Adding, subtracting, multiplying and dividing positive and negative integers

At some point, somebody has told you that a minus and a minus make a plus. Somebody else has told you that is nonsense. Two things below zero do not add up to something above it. They are both right. Minus eight add minus five is minus thirteen. Colder still. But minus eight times minus five is forty. That is above zero, and it came from the same two numbers. Same two numbers, same two minus signs, and the answers land on opposite sides of zero. Nothing is broken. Adding and multiplying are doing two different jobs with that sign, and this video sorts out which is which.

There are two lanes in this topic, and they behave differently. Adding and subtracting live in one. Multiplying and dividing live in the other. Start with the adding lane. Here is the number line. Zero in the middle, the cold side stretching away to the left, the warm side to the right. Which side of zero you sit on is what your number means. Which way the sign moves you is what the sum does. Adding and subtracting are a walk along this line. Adding moves you further right. Start at minus seven, add four, and you walk four steps right onto minus three. Subtracting moves you further left. Start at one, subtract four, and you walk four steps left, straight past zero, onto minus three as well. Your go. Start at minus three and add five. Do you land on minus eight, on two, or on eight? Have a think. I will wait. The answer is two. From minus three you walk five steps right. Three of those steps take you up to zero, and the two left over carry you past it. Two. Now the bit that trips people up. Sometimes the minus sign is stuck to the number, not to the sum. Adding minus four means adding a step downwards, so you move left. Minus two add minus four is minus six. And subtracting minus four means taking a downward step away. Take a drop away and you end up higher, so you move right. Minus two subtract minus four is two. A minus in front of the number flips the direction you were about to walk.

Real questions rarely hand you a tidy sum. They hand you a sentence of English, and you write the sum yourself. Here is one of those. The temperature in Oslo is minus six degrees. The temperature in Cairo is twenty-four degrees warmer than Oslo. What is the temperature in Cairo? Before any arithmetic, turn that into a number sentence. Start at minus six. Warmer means move right. Twenty-four degrees warmer means twenty-four steps right. So the sum is minus six add twenty-four. Write that on the page before you calculate anything. Now stand the number line upright and it is a thermometer. Same line, same order, just turned. Start at minus six. Six steps get you up to zero, with eighteen still to go. You land on eighteen. The digits are ordinary work. Six of the twenty-four steps were used getting to zero, so what is left is twenty-four take away six. Set it out in columns. Four take away six will not go, so borrow ten from the twenty. Fourteen take away six is eight. One ten is left, with nothing to take from it, so the one stays. Eighteen. So Cairo is eighteen degrees. Write it as eighteen with the degrees unit, and no plus sign in front. A number written with no sign is already positive. The line decided the direction, and the columns did the digits.

The other lane does not walk anywhere. Multiplying and dividing settle one thing first: which side of zero the answer lands on. After that it is ordinary times tables. Here is why those signs behave the way they do. Minus five times three is minus fifteen. That is three lots of a drop of five. Minus five times two is minus ten. Times one is minus five. Times zero is zero. Look at what the answers are doing. Every time the multiplier drops by one, the answer climbs by five. Minus fifteen, minus ten, minus five, zero. Carry the pattern on and the next one has to be five. So minus five times minus one is five. The pattern crosses zero and keeps going up. That gives you the grid. Same signs, positive answer. Different signs, negative answer. Plus times plus is positive. Minus times minus is positive. A plus and a minus together give a negative. Dividing obeys exactly the same grid. Back to the pair from the opening. Work out minus eight times minus five. Digits first: eight fives are forty. Then the sign: both numbers carry a minus, so same signs, so the answer is positive. Forty, with no sign written in front. Your turn on the dividing one. Minus thirty divided by minus six. Is it five, or minus five? Have a think about that one. I will wait. The answer is five. Thirty divided by six is five, and both numbers carry a minus, so same signs, so positive. Five, written plainly. Four words to carry into the exam hall. Line walks. Grid signs. Adding and subtracting walk you along the line. Multiplying and dividing send you to the grid to settle the sign first.

Two things worth knowing about how this behaves on a paper. Here is one line from an examiner report, about a question where negative numbers had to be handled. Overall, the errors made were usually down to poor arithmetic skills when dealing with negative numbers. Not the topic. Not the wording. The arithmetic, once negative numbers were in it. That is why the number sentence matters. Put minus six add twenty-four on the page, and whatever story it arrived in loses its power to confuse you. Second thing. Take a question like this: minus three times minus four. Twelve is the accepted answer. Minus twelve is not, and minus twelve is what comes out if you bring the adding rule into the multiplying lane. Two minus signs added stay below zero. Two minus signs multiplied do not.

Two to try now, one from each lane. Paper and pen, no calculator. Work out minus seven subtract minus three. Then work out minus nine divided by three. Take your time with those two. I will wait. First one. Minus seven subtract minus three. Subtracting a minus takes a drop away, so you move right. Three steps right from minus seven lands you on minus four. Second one. Minus nine divided by three. Nine divided by three is three. One number carries a minus and the other does not, so different signs, so the answer is negative. Minus three.

Line walks. Grid signs. Those four words are the summary. Here is the longer version. Adding and subtracting are a walk along the number line. Right for adding, left for subtracting, and a minus stuck in front of the number flips which way you walk. Multiplying and dividing do the ordinary times table, then check the grid for the sign. Same signs give a positive answer. Different signs give a negative one. And when the question arrives dressed up as a sentence about temperature, write the number sentence first. The context never changes the maths.

Next in the chapter: Adding, subtracting, multiplying and dividing decimals. The same four operations, with the decimal points lined up.

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