20
Estimating the error
Textbook: pp. 114–116
GoalEstimate the absolute error when the exact value is unknown; understand the notation x = a ± h.
New words
estimating the error · xatolikni baholashapproximation from below · kami bilan taqribiy qiymatapproximation from above · ortig‘i bilan taqribiy qiymataccuracy up to h · h gacha aniqlik
Explanation
Often the exact value is unknown but we know that it lies between two numbers: a₁ ≤ x ≤ a₂. Then a₁ is an approximation of x from below and a₂ from above. If we take a as the approximate value and |x - a| ≤ h, we say x equals a up to h and write x = a ± h. This means the same as the double inequality a - h ≤ x ≤ a + h. For example, if the divisions of a measuring instrument are 1 unit apart, we take the nearest division as the approximate value and the error is at most 0.5.
Worked examples
A board’s length lies between the marks 36 cm and 37 cm. Taking a = 36.5, from 36 ≤ x ≤ 37 we get -0.5 ≤ x - 36.5 ≤ 0.5, that is x = 36.5 ± 0.5 cm.
The notation x = 7.5 ± 0.2 means 7.5 - 0.2 ≤ x ≤ 7.5 + 0.2, that is 7.3 ≤ x ≤ 7.7; 7.3 is the approximation from below and 7.7 from above.
Class activity
“Measuring with a ruler”: each pupil measures a notebook with a millimetre ruler and writes the approximate value and error bound as x = a ± h.
Practice
1
Write x = 15 ± 2 as a double inequality.
13 ≤ x ≤ 17
2
Write 8.1 ≤ y ≤ 8.5 in the form y = a ± h.
y = 8.3 ± 0.2
3
A length measured with a ruler lies between the 4 cm and 5 cm marks. If 4.5 cm is taken as the approximation, what is the error bound?
0.5 cm
4
Why does x = a ± h mean that x lies in an interval rather than being exact?
Because it means |x - a| ≤ h, which allows many (infinitely many) values of x.