Question 10 of 21intermediate💡 UnderstandShort Answer2 marks

Is there a number that cannot be represented in our base-5 system above? Why or why not?

Correct Answer

No, there is no number that cannot be represented in the base-5 system.

Explanation: Every positive integer can be expressed as a sum of powers of 5 with coefficients from 0 to 4.

Why this works:

  • The landmark numbers (powers of 5: 1, 5, 25, 125, ...) form a complete basis for representing all positive integers.

  • The division algorithm guarantees that any number can be uniquely expressed in this form.

∴ The base-5 system can represent all positive integers.

Exercise: Figure it Out (Section 5) | Q: 2 | (Page Page 17)
For More Understanding

Explanation

The chapter explains that a base-n number system has landmark numbers that are powers of n. The base-5 system can represent any positive integer because: (1) Powers of 5 (1, 5, 25, 125, 625, ...) continue infinitely, covering arbitrarily large numbers. (2) Any number can be decomposed into a unique sum of powers of 5 with coefficients 0-4. (3) This is analogous to how the Hindu base-10 system can represent all numbers using digits 0-9.

Solution Steps

  1. Recognize that a base-n system uses landmark numbers that are powers of n.

  2. In base-5, the landmark numbers are powers of 5: 1, 5, 25, 125, 625, ...

  3. Any positive integer can be decomposed into a unique sum of powers of 5 with coefficients 0-4.

  4. The division algorithm guarantees unique representation for all numbers.

  5. Conclude: No number cannot be represented in the base-5 system.

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