Addition of whole numbers refers to combining two or more whole numbers to find the total.
The numbers that are added are referred to as the addends, and the result or answer is called the total or sum. The symbol ‘+’ denotes addition.
For example, 9 + 5 refers to adding 9 and 5. This is read as ‘nine plus five’.
Follow these steps to add whole numbers:
Perform the following additions:
3,514 + 245
8,578 + 3,982 + 564 + 92
3,514 + 245
The sum of the digits in the ones column is 9 since 4 + 5 = 9. There are 9 ones, so we write 9 in the ones column below the horizontal line.
The sum of the digits in the tens column is 5 since 10 + 40 = 50. There are 5 tens, so we write 5 in the tens column below the horizontal line.
The sum of the digits in the hundreds column is 7 since 500 + 200 = 700. There are 7 hundreds, so we write 7 in the hundreds column below the horizontal line.
As only 3 is in the thousands column, the sum of the digits in the thousands column is 3. Write 3 in the thousands column below the horizontal line.
Therefore, adding 3,514 and 245 results in 3,759.
8,578 + 3,982 + 564 + 92
The sum of the digits in the ones column is 16 since 8 + 2 + 4 + 2 = 16, which is 1 ten and 6 ones. Write 6 in the ones column below the horizontal line and carry the 1 above the tens column.
The sum of the digits in the tens column is 31 since 70 + 80 + 60 + 90 + 10 (carried from the ones column) = 310, which is 3 hundreds and 1 ten. Write 1 in the tens column below the horizontal line and carry the 3 above the hundreds column.
The sum of the digits in the hundreds column is 22 since 500 + 900 + 500 + 300 (carried from the tens column) = 2,200, which is 2 thousands and 2 hundreds. Write 2 in the hundreds column below the horizontal line and carry the 2 above the thousands column.
The sum of the digits in the thousands column is 13 since 8,000 + 3,000 + 2,000 (carried from the hundreds column) = 13,000, which is 1 ten thousand and 3 thousands. Write 3 in the thousands column below the horizontal line and 1 in the ten thousands column.
Therefore, adding 8,578, 3,982, 564, and 92 results in 13,216.
Subtraction of whole numbers refers to finding the difference between numbers. This is the reverse process of addition.
The number from which another number is subtracted is referred to as the minuend and the number that is being subtracted is referred to as the subtrahend. The result or answer is called the difference. The symbol ‘–’ denotes subtraction.
For example, 8 – 5 refers to subtracting 5 from 8. This is read as ‘eight minus five’.
Follow these steps to subtract a whole number from another whole number:
Note: To check your answer, add the difference to the number subtracted (subtrahend); the result should be the number from which it was subtracted (minuend). e.g., 8 – 5 = 3; therefore 3 + 5 = 8
Perform the following subtractions:
Subtract 1,314 from 3,628
Subtract 789 from 8,357
Subtract 1,314 from 3,628
The difference of the digits in the ones column is 4 since 8 – 4 = 4. There are 4 ones, so we write 4 in the ones column below the horizontal line.
The difference of the digits in the tens column is 1 since 20 − 10 = 10. There is 1 ten, so we write 1 in the tens column below the horizontal line.
The difference of the digits in the hundreds column is 3 since 600 − 300 = 300. There are 3 hundreds, so we write 3 in the hundreds column below the horizontal line.
The difference of the digits in the thousands column is 2 since 3,000 − 1,000 = 2,000. There are 2 thousands, so we write 2 in the thousands column below the horizontal line.
Therefore, subtracting 1,314 from 3,628 results in 2,314.
Subtract 789 from 8,357
In the ones column, the ones digit on the top (7) is smaller than the ones digit on the bottom (9). Borrow one ten from the tens digit on the top row and add it to the 7 ones to get 17 ones. 17 − 9 = 8, so we write 8 in the ones column below the horizontal line.
In the tens column, the tens digit on the top (4, after borrowing 1 for the ones column) is smaller than the tens digit on the bottom (8). Borrow one hundred from the hundreds digit on the top row and add it to the 4 tens to get 14 tens. 14 − 8 = 6, so we write 6 in the tens column below the horizontal line.
In the hundreds column, the hundreds digit on the top (2, after borrowing 1 for the tens column) is smaller than the hundreds digit on the bottom (7). Borrow one thousand from the thousands digit on the top row and add it to the 2 hundreds to get 12 hundreds. 12 − 7 = 5, so we write 5 in the hundreds column below the horizontal line.
As only 7 (after borrowing 1 for the hundreds column) is in the thousands column, we write 7 in the thousands column below the horizontal line.
Therefore, subtracting 789 from 8,357 results in 7,568.
Note: If a larger whole number is subtracted from a smaller whole number, the result will be a negative number. To do this, reverse the question to deduct the smaller number from the larger number following the above steps, and set the answer to be negative.
Multiplication is the process of finding the product of two numbers. Multiplication of whole numbers can be thought of as repeated additions. The symbol ‘×’ denotes multiplication.
For example, 5 × 4 refers to repeatedly adding 5, four times. This is read as ‘five times four’, and can also be written as 5 ∙ 4 or 5(4).
5 × 4 can be represented pictorially as:
Here, the size of the set is 5 and it is repeated 4 times:
5 + 5 + 5 + 5 = 20
This can also be viewed as 4 × 5:
Here, the size of the set is 4 and it is repeated 5 times:
4 + 4 + 4 + 4 + 4 = 20
The numbers that are being multiplied are referred to as factors and the result is referred to as the product.
In this example, 5 and 4 are factors of 20.
Follow these steps to multiply one whole number by another whole number:
The above steps for multiplying numbers are provided in detail in Example 1.2-c below.
Perform the following multiplications:
Multiply 38 by 6
Multiply 36 by 24
Multiply 263 by 425
Multiply 38 by 6
Multiplying 8 ones by 6 results in 48 ones. This is 4 tens and 8 ones. Write 8 in the ones column below the horizontal line and 4 above the tens column.
Multiplying 3 tens by 6 results in 18 tens. Add the 4 tens carried from the previous step to 18 to obtain 22 tens. Write 2 in the tens column and 2 in the hundreds column below the horizontal line.
Therefore, multiplying 38 by 6 results in 228.
Multiply 36 by 24
Multiply 36 by 4 ones, as shown, to obtain 144.
Multiply 36 by 2 tens. To do this, write a ‘0’ under the horizontal line in the ones column (or simply leave it blank, as shown) and multiply 36 by 2 to obtain 72.
Therefore, multiplying 36 by 24 results in 864.
Multiply 263 by 425
Multiply 263 by 5 ones.
Multiply 263 by 2 tens.
Multiply 263 by 4 hundreds.
Therefore, multiplying 263 by 425 results in 111,775.
Division is the process of determining how many times one number is contained in another. This is the inverse process of multiplication. When a larger number is divided by a smaller number, this division can be thought of as repeated subtractions. The symbol ‘÷’ denotes division.
For example, 20 ÷ 5 refers to repeatedly subtracting 5 from 20. This is read as ‘twenty divided by five’, and can also be written as 5 |20.
In a set of 20 items, we can create four groups of 5:
Therefore, 20 ÷ 5 = 4.
The number that is being divided is referred to as the dividend, and the number by which the dividend is divided is referred to as the divisor. The result or answer is called the quotient. If the dividend cannot be divided evenly by the divisor, the number left over is referred to as the remainder.
For example, consider 25 ÷ 7:
The following relationship exists between the four components of a division problem:
Dividend = Divisor × Quotient + Remainder
25 = 7 × 3 + 4
The steps to be followed in dividing whole numbers are provided in detail in Example 1.2-d below.
Perform the following divisions and state the quotient and remainder:
Divide 76 by 3
Divide 637 by 25
Divide 6,543 by 12
Divide 76 by 3
7 can be divided by 3. Therefore, determine the number of multiples of 3 there are in 7.
There are two 3’s in 7. Write 2 in the quotient area above 7.
Multiply 2 by 3 (= 6) and subtract this from 7. Write the remainder 1.
Bring down the 6 from the dividend and determine the number of multiples of 3 there are in 16.
There are five 3’s in 16. Write 5 in the quotient area above 6.
Multiply 5 by 3 (= 15) and subtract this from 16 to get the final remainder of 1.
Therefore, the quotient is 25 and the remainder is 1.
Divide 637 by 25
6 cannot be divided by 25. Therefore, determine the number of multiples of 25 there are in 63.
There are two 25’s in 63. Write 2 in the quotient area above 3.
Multiply 2 by 25 (= 50) and subtract this from 63. Write the remainder 13.
Bring down the 7 from the dividend and determine the number of multiples of 25 there are in 137.
There are five 25’s in 137. Write 5 in the quotient area above 7.
Multiply 5 by 25 (= 125) and subtract this from 137 to get the final remainder of 12.
Therefore, the quotient is 25 and the remainder is 12.
Divide 6,543 by 12
6 cannot be divided by 12. Therefore, determine the number of multiples of 12 there are in 65.
There are five 12’s in 65. Write 5 in the quotient area above 5. Multiply 5 by 12 (= 60) and subtract this from 65. Write the remainder 5. Bring down the 4 from the dividend and determine the number of multiples of 12 there are in 54.
There are four 12’s in 54. Write 4 in the quotient area above 4. Multiply 4 by 12 (= 48) and subtract this from 54. Write the remainder 6. Bring down the 3 from the dividend and determine the number of multiples of 12 there are in 63.
There are five 12’s in 63. Write 5 in the quotient area above 3. Multiply 5 by 12 (= 60) and subtract this from 63 to get the final remainder of 3.
Therefore, the quotient is 545 and the remainder is 3.
Operation | Description | Examples |
---|---|---|
Addition | When 0 is added to a number, or when a number is added to 0, there will be no change to that number. | 25 + 0 = 25 0 + 25 = 25 |
Subtraction | When 0 is subtracted from a number, there will be no change to that number. | 16 – 0 = 16 |
When a number is subtracted from 0, the answer will be the negative value of that number. | 0 – 16 = –16 | |
Multiplication | When 0 is multiplied by a number, or when a number is multiplied by 0, the answer will be 0. | 0 × 35 = 0 35 × 0 = 0 |
Division | When 0 is divided by a number, the answer will be 0. | 0 ÷ 25 = 0 |
When a number is divided by 0, the answer is undefined. | 25 ÷ 0 = Undefined |
Operation | Description | Examples |
---|---|---|
Multiplication | When 1 is multiplied by a number, or when a number is multiplied by 1, there will be no change to that number. | 1 × 12 = 12 12 × 1 = 12 |
Division | When 1 is divided by a number, the answer is the reciprocal of that number. | 1 ÷ 35 = \(\displaystyle{\frac{1}{35}}\) |
When a number is divided by 1, there will be no change to that number. | 1 ÷ 35 = \(\displaystyle{\frac{1}{35}}\) = 35 |
We learned previously that multiplication is a shorter way to write repeated additions of a number. Similarly, when a number is multiplied by itself repeatedly, we can represent this repeated multiplication using exponential notation.
When 2 is multiplied 5 times, in repeated multiplication, it is represented by:
2 × 2 × 2 × 2 × 2
However, it can be tedious to represent repeated multiplication using this notation. Instead, exponential notation can be used.
When 2 is multiplied 5 times, in exponential notation, it is represented by:
In this example, 2 is known as the base, 5 is known as the exponent, and the whole representation 2^{5} is known as the power. The exponent is written in superscript to the right of the base, and represents the number of times that the base is multiplied by itself. The whole representation is read as “2 raised to the power of 5” or “2 to the 5^{th} power”.
Expand 8^{2} to show the repeated multiplication and evaluate.
\(8^2 + 8 × 8 = 64\)
Express the repeated multiplication 9 × 9 × 9 × 9 × 9 × 9 in exponential notation.
The number 9 is being multiplied repeatedly 6 times.
In exponential notation, the base is 9 and the exponent is 6.
Therefore, 9 × 9 × 9 × 9 × 9 × 9 = \(9^6\).
Express the following in standard form and then evaluate:
\(5 × 3^4\)
\(2^3 × 3^2\)
\(5^4 × 2^2\)
\(5^2 × 4^2\)
\(5 × 3^4 = 5 × [ 3 × 3 × 3 × 3 ] = 5 × 81 = 405\)
\(2^3 × 3^2 = [ 2 × 2 × 2 ] × [ 3 × 3 ] = 8 × 9 = 72\)
\(5^4 × 2^2 = [ 5 × 5 × 5 × 5 ] + [ 2 × 2 ] = 625 + 4 = 629\)
\(5^2 × 4^2 = [ 5 × 5 ] – [ 4 × 4 ] = 25 – 16 = 9\)
Use the equation \(6^5 = 7,776\) to evaluate the following expressions:
\(6^6\)
\(6^4\)
\(6^5 = 6 × 6 × 6 × 6 × 6 = 7,776\)
\(6^6 = 6 × 6 × 6 × 6 × 6 × 6\)
\(= 6^5 × 6\)
\(= 7,776 × 6\)
\(= 46,656\)
\(6^5 = 6 × 6 × 6 × 6 × 6 = 7,776\)
\(6^5 = 6^4 × 6\)
\(7,776 = 6^4 × 6\)
\(\displaystyle{\frac{7,776}{6} = 6^4}\)
\(6^4 = 1,296\)
Powers of 2 | Powers of 3 | Powers of 4 | Powers of 5 | |
---|---|---|---|---|
\(1^2 = 1\) | \(11^2 = 121\) | \(1^3 = 1\) | \(1^4 = 1\) | \(1^5 = 1\) |
\(2^2 = 4\) | \(12^2 = 144\) | \(2^3 = 8\) | \(2^4 = 16\) | \(2^5 = 32\) |
\(3^2 = 9\) | \(13^2 = 169\) | \(3^3 = 27\) | \(3^4 = 81\) | \(3^5 = 243\) |
\(4^2 = 16\) | \(14^2 = 196\) | \(4^3 = 64\) | \(4^4 = 256\) | \(4^5 = 1,024\) |
\(5^2 = 25\) | \(15^2 = 225\) | \(5^3 = 125\) | \(5^4 = 625\) | \(5^5 = 3,125\) |
\(6^2 = 36\) | \(16^2 = 256\) | \(6^3 = 216\) | \(6^4 = 1,296\) | \(6^5 = 7,776\) |
\(7^2 = 49\) | \(17^2 = 289\) | \(7^3 = 343\) | \(7^4 = 2,401\) | \(7^5 = 16,807\) |
\(8^2 = 64\) | \(18^2 = 324\) | \(8^3 = 512\) | \(8^4 = 4,096\) | \(8^5 = 32,768\) |
\(9^2 = 81\) | \(19^2 = 361\) | \(9^3 = 729\) | \(9^4 = 6,561\) | \(9^5 = 59,049\) |
\(10^2 = 100\) | \(20^2 = 400\) | \(10^3 = 1,000\) | \(10^4 = 10,000\) | \(10^5 = 100,000\) |
Any whole number base with an exponent of 2 is known as a perfect square. For example, 9 is the perfect square produced by 3 raised to an exponent 2. The first twenty perfect squares are shown in Table 1.2-c under Powers of 2.
Finding the square root of a perfect square is the inverse of raising a whole number to the power of 2.
For example,
2 is the square root of 4 because \(2^2 = 4\) or \(2 × 2 = 4\)
5 is the square root of 25 because \(5^2 = 25\) or \(5 × 5 = 25\)
That is, a whole number multiplied by itself results in a perfect square. The whole number multiplied by itself to get that perfect square is called the square root of that perfect square.
For example,
16 is a perfect square because it is the product of two identical factors of 4; i.e., 16 = 4 × 4.
Therefore, 4 is the square root of 16.
The radical sign \(\sqrt{ }\) indicates the root of a number (or expression). The square root of 16, using the radical sign, is represented by \(\sqrt[2]{16}\) , where 2 is the index. The index indicates which root is to be taken, and is written as a small superscript number to the left of the radical symbol. For square roots, the index 2 does not need to be written as it is understood to be there; i.e. \(\sqrt[2]{16}\) is written as \(\sqrt{16}\).
Note: In future chapters, we will discuss roots with indexes other than 2.
Determine the square root of the following:
\(\sqrt{36}\)
\(\sqrt{81}\)
\(\sqrt{144}\)
\(\sqrt{36} = \sqrt{6 × 6} = 6\)
(Using 36 = 6 × 6)
\(\sqrt{81} = \sqrt{9 × 9} = 9\)
(Using 81 = 9 × 9)
\(\sqrt{36} = \sqrt{12 × 12} = 12\)
(Using 144 = 12 × 12)
For the addition Problems 1 to 8, (i) estimate the answer by rounding each number to the nearest ten, and (ii) calculate the exact answer.
For the addition Problems 9 to 14, (i) estimate the answer by rounding each number to the nearest hundred, and (ii) calculate the exact answer.
For the subtraction Problems 15 to 22, (i) estimate the answer by rounding each number to the nearest ten, and (ii) calculate the exact answer.
For the subtraction Problems 23 to 28, (i) estimate the answer by rounding each number to the nearest hundred, and (ii) calculate the exact answer.
For the multiplication Problems 29 to 34, (i) estimate the answer by rounding each number to the nearest ten, and (ii) calculate the exact answer.
For the division Problems 35 to 44, (i) estimate the answer by rounding each number to the nearest ten, and (ii) perform the exact division and state the quotient and the remainder.
For Problems 45 to 48, express the repeated multiplication in exponential notation.
For Problems 49 to 52, write the base and exponent for the powers.
For Problems 53 and 54, express the powers in standard notation and then evaluate.
For Problems 53 and 54, express the powers in standard notation and then evaluate.