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I came to know that a human breathes 21,600 times per day according to Yoga sastra.

Which shloka explicitly states it?

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    Yes... per minute 15 per hour 900... per day 21600.. that's why you see Kalasa of Nataraj temple is 21600 tall with 9 entry points. Through Pranayama if you reduce your breathing your life span extends – Parabrahman Jyoti Jan 21 at 15:58
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    Is it for only humans or all living beings.?because the animals like tortoise lives hundreds of years which means their breathing rate is low? – Soma Sekhar Mar 22 at 6:31
  • @SomaSekhar, and animals like dogs which breathe very fast live only upto 20 – ram Apr 14 at 3:37
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Hamsa Upanishad, Yoga Upanishad which belongs to Shukla Yajurveda, mentions that.

अथ हंस ऋषिः ।
अव्यक्ता गायत्री छन्दः ।
परमहंसो देवता ।
अहमिति बीजम् ।
स इति शक्तिः । सोऽहमिति कीलकम् ।
षट् सङ्ख्यया अहोरात्रयोरेकविंशतिसहस्राणि षट् शतान्यधिकानि भवन्ति

English Translation by K. Narayanasvami Aiyar:

(Here the performance of Ajapā Gāyaṭrī is given).

"Now Hamsa is the ṛshi; the metre is Avyakṭā Gāyaṭrī; Paramahamsa is the ḍevaṭā (or presiding deity) 'Ham' is the bīja; 'Sa' is the śakṭī; So’ham is the kīlaka. Thus there are six. There are 21, 600 Hamsas (or breaths) in a day and night.

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This is mentioned in Yogic texts as well as in numerous Tantras.

For example the Vigyana Bhairava Tantra says:

The breath is exhaled with the sound 'Ha' and inhaled again with the sound 'Sa'. Thus the individual always repeats this particular mantra Hamsa. (155)

This japa of the Devi which was previously indicated, (being repeated) twenty-one thousand six hundred times during the day and night, is easily available and difficult only for the ignorant. (156)

I am quoting from p 68 of the PDF.

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Yet another reference for the same from Dhyana Bindu Upanishad linked with the Krishna Yajur Veda:

61(b)-63. The Jiva comes out with the letter ‘Ha’ and gets in again with the letter ‘Sa’. Thus Jiva always utters the Mantra ‘Hamsa’, ‘Hamsa’. The Jiva always utters the

Mantra twenty-one thousand and six hundred times in one day and night. This is called Ajapa Gayatri and is ever the bestower of Nirvana to the Yogins.

64-66(a). Through its very thought, man is freed from sins. Neither in the past nor in the future is there a science equal to this, a Japa equal to this or a meritorious action equal to this. Parameshvari (viz., Kundalini Sakti) sleeps shutting with her mouth that door which leads to the decayless Brahma-hole

A related and interesting article which you can read is: Time and the Kalachakra.

  • Hamsa Upanishad also mentions same Ajapa Gayatri. – The Destroyer Jan 21 at 16:41
  • Yes saw and upvoted ur answer .. actually numerous Tantras talk about it too .. that article I linked is interesting u can read it...@TheDestroyer – Rickross Jan 21 at 16:43
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Apart from various Tantra as mentioned in other answers, Garuda Purana, Chapter 15 also says that there are twenty-one thousand six hundred breaths in one day.

77-80. It is said by the wise that the subtle movements of the breath in one day and night number twenty-one thousand six hundred.

It goes out with the sound of "ha," and enters again with the sound of "sa." The individual is, indeed, always repeating the mantra. "Haṁsa, haṁsa,"--

Six hundred for Gaṇeśa; six thousand for Vedhas; six thousand for Hari; six thousand for Hara.

A thousand for the Jīvātman; a thousand for Guru; a thousand or the Chidātman;--thus one should understand the respective numbers of the repetitions.

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