Scientific Notation and Powers of Ten

Why Use Scientific Notation?

Some quantities are inconvenient to write with ordinary decimal notation. The distance from Earth to the Sun is about \(150{,}000{,}000\) kilometres, while the width of a red blood cell is about \(0.000008\) metres. Scientific notation writes each using a short coefficient and a power of ten.

The exponent records the general scale of the quantity, while the coefficient gives its position within that scale. This makes scientific notation useful not only for shortening long strings of zeros, but also for calculating and comparing quantities that differ greatly in size.

The exponent \(n\) may be positive, zero, or negative. For example, \(4.2\times10^5\) is a large number, \(4.2\times10^0=4.2\), and \(4.2\times10^{-5}\) is a small positive number. A negative exponent does not make the value negative: the sign of the number comes from its coefficient.

Converting a Decimal to Scientific Notation

To convert a nonzero decimal, move its decimal point until exactly one nonzero digit is to its left. The number of places moved determines the power of ten. If the original number is at least \(10\), moving the decimal point left gives a positive exponent. If its absolute value is less than \(1\), moving the decimal point right gives a negative exponent.

This procedure follows from the fact that moving a decimal point one place left divides the written coefficient by \(10\), so the product is preserved by multiplying by \(10\). Moving it one place right multiplies the coefficient by \(10\), so the product is preserved by dividing by \(10\). Repeating these changes gives the required power.

For a negative number, first write its magnitude in normalized form and then retain the negative sign. For instance, \(-0.0036=-3.6\times10^{-3}\). The coefficient has absolute value \(3.6\), which meets the normalization condition.

Why the Normalized Form Is Unique

For each nonzero positive number, there is exactly one normalized coefficient and integer exponent. The decimal-point procedure gives a representation, and the normalization condition rules out competing choices.

To see why, suppose a positive number had two normalized representations \(a\times10^m=b\times10^n\), with \(1\leq a<10\) and \(1\leq b<10\). If \(m<n\), then \(n\geq m+1\), so \(b\times10^n\geq10\times10^m\). But \(a\times10^m<10\times10^m\). The two representations could not be equal. The case \(n<m\) gives the same contradiction with the roles exchanged. Thus \(m=n\), and dividing the equality by \(10^m\) gives \(a=b\). For a negative number, apply the same argument to its positive magnitude and attach the negative sign.

Calculating with Powers of Ten

The exponent laws give efficient rules for multiplying and dividing scientific-notation expressions. For nonzero coefficients \(a\) and \(b\), and integers \(m\) and \(n\), the relevant laws are

In multiplication, multiply the coefficients and add the exponents. In division, divide the coefficients and subtract the denominator exponent from the numerator exponent. A resulting coefficient may fall outside the normalized range, so check it and adjust if needed.

Addition and subtraction work differently: the exponents cannot simply be added or subtracted. First express both numbers using the same power of ten, then add or subtract their coefficients.

Interpreting the Scale

The exponent gives a useful first indication of scale, but the coefficient matters when comparing values. If two positive normalized numbers have the same exponent, the one with the larger coefficient is larger. If their exponents differ by more than one, the larger exponent does guarantee the larger number. If the exponents differ by exactly one, the larger exponent still means the number is larger, but it does not guarantee a tenfold difference.

In fact, let \(a\times10^n\) and \(b\times10^{n+1}\) be positive normalized numbers, where \(1\leq a<10\) and \(1\leq b<10\). Their ratio, larger-exponent number divided by smaller-exponent number, is

Because \(b\geq1\) and \(a<10\), this ratio is greater than \(1\). Because \(b<10\) and \(a\geq1\), it is less than \(100\). Thus the larger-exponent number is greater, but the ratio can be less than \(10\), equal to \(10\), or greater than \(10\).

For example, \(\frac{1\times10^6}{9\times10^5}=\frac{10}{9}\), which is greater than \(1\) but less than \(10\). The exponents differ by one, yet the larger number is not ten times as large. This is why scale comparisons should use both coefficient and exponent.

Common Pitfall: Ignoring the Coefficient or the Sign of the Shift

A frequent error is to assume that a one-unit increase in exponent always makes a normalized positive number ten times as large. As the ratio calculation shows, that is not generally true: coefficients can offset much of the change in exponent.

Another common error is to use the wrong sign when converting a small decimal. For example, writing \(0.00042=4.2\times10^5\) is incorrect: the right-hand side is \(420{,}000\), not a small number. The decimal point moves five places right to make \(4.2\), so the correct power is negative:

Indeed, \(10^{-4}=0.0001\), and \(4.2\times0.0001=0.00042\). Always check both the direction of the decimal shift and the value obtained by expanding the power of ten.

Why Scientific Notation Matters

Scientific notation makes quantities easier to read, compare, and calculate. A value such as \(6.02\times10^{23}\) can be handled through its coefficient and exponent without writing all its zeros. The notation also communicates scale: \(10^{-6}\) represents one millionth, while \(10^6\) represents one million.

For measured quantities, the coefficient can also convey the precision of the reported value. The form \(3.2\times10^5\) and the form \(3.200\times10^5\) have the same numerical value, but the latter may indicate a more precise measurement when significant digits are being used. Scientific notation alone does not establish measurement precision; it is important to distinguish the written number from what is known about the measurement.