mathematics

Integer square root

Integer square root 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 Integer square root rather than just read about it. In short: In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n, isqrt ⁡ ( n ) = ⌊ n ⌋ . {\displaystyle \operatorname {isqrt} (n)=\lfloor {\sqrt {n}}\rfloor .} For example, isqrt ⁡ ( 27 ) = ⌊ 27 ⌋ = ⌊ 5.19615242270663... ⌋ = 5. {\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor =\lf…

Key takeaways

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

Reference excerpt

In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n,

isqrt ⁡ ( n ) = ⌊ n ⌋ . {\displaystyle \operatorname {isqrt} (n)=\lfloor {\sqrt {n}}\rfloor .}

For example, isqrt ⁡ ( 27 ) = ⌊ 27 ⌋ = ⌊ 5.19615242270663... ⌋ = 5. {\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor =\lfloor 5.19615242270663...\rfloor =5.}

Introductory remark Let y {\displaystyle y} and k {\displaystyle k} be non-negative integers. Algorithms that compute (the decimal representation of) y {\displaystyle {\sqrt {y}}} run forever on each input y {\displaystyle y} which is not a perfect square. Algorithms that compute ⌊ y ⌋ {\displaystyle \lfloor {\sqrt {y}}\rfloor } do not run forever are nevertheless capable of computing y {\displaystyle {\sqrt {y}}} up to any desired accuracy k {\displaystyle k} . Choose any k {\displaystyle k} and compute ⌊ y × 100 k ⌋ {\textstyle \lfloor {\sqrt {y\times 100^{k}}}\rfloor } . For example (setting y = 2 {\displaystyle y=2} ):

k = 0 : ⌊ 2 × 100 0 ⌋ = ⌊ 2 ⌋ = 1 k = 1 : ⌊ 2 × 100 1 ⌋ = ⌊ 200 ⌋ = 14 k = 2 : ⌊ 2 × 100 2 ⌋ = ⌊ 20000 ⌋ = 141 k = 3 : ⌊ 2 × 100 3 ⌋ = ⌊ 2000000 ⌋ = 1414 ⋮ k = 8 : ⌊ 2 × 100 8 ⌋ = ⌊ 20000000000000000 ⌋ = 141421356 ⋮ {\displaystyle {\begin{aligned}&k=0:\lfloor {\sqrt {2\times 100^{0}}}\rfloor =\lfloor {\sqrt {2}}\rfloor =1\\&k=1:\lfloor {\sqrt {2\times 100^{1}}}\rfloor =\lfloor {\sqrt {200}}\rfloor =14\\&k=2:\lfloor {\sqrt {2\times 100^{2}}}\rfloor =\lfloor {\sqrt {20000}}\rfloor =141\\&k=3:\lfloor {\sqrt {2\times 100^{3}}}\rfloor =\lfloor {\sqrt {2000000}}\rfloor =1414\\&\vdots \\&k=8:\lfloor {\sqrt {2\times 100^{8}}}\rfloor =\lfloor {\sqrt {20000000000000000}}\rfloor =141421356\\&\vdots \\\end{aligned}}}

Compare the results with 2 = 1.41421356237309504880168872420969807856967187537694... {\displaystyle {\sqrt {2}}=1.41421356237309504880168872420969807856967187537694...}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Integer square root

Start with the simplest possible case. Write down what Integer square root 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 Integer square root 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 Integer square root 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 Integer square root

In research: Integer square root 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 Integer square root 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: Integer square root is common in secondary-school and first-year university syllabi. It links to neighbouring topics Number theoretic algorithms, Number theory, Root-finding algorithms, so understanding it makes those chapters shorter.
In everyday life: Look for Integer square root 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Integer square root in 20 minutes

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

Frequently asked questions

What is Integer square root in simple terms?

In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n, isqrt ⁡ ( n ) = ⌊ n ⌋ . {\displaystyle \operatorname {isqrt} (n)=\lfloor {\sqrt {n}}\rfloor .} For example, isqrt ⁡ (…

Why does Integer square root 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 Integer square root?

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 Integer square root.