Negative numbers and distance from zero

print(-7 // 2)

The number line

Picture a ruler that carries on to the left of zero. Numbers to the right are bigger; numbers to the left are smaller. So −1 is bigger than −3, because it sits further right, closer to zero. A temperature of −1 is warmer than −3, and a bank balance of −$1 is better than −$3.

Adding moves you right and subtracting moves you left, so subtracting more than you have takes you below zero:

print(-3 < -1)       # True: -3 is further left
print(3 - 10)        # 7 steps left of zero
print(-5 + 8)        # start at -5, move 8 right

A minus sign in front of a number, like -5, just says which side of zero it's on. It's easy to confuse with subtraction, because it's the same symbol: -5 is a number, while 3 - 5 is a calculation that happens to give a negative number. Zero itself is neither positive nor negative; it's the point where the two sides meet.

Negatives show up wherever something can go either way: profit and loss, a balance that's overdrawn, a change that's a fall rather than a rise, a position below sea level, a score below average.

The sign rules

Four rules cover almost everything:

• Adding a negative is the same as subtracting: 10 + (−4) = 6.
• Subtracting a negative is the same as adding: 10 − (−4) = 14.
• Multiply or divide two numbers with the same sign, and the result is positive: −3 × −4 = 12.
• Multiply or divide numbers with different signs, and the result is negative: −3 × 4 = −12.

Brackets help when a negative number follows an operator, both on paper and for readers of your code: 10 − (−4) is easier to read than 10 − −4, though Python accepts both.

The second rule sounds odd until you think of money. A negative number is a debt. Taking away a debt of $4 leaves you $4 better off, which is exactly what adding $4 does.

For multiplying, a pattern shows why two negatives make a positive. Multiply −3 by 2, 1, 0 and −1 in turn: −6, −3, 0, then 3. Each step down in the second number adds 3 to the result, so the pattern carries straight on past zero into the positives.

print(10 + -4, 10 - -4)
print(-3 * -4, -3 * 4, -12 / -3)

balance = 50
refund = -20
print(balance - refund)

high = -2
low = -9

swing = 0     # how far the temperature rose, low to high
middle = 0    # the average of high and low

print(swing, middle)

Distance from zero: abs()

Sometimes the sign doesn't matter, only the size. A forecast that's 3 degrees too high and one that's 3 degrees too low are both 3 degrees off. The absolute value of a number is its distance from zero, always zero or more. Python's abs() gives it:

print(abs(-7), abs(7), abs(0))
predicted = 120
actual = 104
error = actual - predicted
print(error, abs(error))

Pick one order for subtracting and stick to it; this course uses actual minus predicted. The error is −16: the prediction was 16 too high. Its absolute value, 16, is how far off it was. You'll meet this again as the mean absolute error, one of the most common ways to score a model: the average of the absolute errors, so misses in both directions count instead of canceling out.

abs() also measures the gap between any two numbers, whichever is bigger: abs(a - b) is the distance between them on the number line. The gap between −2 and 5 is 7, and so is the gap between 5 and −2. That's how you'd check whether two readings are "within 3 degrees of each other" without caring which one is higher.

miss = 47 - 52
print(miss, abs(miss))

forecast = 120
actual = 104

error = 0
size = 0
percent_off = 0

print(error, size, percent_off)

// and % with negative numbers

With positive numbers, // and % behave as you'd expect. With negatives, Python follows one rule that keeps them consistent: // always rounds down, toward the left of the number line, and % gives whatever is left over so that the two fit back together.

print(-7 // 2, -7 % 2)            # -4 and 1, because -4 * 2 + 1 == -7
print((-7 // 2) * 2 + (-7 % 2))   # always gets back to -7

That rule makes % perfect for anything that wraps around, like a clock. Five hours before 2 a.m. is (2 − 5) % 24, which is 21: nine in the evening the day before. Days of the week, months and angles wrap the same way. (Many other languages round toward zero instead, so this is a classic surprise when moving code between them.)

Rounding toward zero does exist in Python, just not in //. int() chops off the decimals, which moves a negative number up toward zero: int(-3.5) is −3, while -7 // 2 is −4. When you need a whole number from a negative value, decide which one you mean, and pick the tool that does it.

print((2 - 5) % 24)

landing_hour = 3
flight_hours = 7

departure_hour = 0

print("Left at hour", departure_hour)

Negative numbers in AI

You'll see negatives all through the course. A model's raw scores, called logits, are often negative. A model's weights can be negative, which means the feature pushes the prediction down. During training, the direction to nudge each weight comes out as a positive or negative number, and the sign is the whole message: up or down. And the logarithm of a probability is always negative, which you'll meet when models are scored.

Reading the sign first is a habit worth building: before asking "how big?", ask "which way?" And when you compare negative numbers, remember the number line: a loss of −40 is worse than a loss of −10, even though 40 is the bigger number. Sorting a list of profits puts the biggest losses first, which is exactly what you want when you're looking for the products that are hurting most.

old_rate = 8
new_rate = 6
print(new_rate - old_rate)