Scientific notation with significant figures

Compare 4.5 and 4.50. Which statement is true?

Move the value from 9.91 to 9.99. The table shows numerical values after rounding to two significant figures, before normalising. Find where the rounded value reaches ten

What happens at 9.95 and above?

Check value, form and precision

For a nonzero signed value, normalise its magnitude and retain its sign. Round the coefficient once, then normalise again if a carry reaches ten

1. 1Normalised coefficient Its magnitude is at least one and less than ten. The exponent is an integer \(\displaystyle 1\le |a|<10\)
2. 2Significant digits Start at the first nonzero digit. Internal zeros and required trailing decimal zeros count \(\displaystyle 0.00450=4.50\times10^{-3}\)
3. 3Round once Use the next digit of the original value. At an exact halfway value, round the magnitude up and restore the sign \(\displaystyle -7.345\approx-7.35\quad(3\text{ s.f.})\)

Given

\(\displaystyle 0.00450\)

How many significant figures are recorded?

Given

\(\displaystyle 0.004507=a\times10^n\)

First normalise without rounding. Enter the coefficient

Given

\(\displaystyle 0.004507=4.507\times10^n\)

Now enter the integer exponent

Given

\(\displaystyle 4.507\)

Round the coefficient to three significant figures

Given

\(\displaystyle 0.004507\)

Choose the normalised result to three significant figures

Given

\(\displaystyle -73{,}460=a\times10^n\)

Normalise the signed number without rounding. Enter the coefficient

Given

\(\displaystyle -7.346\)

Round this coefficient to three significant figures

Given

\(\displaystyle -73{,}460\approx-7.35\times10^n\)

Enter the exponent for the rounded result

Equal value does not mean equal stated precision

Do not call equal numbers unequal because one omits a significant zero. Instead check whether the answer records the requested precision

1. 1Same value These coefficients are numerically equal \(\displaystyle 6.20=6.2\)
2. 2Different stated precision 6.20 records three significant figures. 6.2 records two \(\displaystyle 6.204\approx6.20\quad(3\text{ s.f.})\)

Given

\(\displaystyle 6.204\)

Round to three significant figures. Keep the required final decimal zero

Given

\(\displaystyle 620{,}400\)

Choose scientific notation with exactly three significant figures

A carry can change the exponent

Rounding is followed by normalisation. Changing the coefficient by a factor of ten requires the opposite change to the power of ten

1. 1Round the value At two significant figures, the rounded magnitude is ten \(\displaystyle 9.96\approx10\)
2. 2Normalise and record precision Write 1.0 to record the requested two significant figures \(\displaystyle 10\times10^4=1.0\times10^5\)

Given

\(\displaystyle 9.96\)

Round this coefficient to two significant figures. Enter its numerical value before normalising

Given

\(\displaystyle 10\times10^4=1\times10^n\)

Normalise the rounded value. Enter the new exponent

Given

\(\displaystyle 10\times10^4=a\times10^5\)

Enter the normalised coefficient with exactly two significant figures

Given

\(\displaystyle 99{,}600\)

Choose the normalised result to two significant figures

Given

\(\displaystyle 0.0000996\)

Write this small number in scientific notation to two significant figures. Enter only the coefficient

Given

\(\displaystyle 0.0000996\approx1.0\times10^n\)

Enter the exponent of the rounded, normalised result

Given

\(\displaystyle -0.009996\)

Choose scientific notation to three significant figures

Given

\(\displaystyle -3.04\times10^{-4}\)

Convert to an ordinary decimal numeral

Given

\(\displaystyle 7.20\times10^0\)

Convert to an ordinary numeral. Only the value is asked for here

Given

\(\displaystyle 0.0005984\)

Independent check: express the number in scientific notation to three significant figures. Enter the coefficient

Given

\(\displaystyle 0.0005984\approx5.98\times10^n\)

Finish the independent check. Enter the exponent