# Negative numbers are direction

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/2/tee-c2-s1

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c2-s1/

### 1. The minus sign can describe a number

Zero is the reference point. Positive numbers such as 1, 2 and 3 lie to its right on a number line. Negative numbers such as −1, −2 and −3 lie to its left. The number −3 is three units below zero; it is not an instruction to subtract until another number is supplied.

![Read left as smaller and right as larger. Follow one signed movement at a time.](https://deep-learning-study-room.vercel.app/assets/figures/v_negative.svg)

Diagram: Read left as smaller and right as larger. Follow one signed movement at a time.. Labels: Negative numbers are positions and directions | Negative numbers are positions and directions | Left means smaller. Right means larger. | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 0 + (−2) = −2 | −2 + 2 = 0 | Subtracting a negative reverses its direction: 5 − (−2) = 5 + 2 = 7.

Read left as smaller and right as larger. Follow one signed movement at a time. Tap the diagram to enlarge.

In 5 − 2, the minus sign is an operation: subtract 2 from 5. In 5 + (−2), the sign inside the brackets belongs to the number negative 2. Both expressions finish at 3. Brackets keep the signed number together.

### 2. Adding a negative moves left

Start at 5. Adding positive 2 moves two places right, to 7. Adding negative 2 moves two places left, to 3.



$$
\begin{aligned}5+2&=7\\5+(-2)&=3\end{aligned}
$$



If both numbers are negative, keep moving left. Start at −3 and add −2: the answer is −5.



$$
(-3)+(-2)=-5
$$



### 3. Subtracting reverses that movement

Subtracting positive 2 moves left. Subtracting negative 2 reverses a leftward movement, so it moves right. A money example is removing a debt of 2: your balance increases by 2.



$$
5-(-2)=5+2=7
$$



The shortcut “two negatives make a positive” is not a rule for every calculation. Negative 3 plus negative 2 is still −5. First identify the operation: addition, subtraction, multiplication or division.

### 4. Multiplication has its own sign rule

For multiplication and division, equal signs give a positive result and different signs give a negative result. Calculate the ordinary size, then attach the correct sign.

| Signs being multiplied | Complete calculation | Result sign |
| --- | --- | --- |
| Positive × positive | 3 × 2 = 6 | Positive |
| Negative × positive | (−3) × 2 = −6 | Negative |
| Positive × negative | 3 × (−2) = −6 | Negative |
| Negative × negative | (−3) × (−2) = 6 | Positive |

Division uses the same sign pattern: (−6) ÷ 2 = −3 and (−6) ÷ (−2) = 3. The divisor must be nonzero.

### 5. Fully worked arithmetic

**Question.** Calculate −3 + 2, −3 − 2 and −3 − (−2).

**Answer — step by step.**

### Given

Expressions: −3+2; −3−2; −3−(−2).

### To find

Evaluate the three signed-number expressions.

### Formula — which one and why

**Order of operations.** Grouping decides which operation is performed first. Evaluate brackets, then powers, then multiplication/division, then addition/subtraction.

A minus sign inside brackets belongs to that number. Subtracting a negative changes the operation to addition.



$$
a-(-b)=a+b
$$



**Yaad rakhne ka cue:** Bracket pehle; phir power; phir multiply/divide; aakhir mein plus-minus.

### Solution — step by step

**Step 1: evaluate the first expression.** start at −3 and move two places right. The answer is −1.



$$
-3+2=-1
$$



**Step 2: evaluate the second expression.** start at −3 and move two places left. The answer is −5.



$$
-3-2=-5
$$



**Step 3: evaluate the third expression.** replace subtract-negative-2 with add-positive-2. The answer is −1.



$$
\begin{aligned}-3-(-2)&=-3+2\\&=-1\end{aligned}
$$



Later, a weight update subtracts a correction that can itself be negative. The third expression explains exactly why that update can increase a weight.