190 CHAPTER 4 Systems of Numeration Example 1 Converting from Base 5 to Base 10 Convert 4325 to base 10. Solution In base 5, the positional values are ,… 5 ,5 ,5,1. 3 2 In expanded form, 432 (4 5) (3 5) (2 1) (4 25) (3 5) (2 1) 100 15 2 117 5 2 = × + × + × = × + × + × = + + = 7 Now try Exercise 5 Example 2 Converting from Base 8 to Base 10 Convert 35148 to base 10. Solution In base 8, the positional values are ,… 8 ,8 ,8,1. 3 2 In expanded form, 3514 (3 8) (5 8) (1 8) (4 1) (3 512) (5 64) (1 8) (4 1) 1536 320 8 4 1868 8 3 2 = × + × + × + × = × + × + × + × = + + + = 7 Now try Exercise 7 Example 3 Converting from Base 2 to Base 10 Convert 1100102 to base 10. Solution 110010 (1 2) (1 2) (0 2) (0 2) (1 2) (0 1) (1 32) (1 16) (0 8) (0 4) (1 2) (0 1) 32 16 0 0 2 0 50 2 5 4 3 2 =× +× +× +× +×+× =× +× +×+×+×+× = + + + + + = 7 Now try Exercise 9 To change a base 10 numeral to a different base, we will use a procedure similar to the one we used to convert base 10 numerals to Babylonian and Mayan numerals, as was explained in Section 4.2. Divide the base 10 numeral by the highest power of the new base that is less than or equal to the given base 10 numeral. Record this quotient. Then divide the remainder by the next smaller power of the new base and record this quotient. Repeat this procedure until the remainder is less than the new base. The answer is the set of quotients listed from left to right, with the remainder on the far right. This procedure is illustrated in Examples 4 and 5. RECREATIONAL MATH The Price Is Right $ ____ ____ ____ ____ ____ Place the digits 0, 4, 5, 3, in the correct order to match the price of the moped. The long-running television game show The Price Is Right has many different games that require the player to understand the place values used in our decimal system. For example, in one game the host gives the player the digits that make up the price of an item but does not tell the player the correct order of the digits. If the player places the digits in the correct order, the player wins the fabulous prize! For example, the price of a 2023 Amigo Avenza 150 Vespa scooter contains the digits 0, 1, 8, 9. See if you can determine the correct order in which to place these digits to match the price of the motorcycle. The answer is upsidedown below. See Exercise 70 for another example. $1980 Example 4 Converting from Base 10 to Base 8 Convert 486 to base 8. Solution We are converting a numeral in base 10 to a numeral in base 8. The positional values in the base 8 system are , 8 , 8 , 8, 1, 3 2 … or , 512, 64, 8, 1. … The highest power of 8 that is less than or equal to 486 is 8 ,2 or 64. Divide 486 by 64. First digit in answer 486 64 7 with remainder 38 ÷ = In Example 1, the units digit in 4325 is 2. Notice that 25 has the same value as 2 , 10 since both are equal to the number 2. That is, 2 2 . 5 10 = If n is a digit less than the base b, and the base b is less than or equal to 10, then n n . b 10 = Jack Jelly/Shutterstock
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