# Drill 1 Signs and brackets

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/17/tee-c17-s1

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c17-s1/

## Question

**Question.** Evaluate 2−0.1 × (−3), then (2−0.1) × (−3). Explain why they differ.

## Answer — step by step

### Given

Expressions: 2−0.1×(−3) and (2−0.1)×(−3).

### To find

Evaluate both and explain the effect of grouping.

### 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

### Meaning of the calculation

A minus sign can mean subtraction or a negative number. Brackets tell us which part belongs together. These two expressions have different brackets, so they describe different calculations.

### Rule used

Do brackets first, then multiplication, then addition or subtraction. Subtracting a negative number means adding its positive size.

**Step 1: First expression: multiply before subtracting.**



$$
2-0.1\times(-3)
$$



First multiply: a positive number times a negative number is negative.



$$
0.1\times(-3)=-0.3
$$



Put this complete result back into the expression. Keep its negative sign inside brackets.



$$
2-(-0.3)=2+0.3=2.3
$$



**Step 2: Second expression: finish the bracket first.**



$$
2-0.1=1.9
$$



Now multiply the whole bracket result by negative 3.



$$
1.9\times(-3)=-5.7
$$



**Final answer:** first expression = 2.3; second expression = −5.7. In a weight update, multiply learning rate by gradient first, then subtract that product from the old weight.