Unit 5 · Topic 06 · Fractions and Decimals
Kabir wrote 3.4 and 3.40 and insisted they were different numbers, since one had an extra zero.
Anaya lined up their place values, column by column, tenths under tenths and hundredths under hundredths.
The extra zero turned out to change absolutely nothing about the number's actual value.
"After the decimal point," said Ms. Rao, "each position has a name, just like whole-number place values, but shrinking instead of growing. The first digit after the point is the TENTHS place, worth 1/10. The second is the HUNDREDTHS place, worth 1/100. The third is the THOUSANDTHS place, worth 1/1000."
She wrote 3.45. "The 4 is in the tenths place, worth 4/10. The 5 is in the hundredths place, worth 5/100." Kabir worked it out as a sum: 3 + 4/10 + 5/100 = 3 + 0.4 + 0.05 = 3.45. "So each decimal digit is just another fraction, added on, getting smaller each step right."
"Now compare 3.4 and 3.40," said Ms. Rao. Anaya lined up the columns: 3.40 has a 4 in tenths and a 0 in hundredths. 3.4 has a 4 in tenths and NOTHING written in hundredths — but an unwritten digit is the same as writing 0 there. "They're exactly the same value," she concluded. "Trailing zeros after the decimal point never change a number's value — 3.4 = 3.40 = 3.400, all identical."
"This matters when COMPARING decimals too," Ms. Rao added. "To compare 3.4 and 3.45, line up the columns first: 3.40 vs 3.45. Same whole part, same tenths digit (4), but 3.45 has 5 in hundredths versus 0 — so 3.45 is bigger." Comparing decimal digits column by column, exactly like whole numbers, avoided every trap of just 'more digits looks bigger.'
The first digit after a decimal point is the tenths place (1/10); the second is hundredths (1/100); the third is thousandths (1/1000).
A decimal number equals its whole-number part plus the sum of each decimal digit's place value.
Trailing zeros after the last nonzero decimal digit never change a number's value: 3.4 = 3.40 = 3.400.
To compare decimals correctly, line up the decimal points and compare digit by digit, place by place.
Write 5.207 as a sum of its whole part and place values.
Compare 6.5 and 6.45 by lining up their decimal columns.
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