Compute the landmark numbers of a base-7 system. In general, what are the landmark numbers of a base-n system?

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Homi · Accepted Answer

Landmark numbers help us understand the value of each position in a number system.

Step 1 — Understanding Place Values

Every number system uses a base. The base tells us how many unique digits are used. For example, our usual number system is base 10. It uses digits from 0 to 9. In base 10, the positions of digits have values like ones, tens, hundreds, and so on. These values are powers of the base, 10. The ones place is $10^0$ . The tens place is $10^1$ . The hundreds place is $10^2$ . These values are called landmark numbers because they mark the value of each position.

Step 2 — Calculating Base-7 Landmark Numbers

For a base-7 system, the base is 7. This means the position values will be powers of 7. We start with the power of 0.

The first landmark number is $7^0$ . $7^0 = 1$

The second landmark number is $7^1$ . $7^1 = 7$

The third landmark number is $7^2$ . $7^2 = 7 \times 7$

$\boxed{49}$

The fourth landmark number is $7^3$ . $7^3 = 7 \times 7 \times 7$ $= 49 \times 7$

$\boxed{343}$

The fifth landmark number is $7^4$ . $7^4 = 7 \times 7 \times 7 \times 7$ $= 343 \times 7$

$\boxed{2401}$

So, the landmark numbers for base 7 are 1, 7, 49, 343, 2401.

Step 3 — Generalizing for Base-n

We can apply the same idea to any base, which we call 'n'. The landmark numbers will be powers of 'n'. We start with 'n' raised to the power of 0. Then we take 'n' raised to the power of 1, then 2, and so on.

The landmark numbers for a base-n system are: $n^0, n^1, n^2, n^3, \dots$

We know that any number raised to the power of 0 is 1. So, $n^0$ is always 1. The landmark numbers are 1, n, n^2, n^3, ...

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(i) The landmark numbers of a base-7 system are 1, 7, 49, 343, 2401. (ii) The landmark numbers of a base-n number system are the powers of n starting from $n^0 = 1$ , which are 1, n, n^2, n^3, ...

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