Calculations with integers

2.2 Calculations with integers

Revise addition and subtraction with integers

When calculating with small integers, you can use the number line:

• Count to the left when subtracting a positive number.
• Count to the right when adding a positive number.

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Understanding addition and subtraction with integers

The following statement is true if the unknown number is \(5\):

\(15\) − (a certain number) = \(10\)

Another number will make the following statement true:

\(15\) + (a certain number) = \(10\)

If you add this number, it should have the same effect as to subtract \(5\).

That number is “negative \(5\)”. It has the property that if you add it to another number, the effect will be the same as when you subtract its positive opposite:

\(15 + ( - 5)\) is equal to \(15 - 5\).

Adding a negative number has the same effect as subtracting a positive number.

For example: \(20 + ( - 15) = 20 - 15 = 5\).

Exercise 2.1 Understanding addition and subtraction of integers

Calculate the following:

1. \(30 + ( - 10)\)
2. \(25 + ( - 15)\)
3. \(10 + ( - 5) - 3\)

1. \(30 + ( - 10) = 30 - 10 = 20\)
2. \(25 + ( - 15) = 25 - 15 = 10\)
3. \(10 + ( - 5) - 3 = 10 - 5 - 3 = 5 - 3 = 2\) or \(10 + ( - 5) - 3 = 10 + ( - 5 - 3) = 10 + ( - 8) = 10 - 8 = 2\)

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Multiply with integers

When multiplying with integers, the following rules apply:

• The product of two positive numbers is a positive number, for example \(5 \times 6 = 30\).
• The product of a positive number and a negative number is a negative number, for example \(5 \times ( - 6) = - 30\).
• The product of a negative number and a positive number is a negative number, for example \(( - 5) \times 6 = - 30\).
• The product of two negative numbers is a positive number, for example \(( - 5) \times ( - 6) = 30\).

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Divide with integers

When dividing with integers, the following rules apply:

• The quotient of two positive numbers is a positive number, for example \(15 \div 3 = 5\).
• The quotient of a positive number and a negative number is a negative number, for example \(50 \div ( - 5) = - 10\).
• The quotient of a negative number and a positive number is a negative number, for example \(( - 50) \div 10 = - 5\).
• The quotient of two negative numbers is a positive number, for example \(( - 10) \div ( - 2) = 5\).

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Perform calculations involving all four operations with integers

Exponents and roots of integers

When finding the square of an integer, we multiply an integer by itself, keeping its sign in mind. For example, \(12^{2} = 12 \times 12 = 144\) and \(( - 12)^{2} = ( - 12) \times ( - 12) = + 144\).

The square of an integer is always a positive number.

When finding a cube of an integer, we multiply an integer by itself, and by itself again. For example, \(2^{3} = 2 \times 2 \times 2 = 8\) and \(( - 2)^{3} = ( - 2) \times ( - 2) \times ( - 2) = - 8\).

The cube of an integer is positive if the integer is positive, and negative if the integer is negative.

Always pay attention to where the minus sign is placed: \(- 5^{2} \neq ( - 5)^{2}\).

\[- 5^{2} = - (5 \times 5) = - 25\] \[( - 5)^{2} = ( - 5) \times ( - 5) = 25\]

The minus sign of a negative integer raised to a power will always be inside the brackets.

Now have a look at the square roots of integers: \(10^{2}\) is \(100\) and \(( - 10)^{2}\) is also \(100\).

Both \(10\) and (\(−10\)) are called square roots of \(100\). \(10\) is the positive square root of \(100\) and (\(−10\)) is the negative square root of \(100\):

\(\sqrt{100} = 10\) and \(\sqrt{100} = - 10\)

Cube roots can also be positive or negative numbers. For example,

\(3^{3}\) is \(27\) and \(( - 5)^{3}\) is \(−125\).

\(3\) is called the cube root of \(27\), because \(3^{3} = 27\). So, \(\sqrt[3]{27} = 3\).

\(( - 5)\) is called the cube root of \(−125\) because \(( - 5)^{3} = - 125\). So, \(\sqrt[3]{- 125} = - 5\).

It is possible to take a cube root of a negative integer, but it is not possible to take a square root of a negative integer.

\(\sqrt[3]{- 8} = - 2\) but \(\sqrt{- 16}\) is undefined

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Order of operations

Remember that, in any numeric or algebraic sentence, the terms are separated by addition and subtraction symbols, and joined into one by multiplication symbols, division symbols, and brackets. To simplify or evaluate an expression, we must:

• separate it into terms
• simplify each term (if needed)
• add or subtract from left to right.

To simplify a term, we must first simplify any parts within brackets. Then we must apply the exponents. If there are no brackets, then the exponent only applies to the base immediately before it, and we must simplify the power first.

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