2.4 Powers of 10

Powers of 10 are the spine of our number system. Recall the expanded form:

58263 = (5\times 10^4) + (8\times 10^3) + (2\times 10^2) + (6\times 10^1) + (3\times 10^0).

With negative powers we can reach beyond the decimal point. Since 10^{-1} = \tfrac{1}{10}, 10^{-2} = \tfrac{1}{100}, and so on:

472.605 = (4\times 10^2) + (7\times 10^1) + (2\times 10^0) + (6\times 10^{-1}) + (0\times 10^{-2}) + (5\times 10^{-3}).

Scientific notation

Some quantities are monstrous. The mass of the Earth is roughly 5{,}972{,}000{,}000{,}000{,}000{,}000{,}000{,}000 \text{ kg}. Try counting those zeros, misjudge even one, and your answer is off by a factor of ten! There is a far steadier route. Any number can be cast as a value between 1 and 10 multiplied by a power of 10:

4800 = 4.8\times 10^3, \qquad 31200 = 3.12\times 10^4, \qquad 90{,}00{,}000 = 9\times 10^6.

Scientific (Standard) Notation Every number can be written as x \times 10^{y}, \qquad 1 \le x < 10, where the coefficient x holds the leading digits and the integer exponent y announces the size (how many places the decimal point has travelled).

The exponent is usually the part that matters most. Writing Mumbai’s population as 2\times 10^7, the 7 outweighs the 2 by far: bumping the 2 to 3 raises the count by half, but bumping the 7 to 8 multiplies it tenfold. How many digits you keep in the coefficient also signals how precisely you actually know the figure, “about 1.42 \times 10^5” claims sharper knowledge than “about 1.4 \times 10^5.”

Worked Example

Write 48{,}60{,}000 in standard form.

Slide the decimal point leftward until a single non-zero digit sits in front: 48{,}60{,}000 = 4.86 \times 10^{6}. The point hopped 6 places, so the exponent is 6. Likewise 72{,}915 = 7.2915\times 10^4 and 90{,}03{,}00{,}00{,}000 = 9.003\times 10^{10}.

Worked Example

Multiply (3\times 10^4) by (2\times 10^3), leaving the answer in scientific notation.

Multiply the coefficients, then add the exponents of 10: (3\times 10^4)\times(2\times 10^3) = (3\times 2)\times 10^{4+3} = 6 \times 10^{7}. Because 6 already lies between 1 and 10, no adjustment is needed: the answer is 6\times 10^7 = 6{,}00{,}00{,}000.

Figure it Out: Scientific Notation

Practice

  1. Write in powers-of-ten expanded form:   (i) 263   (ii) 7041   (iii) 5928
  2. Express in standard (scientific) form:   (i) 72{,}915   (ii) 48{,}060   (iii) 48{,}60{,}000   (iv) 90{,}03{,}00{,}00{,}000
  3. Distances from the Sun: Jupiter 7.785\times10^{11} m, Saturn 1.434\times10^{12} m, Venus 1.082\times10^{11} m. Which distance is the smallest?
    1. 263 = (2\times10^2)+(6\times10^1)+(3\times10^0)   (ii) 7041 = (7\times10^3)+(0\times10^2)+(4\times10^1)+(1\times10^0)   (iii) 5928 = (5\times10^3)+(9\times10^2)+(2\times10^1)+(8\times10^0).
    1. 7.2915\times10^4   (ii) 4.806\times10^4   (iii) 4.86\times10^6   (iv) 9.003\times10^{10}.
  3. The Sun–Venus distance, 1.082\times10^{11} m, is smallest, its exponent 11 is less than the 11 of Jupiter? Note Jupiter is 7.785\times10^{11} and Venus is 1.082\times10^{11}: both share exponent 11, so compare coefficients, 1.082 < 7.785, so Venus is smallest. (Saturn, with exponent 12, is the largest.)