Powers and scientific notation for large and small numbers

One file takes 2,500,000 bytes; another takes 2,500,000,000. How much larger is the second? Writing $2.5\cdot10^6$ and $2.5\cdot10^9$ reveals matching coefficients and three additional factors of ten. Their ratio is $10^{9-6}=10^3=1000$ . Let us see why exponents work this way.

After the lesson on letters in formulas, the symbols $a$ and $n$ are familiar. Here $a$ will be the base of a power and $n$ its integer exponent.

An exponent counts identical factors

$2^4$ means $2\cdot2\cdot2\cdot2=16$ , rather than $2\cdot4$ . For a natural number $n>1$ , the power $a^n$ is a product of $n$ identical factors $a$ ; $a^1=a$ .

If the base is enclosed in parentheses, its sign belongs to it: $(-2)^2=4$ . In $-2^2$ , we evaluate the power first, then change the sign, giving −4. Parentheses change the meaning, just as in other expressions.

When multiplying powers with the same base, we join their factors: $a^3\cdot a^2=a^5$ . When dividing, identical nonzero factors cancel: $a^5/a^2=a^3$ for $a\ne0$ .

Why zero and negative exponents help

The sequence $10^3,10^2,10^1$ means 1000, 100, 10. Each step that decreases the exponent by 1 divides the number by 10. Continuing gives $10^0=1$ , $10^{-1}=0.1$ , $10^{-2}=0.01$ .

For a nonzero base we extend the rules consistently:

a^0=1,\qquad a^{-n}=\frac1{a^n}\quad(n=1,2,3,\ldots).

The zero exponent corresponds to $a^n/a^n=1$ ; a negative exponent gives the reciprocal. It does not mean a negative result: $2^{-3}=1/8$ , while $(-2)^{-3}=-1/8$ . For a zero base, $0^n=0$ for positive integer $n$ , but we do not define $0^0$ or negative powers of zero here. Rules and conditions in OpenStax, §1.2.

Laws with conditions

For nonzero real numbers $a,b$ and integers $m,n$ :

Operation	Rule
Multiplying the same base	$a^m a^n=a^{m+n}$
Dividing the same base	$a^m/a^n=a^{m-n}$
A power of a power	$(a^m)^n=a^{mn}$
A power of a product	$(ab)^n=a^n b^n$
A power of a quotient	$(a/b)^n=a^n/b^n$

Some rules also apply to zero when exponents are positive; the shared nonzero-base condition safely covers every integer exponent. Adding exponents is not a rule for sums: $2^3+2^3=16$ , but $2^{3+3}=64$ .

Length in metres (m)

This is a multiplier scale, not a ruler: each equal step right multiplies the length by 10. From 0.001 m to 1 m is three steps, a factor of 1000.

Scientific notation separates out the scale

A nonzero number can be written in normalized scientific notation as $c\cdot10^k$ , where $k$ is an integer and $1\le|c|<10$ . Here $|c|$ is the distance of $c$ from zero. For a positive number, the coefficient is at least 1 and less than 10. For example, $2500000=2.5\cdot10^6$ and $0.00042=4.2\cdot10^{-4}$ . The expression $25\cdot10^5$ has the same value but is not yet normalized. We write zero simply as 0.

In this course we call $k$ the order in normalized decimal notation. It estimates scale quickly, but coefficients also matter for an exact ratio. For example, $9\cdot10^6$ and $1\cdot10^7$ have adjacent orders, yet differ by only a factor of $10/9\approx1.11$ .

For the original two files, the matching coefficients cancel. In decimal units these are 2.5 MB and 2.5 GB: NIST explicitly defines 1 MB = 10⁶ B and 1 GB = 10⁹ B, where B means byte. Match the units first, then compare the exponents.

Branch · UnitsThis technical detail is optional for understanding decimal powers.

GB and GiB are different byte counts

According to NIST’s table, 1 GiB = 2³⁰ B = 1,073,741,824 B, whereas 1 GB = 1,000,000,000 B. These are different units, not different roundings of one number. Binary-prefix names and symbols were developed to distinguish them. In a particular interface, check what the displayed label means; we assume no specific program’s behaviour here.

Branch · AccuracyMore about reporting measurement results.

Shorter notation does not create a better measurement

A reported measurement of $2.50\cdot10^6$ usually communicates three significant figures: 2, 5 and the trailing 0. Writing $2.5\cdot10^6$ communicates two. OpenStax Chemistry, §1.5 explains how significant figures relate to the precision of a reported result.

As exact mathematical numbers, these two expressions are equal. Adding a zero alone does not improve an instrument or prove a smaller error. Our file sizes are given as exact integer byte counts, so choosing a shorter notation does not itself imply measurement uncertainty.

When you see a long number, keep its unit first, then separate its coefficient from its power of ten. The exponent describes the scale, the coefficient refines the value, and an approximation sign tells you whether anything was rounded.