Which comes first?

What you will be able to do

After this lesson you can

• say which part of a sum Python does first,
• use brackets to change the order, or to make it clear,
• find the wrong order in a formula that gives a wrong answer, and
• write a formula from mathematics in Python without mistakes.

The same order as in mathematics

What is 2 + 3 * 4? A person who reads from left to right says: 2 plus 3 is 5, times 4 is 20. A person who knows the rules of mathematics says: multiplication comes first, 3 times 4 is 12, and 2 plus 12 is 14. Python follows the rules of mathematics.

The order, from first to last:

1. Brackets ( ).
2. Powers **.
3. Multiplication and division * / // %, from left to right.
4. Addition and subtraction + -, from left to right.

A formula with the wrong order

The average of two marks is the sum divided by 2. Look at a mistake that many beginners make.

Why did wrong go wrong? In mark_1 + mark_2 / 2, the division comes first: 90 / 2 is 45, and then 80 + 45 is 125. In (mark_1 + mark_2) / 2, the brackets make the sum first. We meant to add first, but Python does what you wrote.

In mark_1 + mark_2 / 2, only mark_2 is halved, because division comes before addition.

When in doubt, use brackets

Extra brackets never do harm. (2 * 3) + 4 is the same as 2 * 3 + 4, but it shows the reader what you mean. If you cannot remember the order, put in brackets. Experienced programmers do this all the time.

Write a formula from mathematics

Take a formula from science: the distance of a body that starts from rest is s = (1/2) * a * t². In Python:

• s = 0.5 * a * t ** 2 is right. The power t ** 2 is done first, and then the multiplications.
• s = 1 / 2 * a * t ** 2 is also right. 1 / 2 is 0.5, and the rest goes left to right.
• s = (a * t) ** 2 / 2 is wrong. The bracket makes (a * t) squared, which is not the formula.

Always check a formula with a simple example: take a = 2 and t = 3. The right answer is 0.5 * 2 * 9 = 9.

Common mistakes

• Reading a sum from left to right. The multiplication comes first.
• Missing brackets in an average. a + b / 2 is not (a + b) / 2.
• Missing brackets in a fraction. For (a + b) / (c + d), you need both pairs of brackets.
• Forgetting that a power is done before a minus. -3 ** 2 is -9.
• Forgetting that the left-to-right rule applies to division too. 8 / 4 / 2 is 1.0, and not 4.0.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Predict, then check again, and do the second set of sums, which has brackets.
2. Build. Open the problem Order of operations by hand. Work out each sum on paper, before you run it.
3. Stretch. Open the problem A mistake with brackets. Find the wrong formula and fix it.

Remember these

• The order is: brackets, powers, then * / // % from left to right, then + - from left to right.
• 2 + 3 * 4 is 14, and (2 + 3) * 4 is 20.
• -2 ** 2 is -4, because the power comes before the minus.
• Use brackets when you want a different order, and also when you are not sure.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What are 4 + 6 / 2 and (4 + 6) / 2?

2. What is 2 ** 2 ** 3? (The power sign goes from right to left.)

3. Write the average of three marks a, b and c in Python.

4. What is 10 - 2 * 3 + 1?