Worksheet

Worksheet: integer part and fractional part

Published

Open PDF The PDF is in Romanian.

Press “Check” next to an exercise. Your answers stay on this device.

Remember

$\lbrack x\rbrack\ \leq \ x\ < \ \lbrack x\rbrack\ + \ 1$ — the integer part of $x$ is the greatest integer ≤ $x$.

$\{ x\}\ = \ x\ - \ \lbrack x\rbrack$, $0\ \leq \ \{ x\}\ < \ 1$ — the fractional part of $x$.

$x\ = \ \lbrack x\rbrack\ + \ \{ x\}$

$\lbrack x\ + \ n\rbrack\ = \ \lbrack x\rbrack\ + \ n$ and $\{ x\ + \ n\}\ = \ \{ x\}$, for every $n\ \in \ {\mathbb{Z}}$.

Level I – Knowledge and direct calculation

  1. Calculate $\lbrack 7,3\rbrack$ and $\{ 7,3\}$.

  2. Calculate $\lbrack - 4,6\rbrack$ and $\{ - 4,6\}$.

  3. Calculate $\lbrack 5\rbrack$ and $\{ 5\}$ (5 is an integer).

  4. Calculate $\lbrack\sqrt{30}\rbrack$ and $\{\sqrt{30}\}$, given that $5\ < \ \sqrt{30}\ < \ 6$.

  5. Calculate $\lbrack - \sqrt{2}\rbrack$ and $\{ - \sqrt{2}\}$, given that $1\ < \ \sqrt{2}\ < \ 2$.

  6. Calculate $\lbrack\frac{17}{4}\rbrack$ and $\{\frac{17}{4}\}$.

  7. Find the real number $x$, given that $\lbrack x\rbrack\ = \ - 3$ and $\{ x\}\ = \ 0,25$.

Level II – Equations and properties

  1. Solve in $\mathbb{R}$ the equation $\lbrack x\rbrack\ = \ 4$.

  2. Find the real number $x$, given that $\lbrack x\rbrack\ = \ 10$ and $\{ x\}\ = \ 0,6$.

  3. Solve the equation $\lbrack 2x\rbrack\ = \ 7$, $x\ \in \ {\mathbb{R}}$.

  4. Solve the equation $\lbrack x\ - \ 1\rbrack\ = \ 3$.

  5. Show that $\{ x\ + \ 7\}\ = \ \{ x\}$ for every $x\ \in \ {\mathbb{R}}$.

  6. Solve the equation ${\lbrack x\rbrack}^{2}\ = \ 9$, for $x\ \geq \ 0$.

  7. Solve the equation $\lbrack 3x\ - \ 2\rbrack\ = \ 4$.

Level III – Equations, properties and case analysis

  1. Solve the equation ${\lbrack x\rbrack}^{2}\ - \ 5\lbrack x\rbrack\ + \ 6\ = \ 0$.

  2. Solve the equation ${\{ x\}}^{2}\ = \ \{ x\}$.

  3. Solve the equation $\lbrack x\rbrack\ + \ \{ x\}\ = \ 2x\ - \ 3,4$.

  4. Solve the inequality $2\ < \ \lbrack x\rbrack\ < \ 6$, $x\ \in \ {\mathbb{R}}$.

  5. Find the possible values of the expression $E\ = \ \{ x\}\ + \ \{ - x\}$, $x\ \in \ {\mathbb{R}}$.

  6. Show that $\lbrack x\ + \ 1\rbrack\ - \ \lbrack x\rbrack\ = \ 1$ for every $x\ \in \ {\mathbb{R}}$.

  7. Solve the equation $\lbrack x\rbrack\ \cdot \ \{ x\}\ = \ 0$.

Level IV – Advanced and competition problems

  1. Solve the equation $\lbrack x\rbrack\ + \ \lbrack 2x\rbrack\ = \ 10$, $x\ \in \ {\mathbb{R}}$.

  2. Prove Hermite's identity: $\lbrack x\rbrack\ + \ \lbrack x\ + \ \frac{1}{2}\rbrack\ = \ \lbrack 2x\rbrack$ for every $x\ \in \ {\mathbb{R}}$.

  3. Solve the equation $\{ x\}\ + \ \{ 2x\}\ = \ 1$, $x\ \in \ \lbrack 0;\ 1)$.

  4. Find $x\ \in \ {\mathbb{R}}$ such that ${\lbrack x\rbrack}^{2}\ = \ x\ + \ \lbrack x\rbrack$.