated Addition (MIRA) and should be defined as repeated addition. Despite the hundreds of thousands of words written on this topic, it appears nobody has proven MIRA wrong with this ** "Proof by Contradiction"** until now.

Multiplication cannot be defined as repeated addition because the same laws of mathematics mean just as multiplication may INVOLVE repeated addition, multiplication may INVOLVE repeated subtraction.

Multiplication on the integers may involve either repeated addition or repeated subtraction, depending on the sign of the multiplier, yet cannot be defined solely as one or the other.