Represent the following numbers in the Mesopotamian system —

(i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605

Question

Represent the following numbers in the Mesopotamian system — (i) 63 (ii) 132 (iii) 200 (iv) 60 (v) 3605

Homi · Accepted Answer

The Mesopotamian number system is a base-60 system, which means numbers are grouped in sets of 60, similar to how we group numbers in sets of 10 in our usual system.

Step 1 — Understanding the Base-60 System

Our everyday number system is base-10. This means we use powers of 10 like $1, 10, 100, 1000$ , and so on. The Mesopotamian system uses powers of 60. These powers are $60^0 = \mathbf{1}$ , $60^1 = \mathbf{60}$ , $60^2 = \mathbf{3600}$ , and so on. To represent a number, we find how many times each power of 60 fits into it. We start by dividing the number by the largest power of 60 that is smaller than or equal to it. Then we find the remainder and repeat the process for the next smaller power of 60.

Step 2 — Representing 63

We want to represent the number 63 in the Mesopotamian system. The largest power of 60 less than or equal to 63 is $60^1 = 60$ . Let us divide 63 by 60.

$63 \div 60 = 1 \text{ with a remainder of } 3$

This means 63 contains one group of 60. There are 3 units left over. So, we can write 63 as one times 60 plus 3.

$\boxed{\text{(i) } 63 = 1 \times 60 + 3}$

Step 3 — Representing 132

Now, let us represent the number 132. The largest power of 60 less than or equal to 132 is $60^1 = 60$ . Let us divide 132 by 60.

$132 \div 60 = 2 \text{ with a remainder of } 12$

This means 132 contains two groups of 60. The remainder is 12. We can think of 12 as one group of 10 and two units. So, we write 132 as two times 60, plus 10, plus 2.

$\boxed{\text{(ii) } 132 = 2 \times 60 + 10 + 2}$

Step 4 — Representing 200

Next, we represent the number 200. The largest power of 60 less than or equal to 200 is $60^1 = 60$ . Let us divide 200 by 60.

$200 \div 60 = 3 \text{ with a remainder of } 20$

This means 200 contains three groups of 60. There are 20 units left over. So, we can write 200 as three times 60 plus 20.

$\boxed{\text{(iii) } 200 = 3 \times 60 + 20}$

Step 5 — Representing 60

Let us represent the number 60. The largest power of 60 less than or equal to 60 is $60^1 = 60$ . Let us divide 60 by 60.

$60 \div 60 = 1 \text{ with a remainder of } 0$

This means 60 contains exactly one group of 60. There are no units left over. So, we write 60 as one times 60.

$\boxed{\text{(iv) } 60 = 1 \times 60}$

Step 6 — Representing 3605

Finally, we represent the number 3605. First, we list the powers of 60. We know $60^1 = 60$ and $60^2 = 3600$ . The largest power of 60 less than or equal to 3605 is $60^2 = 3600$ . Let us divide 3605 by 3600.

$3605 \div 3600 = 1 \text{ with a remainder of } 5$

This means 3605 contains one group of 3600. The remainder is 5. This 5 is less than 60, so it represents the units place. There are no groups of 60 in the remainder. So, we write 3605 as one times 3600 plus 5.

$\boxed{\text{(v) } 3605 = 1 \times 3600 + 5}$

Answer

(i) $63 = 1 \times 60 + 3$ (ii) $132 = 2 \times 60 + 10 + 2$ (iii) $200 = 3 \times 60 + 20$ (iv) $60 = 1 \times 60$ (v) $3605 = 1 \times 3600 + 5$

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