Real Numbers
Learning objectives
- Define irrational numbers and the set of real numbers.
- Prove that \(\sqrt2\) is irrational.
- Describe how decimals tell rational and irrational numbers apart.
- Estimate square roots between consecutive integers and tenths.
A number that is not a fraction
A right triangle whose two legs have length 1 has a hypotenuse \(c\) satisfying:
So \(c=\sqrt2\approx 1.41421\). Is \(\sqrt2\) a fraction? The ancient Greeks discovered that it is not.
A real number is irrational if it is not rational, that is, it cannot be written as \(\frac{a}{b}\) with integers \(a,b\) and \(b\neq 0\). The real numbers, written \(\mathbb{R}\), are all rational and irrational numbers together; they correspond exactly to the points of the number line.
The number \(\sqrt2\) is irrational.
Show proof
Suppose \(\sqrt2=\frac{a}{b}\) with integers \(a,b\) having no common factor. Squaring gives \(a^2=2b^2\), so \(a^2\) is even. An odd number has an odd square, so \(a\) itself is even: \(a=2k\). Then \(4k^2=2b^2\), i.e. \(b^2=2k^2\), so \(b\) is even too. Now \(a\) and \(b\) are both even, contradicting that they have no common factor. Hence \(\sqrt2\) is not rational. ∎
We assumed the opposite of what we want, followed the consequences, and reached an impossibility. This method is called proof by contradiction.
\(\sqrt3,\ \sqrt5,\ \sqrt{10}\) and \(\pi\) are irrational, while \(\sqrt4=2\) and \(\sqrt{\frac{9}{16}}=\frac34\) are rational.
For a positive integer \(n\), the number \(\sqrt n\) is irrational unless \(n\) is a perfect square.
Decimals of real numbers
Every real number has a decimal expansion. It is rational exactly when the decimal stops or repeats. An irrational number has a decimal that never stops and never repeats:
The number \(0.1010010001\ldots\) (one more zero in each block) never repeats, so it is irrational.
No decimal with finitely many digits equals \(\sqrt2\). A value such as 1.41421 is only an approximation.
Squeezing √2 on the number line
The sliders below pick fractions \(\frac pq\) with numerator and denominator up to 60. Try \(\frac75\), \(\frac{17}{12}\) and \(\frac{41}{29}\): each decimal is closer to \(\sqrt2\approx 1.41421\), but no fraction ever equals it.
Between which integers, and then which tenths, is \(\sqrt7\)?
Since \(2^2=4<7<9=3^2\), we have \(2<\sqrt7<3\). Next \(2.6^2=6.76<7<7.29=2.7^2\), so \(2.6<\sqrt7<2.7\).
Show that \(1+\sqrt2\) is irrational.
Suppose \(1+\sqrt2=\frac ab\). Then \(\sqrt2=\frac ab-1=\frac{a-b}{b}\) would be rational, contradicting the theorem. So \(1+\sqrt2\) is irrational.
Exercises
Classify each as rational or irrational: \(0.25,\ \sqrt{16},\ \sqrt{10},\ 0.\overline{45},\ \pi\).
Show solution
\(0.25=\frac14\), \(\sqrt{16}=4\) and \(0.\overline{45}=\frac{45}{99}=\frac{5}{11}\) are rational. \(\sqrt{10}\) (10 is not a perfect square) and \(\pi\) are irrational.
Prove that \(\sqrt3\) is irrational. Hint: imitate the proof for \(\sqrt2\), using that 3 divides \(a^2\) only if 3 divides \(a\).
Show solution
Suppose \(\sqrt3=\frac ab\) in lowest terms. Then \(a^2=3b^2\), so 3 divides \(a^2\) and hence \(a\); write \(a=3k\). Then \(9k^2=3b^2\), so \(b^2=3k^2\) and 3 divides \(b\). Both \(a\) and \(b\) are divisible by 3, a contradiction.
Find consecutive integers, then consecutive tenths, with \(\sqrt{20}\) between them.
Show solution
\(4^2=16<20<25=5^2\), so \(4<\sqrt{20}<5\). Also \(4.4^2=19.36<20<20.25=4.5^2\), so \(4.4<\sqrt{20}<4.5\).
Is \(2\sqrt2\) rational? Explain.
Show solution
No. If \(2\sqrt2=\frac ab\) then \(\sqrt2=\frac{a}{2b}\) would be rational, contradicting the theorem.
True or false: the sum of two irrational numbers is always irrational.
Show solution
False: \(\sqrt2+(-\sqrt2)=0\), which is rational.
Find a rational number between \(\sqrt2\) and \(\sqrt3\).
Show solution
Since \(\sqrt2\approx1.414\) and \(\sqrt3\approx1.732\), try \(\frac32\): \(\left(\frac32\right)^2=\frac94=2.25\) and \(2<2.25<3\), so \(\sqrt2<\frac32<\sqrt3\).
Summary
- Irrational numbers cannot be written as fractions; \(\sqrt2\) is the classic example.
- Rational decimals stop or repeat; irrational decimals do neither.
- The real numbers are all rationals and irrationals, exactly the points of the number line.
- To prove a number is irrational, assume it is a fraction and reach a contradiction.