# Music and Math Project

## More on the Land of ...

In this section, whenever we write a subscript to a number, it means we should read it as if we were in the Land of that number. For instance $$1328_{12}$$ means $$1328$$ in the Land of Twelve. In this section, if we leave out the subscript, we are in the Land of Ten. We are then dealing with ordinary numbers.

### Examples

#### Decimal Example

Let's have a look at the number $$1328$$, in decimal notation. We can break it up into pieces like this $\begin{array}{lllll} 1328 &= 1000 &+ 300 &+ 20 &+ 8 \\ &= 1\times 1000 &+ 3 \times 100 &+ 2 \times 10 &+ 8 \times 1 \\ &= 1\times 10^3 &+ 3 \times 10^2 &+ 2 \times 10^1 &+ 8 \times 10^0. \end{array}$

#### Translation from the Land of Twelve to the Land of Ten

In the same spirit, in the Land of Twelve, (or in dozenal notation) the number $$1328_{12}$$ means the following $\begin{array}{lllll} 1328_{12} &= 1000_{12} &+ 300_{12} &+ 20_{12} &+ 8_{12} \\ &= 1\times 1000_{12} &+ 3 \times 100_{12} &+ 2 \times 10_{12} &+ 8 \times 1_{12} \\ &= 1\times 12^3 &+ 3 \times 12^2 &+ 2 \times 12^1 &+ 8 \times 12^0 \\ &= 1728 &+ 432 &+ 24 &+ 8 \\ &= 2192. \end{array}$

#### Translation from the Land of Ten to the Land of Two

In the Land of Two, counting goes as follows, $1_2, 10_2, 11_2, 100_2, 101_2, 110_2, 111_2, 1000_2, 1001_2 \dots$ Suppose we would like to know what $$708$$ would be in the Land of Two, that is, in binary form. For that, let's first make a table of powers of $$2$$:
 $$2^0$$ $$1$$ $$2^1$$ $$2$$ $$2^2$$ $$4$$ $$2^3$$ $$8$$ $$2^4$$ $$16$$ $$2^5$$ $$32$$ $$2^6$$ $$64$$ $$2^7$$ $$128$$ $$2^8$$ $$256$$ $$2^9$$ $$512$$ $$2^{10}$$ $$1024$$
Now we follow the following steps:
• Let's find the highest power of $$2$$ that is smaller than our number $$708$$. From the table, we see that this is $$2^9$$.
• Divide $$708$$ by $$2^9$$, keeping track of the remainder. Observe that $$2^9$$ fits $$1$$ time in 708 and that $$708 - 1\times 2^9 = 708 - 512 = 196$$. In other words, $$708 = 1 \times 2^9 + 196$$.
• Repeat the above steps with the number $$196$$. The highest power of $$2$$ smaller than $$196$$ is $$2^7$$
• Division with remainder gives $$196 = 1 \times 2^7 + 68$$. So $708 = 1 \times 2^9 + 196 = 1 \times 2^9 + 1 \times 2^7 + 68.$
• Repeat the above with $$68$$, etc. Finally, you will find $\begin{array}{lllll} 708 &= 1 \times 2^9 &+ 1 \times 2^7 &+ 1 \times 2^6 &+ 1 \times 2^2\\ &= 1000000000_2 &+ 10000000_2&+ 1000000_2 &+ 100_2\\ &= 1011000100_2. \end{array}$

### Exercises

#### Exercise 1

Translate the following numbers from the Land of Twelve to the Land of Ten: $$6_{12}, \mathrm{X}_{12}, \mathrm{YY}_{12}, 1000_{12}, \mathrm{X}3_{12}, 866_{12}$$

#### Exercise 2

Translate the following numbers from the Land of Ten to the Land of Twelve: $$12, 144, 764, 1024$$

#### Exercise 3

In the Land of sixteen, counting is as follows $1, 2, 3, 4, 5, 6, 7, 8, 9, \mathrm{a, b, c, d, e, f}, 10, 11, \dots$ Translate the following numbers from the Land of Ten to the Land of Sixteen: $$16, 25, 300, 4223$$. Also, translate the numbers $$\mathrm{e}\mathrm{a}_{16}, \mathrm{b}2\mathrm{f}_{16}, 10\mathrm{e}4_{16}$$ to the Land of Ten.