mathematics

Strobogrammatic number

Strobogrammatic 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 Strobogrammatic number rather than just read about it. In short: A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated by 180 degrees. In other words, the numeral looks the same right-side up and upside down (e.g., 69, 96, 1001).

Key takeaways

Strobogrammatic 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 Strobogrammatic number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Strobogrammatic number from memory before moving on to harder problems.

Reference excerpt

A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated by 180 degrees. In other words, the numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). A strobogrammatic prime is a strobogrammatic number that is also a prime number, i.e., a number that is only divisible by one and itself (e.g., 11). It is a type of ambigram, words and numbers that retain their meaning when viewed from a different perspective, such as palindromes.

Description

With standard handwriting, the numbers, 0, 1, 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other when rotated 180 degrees. In such a system, the first few strobogrammatic numbers are: 0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, 6699, 6889, 6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, … (sequence A000787 in the OEIS) The first few strobogrammatic primes are:

11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889, … (sequence A007597 in the OEIS) The years 1881 and 1961 were the most recent strobogrammatic years; the next strobogrammatic year will be 6009. Although amateur aficionados of mathematics are quite interested in this concept, professional mathematicians generally are not. Like the concept of repunits and palindromic numbers, the concept of strobogrammatic numbers is base-dependent (expanding to base-sixteen, for example, produces the additional symmetries of 3/E; some variants of duodecimal systems also have this and a symmetrical x). Unlike palindromes, it is also font dependent. The concept of strobogrammatic numbers is not neatly expressible algebraically, the way that the concept of repunits is, or even the concept of palindromic numbers.

Nonstandard systems The strobogrammatic properties of a given number vary by typeface. For instance, in an ornate serif type, the numbers 2 and 7 may be rotations of each other; however, in a seven-segment display emulator, this correspondence is lost, but 2 and 5 are both symmetrical. There are sets of glyphs for writing numbers in base 10, such as the Devanagari and Gurmukhi of India in which the numbers listed above are not strobogrammatic at all. In binary, given a glyph for 1 consisting of a single line without hooks or serifs and a sufficiently symmetric glyph for 0, the strobogrammatic numbers are the same as the palindromic numbers and also the same as the dihedral numbers. In particular, all Mersenne numbers are strobogrammatic in binary. Dihedral primes that do not use 2 or 5 are also strobogrammatic primes in binary. The natural numbers 0 and 1 are strobogrammatic in every base, with a sufficiently symmetric font, and they are the only natural numbers with this feature, since every natural number larger than one is represented by 10 in its own base. In duodecimal, the strobogrammatic numbers are (using inverted two and three for ten and eleven, respectively)

0, 1, 8, 11, 2↊, 3↋, 69, 88, 96, ↊2, ↋3, 101, 111, 181, 20↊, 21↊, 28↊, 30↋, 31↋, 38↋, 609, 619, 689, 808, 818, 888, 906, 916, 986, ↊02, ↊12, ↊82, ↋03, ↋13, ↋83, … Examples of strobogrammatic primes in duodecimal are:

11, 3↋, 111, 181, 30↋, 12↊1, 13↋1, 311↋, 396↋, 3↊2↋, 11111, 11811, 130↋1, 16191, 18881, 1↋831, 3000↋, 3181↋, 328↊↋, 331↋↋, 338↋↋, 3689↋, 3818↋, 3888↋, …

Upside down year The most recent upside down year was 1961, and before that were sequentially 1881 and 1691, unless leading zeroes are allowed to be arbitrarily added. In this case, 02020 would be the most recent upside down year. Before that were 1111 and 1001, and before that were the 3-digit years 986, 916, 906, 888, 818, 808, 689, 619, 609, 181, 111, and 101. If the number 2 is included (in the case of seven-segment displays), then the most recent upside down year is 2002. However, 2002 is not a strobogrammatic number due to the 2 being different in traditional fonts. Using only the digits 0, 1, 6, 8 and 9, the next upside-down year will not occur until 6009. Allowing for the numbers 2, 5 and 7, the next such year will be 2112. Mad magazine parodied the upside down year in March 1961.

References

Illustrations

Worked examples

Example 1 — a first encounter with Strobogrammatic number

Start with the simplest possible case. Write down what Strobogrammatic 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 Strobogrammatic 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 Strobogrammatic 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 Strobogrammatic number

In research: Strobogrammatic 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 Strobogrammatic 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: Strobogrammatic number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Classes of prime numbers, Integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Strobogrammatic 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 Strobogrammatic number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Strobogrammatic 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 Strobogrammatic number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Strobogrammatic number in simple terms?

Why does Strobogrammatic 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 Strobogrammatic 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 Strobogrammatic number.