Negative Numbers and Linear Equations

Below Zero #

The number line carries on past zero.

The number line carries on below zero, and the minus sign names that side

Zero is not the end of the scale. It carries on below — that is where negatives live.

3 below zero is −3. The minus sign says which side of zero you are on.

Start at −3 and add 4. Adding warms you up, right past zero, and lands you on 1.

Ordering Negative Numbers #

Bigger means higher, not bigger-looking.

Negatives order by depth below zero, so the bigger-looking number can be the smaller

−2 sits two rungs below zero. Where does −7 go? Five rungs deeper still.

−7 hangs below −2, so −7 is smaller. Bigger means higher up, not bigger-looking.

Sideways it still holds: −6, then −1, then 3 — position is the whole story.

So −2 > −3: the higher of two numbers is the larger, however they look.

Adding and Subtracting Across Zero #

Walks that pass through zero.

Adding climbs and subtracting drops, and either walk can carry you across zero

Take away more than you have: 5 − 9 drops nine rungs, straight past zero to −4.

Two falls stack: −3 + (−4) starts below zero and drops 4 more, landing on −7.

Adding still climbs: −3 + 5 rises through zero and comes out at 2.

Subtracting Negative Numbers #

Taking away a debt leaves you better off.

Subtracting a negative adds, because taking away a debt leaves you better off

Subtract less each row and the answer climbs by 1: 5 − 2 = 3, 5 − 1 = 4, 5 − 0 = 5.

Carry the pattern one row below zero. The answer has to climb by 1 once more.

So 5 − (−1) = 6 and 5 − (−2) = 7. Subtracting a negative adds it on.

Owe 2 and have the debt wiped: you end 2 better off. That is 5 − (−2) = 5 + 2.

Absolute Value #

How far a number sits from zero.

The absolute value of a number is its distance from zero, so it is never negative

−5 and 5 sit the same distance from zero: five steps, one going each way.

Write that distance as |x|. The bars keep the size, so |x| is never below zero.

Two numbers now. The gap from −2 up to 6 is eight steps of the same line.

The distance between two numbers is |p − q| — subtract either way round.

Multiplying and Dividing Negative Numbers #

Two minus signs multiply to a plus.

Two negatives multiplied give a positive, and dividing follows the same sign rule

Down each row the left number drops by 1, and the answer climbs by 2.

Carry the pattern below zero. The answer has to climb by 2 once more.

So −1 × −2 is 2, and −2 × −2 is 4. Two negatives multiplied give a positive.

Division reads a multiplication backwards, so the same sign rule settles it.

Algebraic Notation #

A letter holds a number you do not know yet.

A letter is a placeholder for a number you do not know yet

The scales are perfectly level, so x has to be 5.

Here x + 2 balances 7. The letter still stands for one particular number.

Solving Linear Equations #

Do the same to both sides.

Whatever you do to one side you must do to the other

You want x all on its own. That 2 is in the way.

Take 2 off both pans and the scales stay perfectly level.

x is alone and the pans are still level, so x = 5.

A two-step one: 3x + 2 = 14. Clear the added 2 first, and both pans lose 2.

That leaves 3x = 12. Three lots of x weigh 12, so one x weighs 4.