Integers

Hello, and welcome to your mathematics lesson for today. We are beginning an exciting new chapter all about integers. By the end of this lesson, you will understand what integers are, how to multiply and divide them, the special properties that govern these operations, and how to simplify complex expressions using brackets and the DMAS rule.

Let us start with the basics. What exactly are integers?

Integers are the collection of all whole numbers, their negatives, and zero. Imagine a number line stretching endlessly in both directions. To the right of zero, we have the positive integers: one, two, three, four, five, and so on, continuing forever. To the left of zero, we have the negative integers: negative one, negative two, negative three, negative four, negative five, and so on, also continuing forever. And right in the middle sits zero itself.

So the complete set of integers looks like this: dot dot dot, negative five, negative four, negative three, negative two, negative one, zero, one, two, three, four, five, dot dot dot. Notice that zero is special — it is neither positive nor negative. It simply marks the boundary between the two sides.

Now, let us explore how we multiply integers. This is where patterns become really interesting.

Here is the first crucial rule. When you multiply two integers with the same sign, the result is always positive. Positive times positive gives positive. Negative times negative also gives positive.

For example, five times four equals twenty. And negative five times negative four also equals twenty. Similarly, seven times six is forty-two, and negative seven times negative six is also forty-two. Think of it this way: two wrongs make a right, or two negatives create a positive.

Now for the second rule. When you multiply two integers with opposite signs, the result is always negative. Positive times negative gives negative. Negative times positive also gives negative.

For instance, five times negative four equals negative twenty. And negative five times four also equals negative twenty. Seven times negative six is negative forty-two, and negative seven times six is negative forty-two.

Let us now examine the properties of multiplication. These properties help us understand why numbers behave the way they do.

First, the closure property. When you multiply any two integers, the result is always another integer. Mathematically, if m and n are integers, then m × n equals mn, which is also an integer. For example, five times six gives thirty, eight times negative twelve gives negative ninety-six, and negative fifteen times negative six gives ninety. The product never escapes the world of integers.

Second, the commutative property. This tells us that the order of multiplication does not matter. If m and n are integers, then m × n equals n × m. Six times seven equals seven times six, both giving forty-two. Negative ten times eight equals eight times negative ten, both giving negative eighty. Even with negative numbers, swapping the order keeps the answer the same.

Third, the associative property. When multiplying three integers, the way you group them does not affect the result. If l, m, and n are integers, then l × (m × n) equals (l × m) × n. Take five times three times two. You could first multiply five times three to get fifteen, then times two to get thirty. Or you could first multiply three times two to get six, then five times six to get thirty. Same answer, different grouping.

Fourth, the distributive property. This connects multiplication with addition. If l, m, and n are integers, then l × (m + n) equals l × m + l × n. Let us verify with numbers. Five times the quantity seven plus two equals five times nine, which is forty-five. Alternatively, five times seven plus five times two equals thirty-five plus ten, which is also forty-five. This property also works with subtraction: l × (m − n) equals l × m − l × n.

Fifth, the multiplicative identity. For every integer a, we have a × 1 = a, and 1 × a = a. The number one is called the multiplicative identity because multiplying any integer by one leaves it unchanged. Five times one is five. Negative twelve times one is negative twelve.

Sixth, let us discuss multiplicative inverses. For a non-zero integer a, its multiplicative inverse would be 1/a, such that a × 1/a = 1. However, except for one and negative one, the multiplicative inverse of an integer is not itself an integer. Therefore, we say that multiplicative inverses do not generally exist within the set of integers.

Here is a fascinating pattern about multiplying many negative numbers together. When you multiply an odd number of negative integers, the result is always negative. Negative two times negative three times negative four equals negative twenty-four. Three negatives, odd count, negative result.

But when you multiply an even number of negative integers, the result is always positive. Negative two times negative three equals positive six. Negative two times negative three times negative four times negative five equals positive one hundred twenty. Four negatives, even count, positive result. This pattern continues no matter how many numbers you multiply.

Now we turn to division of integers. Division is essentially the inverse operation of multiplication. When we ask what is thirty-six divided by nine, we are asking: what integer multiplied by nine gives thirty-six? The answer is four, because four times nine equals thirty-six.

The sign rules for division mirror those for multiplication. When you divide two integers with the same sign, the quotient is always positive. Negative twenty-four divided by negative six equals four. Twenty-four divided by six also equals four.

When you divide two integers with opposite signs, the quotient is always negative. Negative twenty-four divided by six equals negative four. Twenty-four divided by negative six equals negative four.

Let us clarify the terminology. In a division problem, the number being divided is called the dividend. The number doing the dividing is called the divisor. The result is called the quotient. So in sixty-five divided by thirteen equals five, sixty-five is the dividend, thirteen is the divisor, and five is the quotient.

Now, how do the properties work for division? Interestingly, division is much more restrictive than multiplication.

Closure property fails for division. The division of one integer by another is not always an integer. Eighteen divided by five equals eighteen over five, which is not an integer. Similarly, negative fourteen divided by four equals negative seven halves, not an integer.

Commutative property also fails. Fifteen divided by five equals three, but five divided by fifteen equals one-third. Clearly, three does not equal one-third.

Associative property fails as well. If we try to divide twenty-four by the quantity six divided by two, we get twenty-four divided by three, which is eight. But if we group differently as twenty-four divided by six, then divided by two, we get four divided by two, which is two. Eight does not equal two, so the grouping matters enormously.

Distributive property fails too. Division does not distribute over addition or subtraction.

There is no identity element for division. While m ÷ 1 = m, we have 1 ÷ m = 1/m, which is not an integer unless m = 1.

However, some properties do hold. Any non-zero integer divided by itself equals one. Twelve divided by twelve equals one. Negative twenty-five divided by negative twenty-five equals one.

Zero divided by any non-zero integer equals zero. Zero divided by six equals zero. Zero divided by negative eleven equals zero.

But crucially, division by zero is undefined. Eight divided by zero has no meaning in mathematics. Negative twenty-three divided by zero is also undefined. Never divide by zero.

When expressions contain multiple operations, we need a clear order to follow. This is where DMAS comes in.

DMAS stands for Division, Multiplication, Addition, and Subtraction. This tells us the priority of operations. First, perform all divisions. Then, perform all multiplications. Next, perform all additions. Finally, perform all subtractions.

Let us work through an example: 18 − 6 ÷ 3 × 4. Following DMAS, we first do the division: six divided by three equals two. Then we do the multiplication: two times four equals eight. Finally, we do the subtraction: eighteen minus eight equals ten.

Another example: (−10) + (−4) ÷ (−2) × 3. First, division: negative four divided by negative two equals two. Then, multiplication: two times three equals six. Finally, addition: negative ten plus six equals negative four.

Brackets add another layer of complexity. When expressions contain brackets, we simplify the innermost parts first.

The order of removing brackets is important. First, simplify any expression under a vinculum — that is a bar drawn over part of an expression. Next, simplify inside small brackets or parentheses. Then, simplify inside curly brackets. Finally, simplify inside square brackets.

Here is a key rule about brackets preceded by a number. When a number sits just before a bracket containing terms separated by plus or minus signs, you multiply that number by each term inside when removing the bracket.

For example, 3(8 + 11) equals 3 × 8 + 3 × 11, which is 24 + 33, giving 57.

Similarly, 5(3 − 8 + 10) equals 15 − 40 + 50, which simplifies to 25.

When a minus sign appears before a bracket, change the sign of every term inside when removing it. −(a − b + c) equals −a + b − c.

Let us work through a complex example: 30 − [26 − {15 + (8 − 6 − 3)}].

First, inside the small brackets: 8 − 6 − 3 becomes 8 − 3, which is 5. Now we have 30 − [26 − {15 + 5}].

Next, the curly brackets: 15 + 5 equals 20. Now we have 30 − [26 − 20].

Then, the square brackets: 26 − 20 equals 6. Finally: 30 − 6 equals 24.

Let us apply our knowledge to some word problems involving integers.

Suppose the sum of two integers is negative twenty-three, and one of them is fifteen. To find the other, we subtract: −23 − 15 equals −38. The other integer is negative thirty-eight.

Here is another type: by how much does negative six exceed negative eighteen? We calculate (−6) − (−18), which equals −6 + 18, giving 12. So negative six exceeds negative eighteen by twelve.

Notice that on the number line, negative six is to the right of negative eighteen, so it is indeed larger. The difference between them is 12 units.

Let us recap the key takeaways from this lesson.

First, integers include all positive whole numbers, all negative whole numbers, and zero. Zero is neither positive nor negative.

Second, when multiplying or dividing integers: same signs give a positive result, opposite signs give a negative result.

Third, multiplication of integers satisfies closure, commutativity, associativity, and distributivity, and has a multiplicative identity of one.

Fourth, division of integers does not satisfy closure, commutativity, associativity, or distributivity, and has no identity element.

Fifth, when simplifying expressions, follow DMAS: Division, then Multiplication, then Addition, then Subtraction.

Sixth, when removing brackets, work from the innermost outward: first vinculum, then parentheses, then curly brackets, then square brackets. Remember that a minus sign before a bracket changes all signs inside.

You have done excellent work today exploring the world of integers. These concepts form the foundation for much of the mathematics you will encounter in the years ahead. Keep practicing, stay curious, and I look forward to seeing you in our next lesson.