Question

Find the decimal form of √24.75

Answer 1

In this case we are going to use the 'Babylonian Method' to get the square root of any positive number.

We must set an error for the final result. Say, smaller than 0.001. In other words we will try to find the square root value with at least 2 correct decimal places.

So, we can say that the square root of 24.75 is 4.9749 with an error smaller than 0.001

• Step 1:
  Divide the number (24.75) by 2 to get the first guess for the square root .
  First guess = 24.75/2 = 12.375.
• Step 2:
  Divide 24.75 by the previous result. d = 24.75/12.375 = 2.
  Average this value (d) with that of step 1: (2 + 12.375)/2 = 7.1875 (new guess).
  Error = new guess - previous value = 12.375 - 7.1875 = 5.1875.
  5.1875 > 0.001. As error > accuracy, we repeat this step again.
• Step 3:
  Divide 24.75 by the previous result. d = 24.75/7.1875 = 3.4434782609.
  Average this value (d) with that of step 2: (3.4434782609 + 7.1875)/2 = 5.3154891305 (new guess).
  Error = new guess - previous value = 7.1875 - 5.3154891305 = 1.8720108695.
  1.8720108695 > 0.001. As error > accuracy, we repeat this step again.
• Step 4:
  Divide 24.75 by the previous result. d = 24.75/5.3154891305 = 4.6562036705.
  Average this value (d) with that of step 3: (4.6562036705 + 5.3154891305)/2 = 4.9858464005 (new guess).
  Error = new guess - previous value = 5.3154891305 - 4.9858464005 = 0.32964273.
  0.32964273 > 0.001. As error > accuracy, we repeat this step again.
• Step 5:
  Divide 24.75 by the previous result. d = 24.75/4.9858464005 = 4.9640518403.
  Average this value (d) with that of step 4: (4.9640518403 + 4.9858464005)/2 = 4.9749491204 (new guess).
  Error = new guess - previous value = 4.9858464005 - 4.9749491204 = 0.0108972801.
  0.0108972801 > 0.001. As error > accuracy, we repeat this step again.
• Step 6:
  Divide 24.75 by the previous result. d = 24.75/4.9749491204 = 4.9749252507.
  Average this value (d) with that of step 5: (4.9749252507 + 4.9749491204)/2 = 4.9749371856 (new guess).
  Error = new guess - previous value = 4.9749491204 - 4.9749371856 = 0.0000119348.
  0.0000119348 <= 0.001. As error <= accuracy, we stop the iterations and use 4.9749371856 as the square root.

So, we can say that the square root of 24.75 is 4.9749 with an error smaller than 0.001