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Positional notation and distributive property

We have discussed positional notation in the context of number systems in different bases. However, now taking those concepts a bit further we can find other uses for positional notation.

Another place where we can find positional notation is with multiplication and the distributive property. This will help us find a uniform method of mutiplying numbers of any base. This is usefull to us in that we just learn a few basic concepts and we can then apply them to a broad range of problems.

Make sure we pay attention to this part of the journey. Positional notation is such a simple and versatile concept in mathematics. It should be given much more class time in school than it currently does.
But, this is the purpose of our math journey to approach and apply techniques that are often left unexplored.

a(b+c) = ab + ac
3(4 + 5) = 3(4) + 3(5)
= 12 + 15 = 27

To apply this to positional notation I will use an example.
Let’s look at the problem 2 x 23: Putting each of these numbers into positional notation we will get.

Standard Notation Positional Notation
23 2 x 101 + 3 x 100 = 2 x 10 + 3
2 2 x 100 = 2
2 x 23
= 2(2 x 10 + 3)

The next step I shall look at this in terms of abstracting out the base. What this means that I will replace the 10 with a variable to illustrate that it is only there to notate the base we are working with. When we do this we can see how positional notation can help us to perform mathematical operations on numbers from any base with relative ease.

Going back to our example when we remove the 10 and replace it with a variable we will get.

2 x 10 + 3 = 2x + 3
2(2 x 10 + 3) 2(2x + 3)

We can now apply the distributive property to this example.

2(2x + 3) = 2(2x) + 2(3)
= 4x + 6
Substituting the 10 back into x = 4(10) + 6
= 40 + 6
= 46

This is just one of the many examples of the learning opportunities awarded by taking a Math Journey into positional notation.