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.