Business Mathematics: Review of Arithmetic - Whole Numbers

1.  Whole Numbers
Our number system, the decimal or base 10 system, uses the digits 0 through 9 to represent numerical values. A whole number is a string of digits representing, from right to left, how many ones, tens, hundreds, thousands, ten thousands, and so on are included within the number. Commas are often used to separate groups of three digits for visual clarity.

Example 1
The number 247 represents 2 hundreds, 4 tens, and 7 ones.


Example 2
The number 1,547,689 represents 1 million, 5 hundred thousands, 4 ten thousands. 7 thousands, 6 hundreds, 8 tens, and 9 ones.


Whole numbers are added by adding digits in corresponding columns and “carrying” if necessary. The result of an addition is called the sum. Addition is most conveniently done in a vertical format and columns are added from right to left.

Example 3
To add 125, 231, and 122 we list the numbers vertically. We add the ones: 5 +1+2 = 8, the tens: 2 + 3 + 2 = 7, and the hundreds: 1 + 2 + 1 = 4.


Example 4
To add 287, 168, and 271, we add the ones: 7 + 8 + 1 = 16 (write 6 and carry 1 to the tens column), the tens: 1 + 8 + 6 + 7 = 22 (write 2 and carry 2 to the hundreds column), and the hundreds: 2 + 2 + l+ 2 = 7.


Example 5
Find the sum of 1,367, 4,672, 1,258, and 2,116.

Whole numbers are subtracted by subtracting digits in corresponding columns and “borrowing” or “exchanging” if necessary. The result of a subtraction is called the difference.

Example 6
To subtract 237 from 579 we list the numbers vertically with the larger number on top. We subtract the ones: 9-7 = 2, then the tens: 7-3 = 4, and finally the hundreds: 5-2 = 3. In this example exchanging is not necessary.


Example 7
Subtract 257 from 492. Since 7 cannot be subtracted from 1, we exchange 1 ten for 10 ones, giving a total of 12 ones. 7 is then subtracted from 12:12 – 7 = 5 and the 9 in the tens column is reduced to 8. Then 5 is subtracted from 8: 8 – 5 = 3 and finally, in the hundreds column, 2 is subtracted from 4: 4 – 2 = 2.


Example 8
Find the difference between 2,257 and 1,189. Here, two exchanges are necessary. Since 9 cannot be subtracted from 7, 1 ten is exchanged for 10 ones, giving 17 in the ones column. 9 is subtracted from 17: 17 – 9 = 8. Since 8 cannot be subtracted from 4, one hundred is exchanged for 10 tens giving 14 in the tens column. Then 8 is subtracted from 14: 14 – 8 = 6.

To multiply two numbers, multiply the first number (multiplicand) by each digit of the second number (multiplier). Then add. The result is called the product. The symbol for multiplication is either or ×.

Example 9
Multiply 54 by 23. Since the “2” in the number 23 really represents 20 its product with 54 is really 1,080. The rightmost “0” may be omitted and the 108 shifted one column to the left.


Example 10
Multiply 112 by 245.

The process of division is the most complicated of the four basic arithmetic operations. We divide the dividend (the number to be divided) by the divisor (the number you are dividing by) and the result is called the quotient. The next example illustrates the process of “long division.”

Example 11
Divide 345 by 15. We write the problem using the standard division symbol: . First we compare the two-digit divisor, 15, with the first two digits of the dividend, 345. Since 15 goes into 34 twice (15 × 2 = 30 but 15 × 3 is too large), we put the first digit of the quotient, 2, in the position shown.

Next, multiply the partial quotient, 2, by the divisor 15 to get 30. Then subtract, bringing down the next rightmost digit.

Finally, divide 15 into 45. Since 15 goes into 45 three times, we put 3 as the next digit of the quotient and multiply by the divisor, 15. Subtract to get a remainder of 0.


Example 12
Divide 6,247 by 23. In this example, the final subtraction gives 14. Since this number is less than the divisor, 23, the process ends with a remainder of 14.

This answer is often represented as 271 R 14.
In algebra, exponents are used to represent repetitive multiplication by the same number.

This is usually read “a to the power n ”

Example 13
(a) 24 = 2 · 2 · 2 · 2 = 16

(b) 35 = 3 · 3 · 3 · 3 · 3 = 243

A prime is a positive integer which can only be divided by itself and 1. Examples of primes are 2, 3, 5, 7,11, 13, 17, 19, 23, ... (For technical reasons, 1 is not prime.) Every whole number which is not prime can be written as a product of primes. This is called its prime factorization.

A given number has only one prime factorization except for the order in which the primes are written.
The prime factorization of a number can be determined by successively factoring into smaller numbers until only primes remain.

Exponents offer a convenient way to represent prime factors.

Example 14
Find the prime factorizations of (a) 180, (b) 504.



Solved Problems


1.1   Explain the significance of each of the digits in the number 23,456.
Solution
The “2” represents 2 ten-thousands, the “3” represents 3 thousands, the “4” represents 4 hundreds, the “5” represents 5 tens, and “6” represents 6 ones.


5.1   Add 194, 638, and 211.
Solution



1.3   Subtract 672 from 913.
Solution



1.4   Multiply 431 by 25.
Solution



1.5   Divide 2,842 by 58.
Solution


1.6    Determine the prime factorization of 360.
Solution
360 = 36 · 10 = 6 · 6 · 2 · 5 = 2 · 3 · 2 · 3 · 2 · 5 = 23 · 32 · 5