**High School Number Sense Lesson 34: Converting from Base N to Base 10**

Ahhh...bases. We're going to open up a beautiful world of non-decimal numbers today. Operations in bases other than base 10 are frequently required on number sense tests. In fact, I counted 15

*different types*of base concepts on high school tests this year. Today we will focus on the most common base concept, which showed up

**12 times**with an average spot of

**question # 32**.

**Number Dojo Level: 179**

Let me start by saying that numbers aren't always what they appear to be. We spend years learning math concepts in school. Then one day we are told that all along, we've been skipping happily along in the decimal number system (base 10), where every place value of a number represents a power of 10. For example,

**1234.567**is really:

^{3}), plus

* 2 hundreds (2 x 10

^{2}), plus

* 3 tens (3 x 10

^{1}), plus

* 4 ones (4 x 10

^{0}), plus

* 5 tenths (5 x 10

^{-1}), plus

* 6 hundredths (6 x 10

^{-2}), plus

* 7 thousandths (7 x 10

^{-3})...

*and this could go on in either direction!*

**binary**or

**hexadecimal**or even

**octal**, and our minds are blown.

To put it simply, a number may be given in any base, where each place value represents a power of that base. It can be noted as:

**12345 base 8, or**

12345 base eight, or

12345

12345 base eight, or

12345

_{8}**Just trust me on this:**12345 base 8 really means:

^{4}), plus

* 2 '512's (2 x 8

^{3}), plus

* 3 '64's (3 x 8

^{2}), plus

* 4 '8's (4 x 8

^{1}), plus

* 5 '1's (5 x 8

^{0})

**For a much more in-depth study of number systems, please see Khan Academy's site.**

**How to Solve:**

- Determine the place value of each digit in the number.
- Multiply each digit by the appropriate base "to the power of" the associated place value.
- Add all these values together.

**convert 314 base 5 to base 10:**

* In base 5, the three digits represent:

**3 x 5**, plus

^{2}**1 x 5**, plus

^{1}**4 x 5**

^{0}**75**, plus

1 x 5 =

**5**, plus

4 x 1 =

**4**,

**84**.

**Solving these the Number Sense way:**

- Multiply the base by the first digit (leftmost digit) of the number.
- Add the 2nd digit to the product in Step 1.
- Multiply this sum by the base.
- Add the 3rd digit to the product in Step 3.
- Continue this pattern until you run out of digits.

**Examples are crucial to understanding this concept.**

**Example 1: 36 base 9 in base 10 is ___**

- Multiply the base (9) by the first digit (3). 9 x 3 =
**27**. - Add the 2nd digit (6) to the product in Step 1 (27). 6 + 27 =
**33**. You have your answer.

**Example 2: 102 base 3 in base 10 is ___**

- Multiply the base (3) by the first digit (1). 3 x 1 =
**3**. - Add the 2nd digit (0) to the product in Step 1 (3). 0 + 3 =
**3**. - Multiply this sum (3) by the base (3). 3 x 3 =
**9**. - Add the 3rd digit (2) to the product in Step 3 (9). 2 + 9 =
**11**. You have your answer.

**Example 3: 324 base 5 in base 10 is ___**

- Multiply the base (5) by the first digit (3). 5 x 3 =
**15**. - Add the 2nd digit (2) to the product in Step 1 (15). 2 + 15 =
**17**. - Multiply this sum (17) by the base (5). 17 x 5 =
**85**. - Add the 3rd digit (4) to the product in Step 3 (85). 4 + 85 =
**89**. You have your answer.

**Example 4: 213 base 4 in base 10 is ___**

- Multiply the base by the first digit. 4 x 2 =
**8**. - Add the 2nd digit. 8 + 1 =
**9**. - Multiply by the base again. 9 x 4 =
**36**. - Add the 3rd digit. 36 + 3 =
**39**. You have your answer.

**Example 5: 215 base 7 in base 10 is ___**

- Multiply the base by the first digit. 7 x 2 =
**14**. - Add the 2nd digit. 14 + 1 =
**15**. - Multiply by the base again. 15 x 7 =
**105**. - Add the 3rd digit. 105 + 5 =
**120**. You have your answer.

**Example 6: 3105 base 8 in base 10 is ___**

- Multiply the base by the first digit. 8 x 3 =
**24**. - Add the 2nd digit. 24 + 1 =
**25**. - Multiply by the base again. 25 x 8 =
**200**. - Add the 3rd digit. 200 + 0 =
**200**. - Multiply by the base again. 200 x 8 =
**1600**. - Add the 4th digit. 1600 + 5 =
**1605**. You have your answer.

**Example 7: 1101101 base 2 in base 10 is ___**

- Multiply the base by the first digit. 2 x 1 =
**2**. - Add the 2nd digit. 2 + 1 =
**3**. - Multiply by the base again. 3 x 2 =
**6**. - Add the 3rd digit. 6 + 0 =
**6**. - Multiply by the base again. 6 x 2 =
**12**. - Add the 4th digit. 12 + 1 =
**13**. - Multiply by the base again. 13 x 2 =
**26**. - Add the 5th digit. 26 + 1 =
**27**. - Multiply by the base again. 27 x 2 =
**54**. - Add the 6th digit. 54 + 0 =
**54**. - Multiply by the base again. 54 x 2 =
**108**. - Add the 7th digit. 108 + 1 =
**109**. You have your answer.**PHEW!**

**Here's a free worksheet to help you practice BaseNtoBase10:**

basentobase10.pdf |

**Up Next for High School: Base10toBaseN**