Numbers that tell a story

Start with a quantity

A number can describe a balance, a distance, or a measurement. Put those numbers on a line: moving right increases the value; moving left decreases it. Zero is the reference point. Negative numbers are ordinary numbers on the other side of that reference.

The real numbers, written $\mathbb R$ , include integers, fractions, and numbers such as $\sqrt{2}$ and $\pi$ . A fraction like $3/4$ divides a whole into four equal pieces and takes three. Its decimal form is $0.75$ , or $75\%$ .

A percentage needs a reference whole. A 20% increase from 50 adds 10; a 20% increase from 100 adds 20.

Predict a pattern of growth

Imagine starting with one square and doubling the number of squares every round. The counts are $1,2,4,8,16,\ldots$ . Adding one to the round number multiplies the count by two.

Explore it yourself

One doubling at a time

An exponent counts repeated multiplication. A logarithm asks you to count backward.

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2³ = 8

Number of doublings n3

2 × 2 × 2 = 8

An exponent is a compact count of repeated multiplication:

2^3=2\times2\times2=8.

A square root reverses squaring: $\sqrt{9}=3$ because $3^2=9$ . The symbol $\sqrt{9}$ means the nonnegative square root. The equation $x^2=9$ , however, has two solutions: $x=3$ and $x=-3$ .

A logarithm reverses exponentiation. If the input is 8, $\log_2 8$ asks, “How many factors of 2 take me from 1 to 8?” The answer is 3.

b^t=u\quad\Longleftrightarrow\quad\log_b u=t,

where $b>0$ , $b\ne1$ , and $u>0$ . The natural logarithm $\ln u$ uses the base $e\approx2.71828$ . You will meet it when working with likelihoods.

Use the rules without losing the meaning

For real exponents, take $b>0$ :

Expression	Meaning	Example
$b^r b^s=b^{r+s}$	Combine the factors	$2^2\cdot2^3=2^5=32$
$b^0=1$	Multiplicative starting point	$7^0=1$
$b^{-r}=1/b^r$	Take the reciprocal	$2^{-3}=1/8$
$(b^r)^s=b^{rs}$	Repeat the exponentiation	$(2^2)^3=2^6$
$\ln(uv)=\ln u+\ln v$	Turn products into sums, for $u,v>0$	$\ln(2\cdot3)=\ln2+\ln3$

Signs and parentheses matter: $(-2)^2=4$ , while $-2^2=-(2^2)=-4$ . Also, squaring does not distribute over addition: $(2+3)^2=25$ , but $2^2+3^2=13$ .

Work through a scale change

A dataset starts with 200 examples. Increasing its size by 50% gives $200(1+0.5)=300$ . Doubling it three times gives $200\cdot2^3=1600$ . These are different operations: a percentage scales a reference amount; an exponent counts repeated scaling.

Make a prediction

A quantity doubles from 1 to 32. How many doublings happened?

Take the idea with you

A root asks for a value. A logarithm asks for an exponent. When you see a complicated expression, name the operation before calculating.