Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Unit Interval

Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product. That is, if $0< x< 1$ and $0< y< 1$ then prove $$x+y< 1+xy$$

## You may also like

### Circles Ad Infinitum

### Climbing Powers

Links to the University of Cambridge website
Links to the NRICH website Home page

Nurturing young mathematicians: teacher webinars

30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

Or search by topic

Age 16 to 18

ShortChallenge Level

- Problem
- Getting Started
- Solutions

Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product. That is, if $0< x< 1$ and $0< y< 1$ then prove $$x+y< 1+xy$$

Did you know ... ?

Pure inequalities such as this one are often used in the analysis of far more difficult mathematics problems: whilst the inequalities might be simple to prove in themselves, they can be surprisingly useful as tools.

Pure inequalities such as this one are often used in the analysis of far more difficult mathematics problems: whilst the inequalities might be simple to prove in themselves, they can be surprisingly useful as tools.

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?