mathematics

Self number

Self number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Self number rather than just read about it. In short: In number theory, a self number in a given number base b {\displaystyle b} is a natural number that cannot be written as the sum of any other natural number n {\displaystyle n} and the individual digits of n {\displaystyle n} . 20 is a self number (in base 10), because no such combination can be found (all n < 15 {\displaystyle n<15} give a result less than 20; all other n {\displaystyle n} give a result greater tha…

Key takeaways

Self number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Self number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Self number from memory before moving on to harder problems.

Reference excerpt

In number theory, a self number in a given number base b {\displaystyle b} is a natural number that cannot be written as the sum of any other natural number n {\displaystyle n} and the individual digits of n {\displaystyle n} . 20 is a self number (in base 10), because no such combination can be found (all n < 15 {\displaystyle n<15} give a result less than 20; all other n {\displaystyle n} give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 using n = 15. These numbers were first described in 1959 by the Indian mathematician D. R. Kaprekar.

Definition and properties Let n {\displaystyle n} be a natural number. We define the b {\displaystyle b} -self function F b : N → N {\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} } for base b > 1 {\displaystyle b>1} to be the following:

F b ( n ) = n + ∑ i = 0 k − 1 d i . {\displaystyle F_{b}(n)=n+\sum _{i=0}^{k-1}d_{i}.}

where k = ⌊ log b ⁡ n ⌋ + 1 {\displaystyle k=\lfloor \log _{b}{n}\rfloor +1} is the number of digits in the number in base b {\displaystyle b} , and

d i = n mod b i + 1 − n mod b i b i {\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}

is the value of each digit of the number. A natural number n {\displaystyle n} is a b {\displaystyle b} -self number if the preimage of n {\displaystyle n} for F b {\displaystyle F_{b}} is the empty set. In general, for even bases, all odd numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers. The set of self numbers in a given base b {\displaystyle b} is infinite and has a positive asymptotic density: when b {\displaystyle b} is odd, this density is 1/2.

Self numbers in specific bases For base 2 self numbers, see (sequence A010061 in the OEIS). (written in base 10) The first few base 10 self numbers are:

1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97, 108, 110, 121, 132, 143, 154, 165, 176, 187, 198, 209, 211, 222, 233, 244, 255, 266, 277, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389, 400, 411, 413, 424, 435, 446, 457, 468, 479, 490, ... (sequence A003052 in the OEIS)

Self primes A self prime is a self number that is prime. The first few self primes in base 10 are

3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, ... (sequence A006378 in the OEIS)

References

Kaprekar, D. R. The Mathematics of New Self-Numbers Devaiali (1963): 19 - 20. R. B. Patel (1991). "Some Tests for k-Self Numbers". Math. Student. 56: 206–210. B. Recaman (1974). "Problem E2408". Amer. Math. Monthly. 81 (4): 407. doi:10.2307/2319017. JSTOR 2319017. Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001. Weisstein, Eric W. "Self Number". MathWorld.

Worked examples

Example 1 — a first encounter with Self number

Start with the simplest possible case. Write down what Self number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Self number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Self number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Self number

In research: Self number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Self number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Self number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic dynamics, Base-dependent integer sequences, Inverse functions, so understanding it makes those chapters shorter.
In everyday life: Look for Self number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Self number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Self number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Self number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Self number in simple terms?

Why does Self number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Self number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Self number.