Rational approximations of irrationals
Rational Approximations of Irrationals
Rational Approximations of Irrationals is a topic in Numbers & Operations in the Texas Essential Knowledge and Skills. It is aligned to Standard 8.2(D), which requires students to approximate irrational numbers with rationals and locate them on a number line.
A rational approximation estimates an irrational number closely enough to place it on a number line and compare it.
Theory
A rational approximation estimates an irrational number closely enough to place it on a number line and compare it.
Bracketing:
How to approximate a root
- Find the perfect squares just below and above.
- The root lies between those integers.
- Square trial decimals to narrow it down.
- Round to the needed place.
\(1<2<4\), so between \(1\) and \(2\).
| \(1\) | < | \(\sqrt2<2\) |
\(1.4^2=1.96\) and \(1.5^2=2.25\), so \(\sqrt2\approx1.4\).
| \(\sqrt2\) | \(\approx\) | \(1.4\) |
\(\sqrt5\approx2.24\), so \(2.3\) is larger.
| \(2.3\) | \(>\) | \(\sqrt5\) |
\(9<10<16\), so between \(3\) and \(4\), near \(3.2\).
| \(\sqrt{10}\) | \(\approx\) | \(3.16\) |
Common pitfalls
Frequently asked questions
How do I approximate a square root?
Bracket it between perfect squares, then refine the decimal.
What is \(\sqrt2\) approximately?
About \(1.41\).
Are approximations exact?
No, only the radical form is exact.
Between which integers is \(\sqrt{10}\)?
\(3\) and \(4\).